1314
Biophysical Journal
Volume 81
September 2001
1314 –1323
A Diffusion–Translocation Model for Gradient Sensing
by Chemotactic Cells
Marten Postma and Peter J. M. Van Haastert
Groningen Biomolecular Sciences and Biotechnology Institute, Department of Biochemistry, University of Groningen, Nijenborgh 4,
9747 AG Groningen, The Netherlands
ABSTRACT Small chemotactic cells like Dictyostelium and neutrophils transduce shallow spatial chemoattractant gradients
into strongly localized intracellular responses. We show that the capacity of a second messenger to establish and maintain
localized signals, is mainly determined by its dispersion range, ␭ ⫽ 公Dm/k⫺1, which must be small compared to the cell’s
length. Therefore, short-living second messengers (high k⫺1) with diffusion coefficients Dm in the range of 0 –5 ␮m2 s⫺1 are
most suitable. Additional to short dispersion ranges, gradient sensing may include positive feedback mechanisms that lead
to local activation and global inhibition of second-messenger production. To introduce the essential nonlinear amplification,
we have investigated models in which one or more components of the signal transduction cascade translocate from the
cytosol to the second messenger in the plasma membrane. A one-component model is able to amplify a 1.5-fold difference
of receptor activity over the cell length into a 15-fold difference of second-messenger concentration. Amplification can be
improved considerably by introducing an additional activating component that translocates to the membrane. In both models,
communication between the front and the back of the cell is mediated by partial depletion of cytosolic components, which
leads to both local activation and global inhibition. The results suggest that a biochemically simple and general mechanism
may explain various signal localization phenomena not only in chemotactic cells but also those occurring in morphogenesis
and cell differentiation.
INTRODUCTION
Many biochemical and cellular processes, such as nuclear
division, lipid transport, and cytoskeletal rearrangement are
nonuniformly distributed within the cell. A pronounced
example is chemotaxis, a process where cells move in the
direction of a chemical gradient. Prokaryotic cells detect a
temporal change in chemoattractant concentration and
change the duration of movement depending on whether
they move toward or away from the chemotactic source
(Stock and Mowbray, 1995; Stock et al., 1989). However,
eukaryotic cells detect a spatial difference of chemoattractant by measuring the difference in concentration between
the ends of the cell, and then they move up the gradient of
chemoattractant (Devreotes and Zigmond, 1988). Examples
are human neutrophils that react to small peptides, such as
fMLP, Dictyostelium that responds to cAMP, and plateletderived growth factor (PDGF) that induces directional
movement of fibroblasts in wound-healing processes (Heldin and Westermark, 1999).
Activation of receptors in the plasma membrane induces
production of second-messenger molecules that, by diffusion, transduce the signal to structures that initiate pseudopod formation. However, diffusion of second-messenger
molecules will also lead to signal dispersion: loss of spatial
information. The diffusion speed of second-messenger mol-
Received for publication 11 January 2001 and in final form 24 May 2001.
Address reprint requests to Peter J. M. Van Haastert, GBB, Dept. of
Biochemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands. Tel.: ⫹31-503634172; Fax ⫹31-503634165;
E-mail: [email protected].
© 2001 by the Biophysical Society
0006-3495/01/09/1314/10 $2.00
ecules can be very different and mainly depends on their
size and location. The observed value may, however, differ
considerably if molecules bind to immobile structures (cytoskeleton or large protein complexes) or if obstacles restrict their path. Small soluble molecules, like cAMP,
cGMP, Ca2⫹, and IP3, generally have diffusion coefficients
well above 100 ␮m s⫺1 (see, for example, Chen et al., 1999;
Allbritton et al., 1992), whereas small membrane-bound
molecules like DAG and PIP3 diffuse at least 100-fold
slower because the diffusion coefficients are in the order of
1 ␮m2 s⫺1 (see Almeida and Vaz, 1995; Korlach et al.,
1999). The diffusion coefficients of free cytosolic proteins
are in the order of 10 –50 ␮m2 s⫺1 (see Arrio-Dupont et al.,
2000), depending on their size, and those of membranebound proteins are ⬃0.1 ␮m2 s⫺1 (see Niv et al., 1999). An
equally important factor in signal dispersion is the second
messenger’s lifetime. In contrast to long lifetimes, second
messengers with short lifetimes preserve spatial information
much better (Haugh et al., 2000). In addition to formation of
second-messenger molecules, receptor activation may also
lead to translocation of cytosolic proteins to docking sites in
the plasma membrane generated by the activated receptors.
Such docking sites may include phosphorylated tyrosines on
growth-factor receptors that bind Src homology 2 (SH2)
domain-containing proteins, or phosphatidyl inositol phospholipids that bind proteins with a pleckstrin homology
(PH) domain (Teruel and Meyer, 2000).
Because cells are able to sense very weak gradients of
chemoattractant, the transduction mechanism must be able
to convert a difference in receptor occupancy between front
and back of the cell in an all-or-nothing response at the
front. Asymmetric amplification of the signal between front
Translocation Model for Gradient Sensing
1315
and back of the cell is essential to transduce weak spatial
information. Thus, two major events have to be understood:
diffusion/degradation of second-messenger molecules for
spatial intracellular communication, and nonlinear amplification to convert weakly localized signals into strongly
localized responses. In this study, we have investigated the
fundamental role of diffusion and degradation of secondmessenger molecules in gradient-sensing mechanisms.
Based on translocation experiments (Parent et al., 1998;
Firtel and Chung, 2000; Servant et al., 2000; Haugh et al.,
2000) and activator-inhibitor models (Meinhardt, 1999;
Meinhardt and Gierer, 2000; Haugh and Lauffenburger,
1997), we propose a diffusion–translocation model that may
provide the required amplification in gradient sensing.
A gradient of second-messenger production
We next consider a spherical cell, with radius r, where production, diffusion, and degradation of the second messenger occur at the inner face of the
cell membrane. We assume that an external gradient of chemoattractant
induces a linear gradient in receptor activity along the x-coordinate; ⫺r ⱕ
x ⱕ r. This linear gradient of receptor activity is given by
x
R*共x兲 ⫽ R៮ * ⫺ ⌬R* ,
r
where R*(x) is the local fraction of active receptors, R៮ * ⫽ (R*f ⫹ R*b)/2
represents the mean fraction of active receptors and ⌬R* ⫽ (R*f ⫺ R*b)/2
represents the difference between the ends of the cell in the fraction of
active receptors. The maximum production rate, kR, will be reached when
all receptors are active, and, hence, the local production rate of second
messenger is given by P(x) ⫽ kRR*(x).
The time and space dependence of the second-messenger concentration
then follows from Eq.1 with the initial condition m(x, 0) ⫽ 0:
MATERIALS AND METHODS
We assume that the proteins synthesizing the second messengers are
immobile and are located in or at the membrane, and are activated by local
receptors. After its production, a second-messenger molecule will diffuse
from its location of synthesis, either into the cytosol or laterally in the
membrane, and, ultimately, it will be degraded. This general reaction–
diffusion process is described by (see, for example, Haugh et al., 2000)
dm
⫽ Dmⵜ2m ⫺ k⫺1m ⫹ P,
dt
(1)
where m is the position-dependent second-messenger concentration (with
unit ␮M for cytosolic molecules or molecules 䡠 ␮m⫺2 for membrane
molecules), Dm the diffusion coefficient of the second messenger, k⫺1 the
degradation rate constant, and P the production rate (unit: ␮Ms⫺1 or
molecules␮m⫺2 s⫺1 for cytosolic and membrane-bound second messengers, respectively). We have studied solutions of Eq. 1 for different cases
of cell geometry and production, diffusion, and degradation rates. All
calculations where done numerically and, when possible, also with analytical solutions. The solutions of Eq. 1 were derived by the Laplace transform method as described in Carslaw and Jaeger (1959) and Crank (1979).
We discuss two cases where analytical solutions of Eq. 1 exist.
A point source of second-messenger production
We assume that second-messenger production takes place at a constant rate
p ⫽ kP at one end face of a cylinder with length L, and that degradation and
diffusion occurs along the axial coordinate of the cylinder. When secondmessenger production starts at t ⫽ 0, i.e., m(x, 0) ⫽ 0, the solution of Eq.
1 is
m共x, t兲 ⫽
kP L cosh共共L ⫺ x兲/␭兲
k⫺1 ␭
sinh共L/␭兲
⫺ kP
冘 ␶ cos冉n␲ Lx 冊e
⬁
n⫽⫺⬁
n
⫺t/␶n
,
(2)
where 0 ⬍ x ⬍ L; the space constant ␭ is given by ␭ ⫽ 公Dm/k⫺1 and the
time constants ␶n by ␶n ⫽ 1/(k⫺1 ⫹ n2␲2Dm/l2). The first term in Eq. 2 is
the steady-state solution and will be reached quickly when the time
constants are small, i.e., at high degradation rates (k⫺1), fast diffusion (Dm)
and in small cells (L).
(3)
m共x, t兲 ⫽
冉
kR
x
R៮ * ⫺ F⌬R*
k⫺1
r
⫺
冉
冊
冊
kR
x
R៮ *e⫺t/␧0 ⫺ F⌬R* e⫺t/␧1 ,
k⫺1
r
(4)
where the factor F ⫽ r2/(2␭2 ⫹ r2), and the time constants are ␧0 ⫽ 1/k⫺1
and ␧1 ⫽ 1/(k⫺1 ⫹ 2Dm/r2), respectively. The time-independent part of Eq.
4 is the steady-state solution.
The concentration of second messenger in the sphere’s volume can also
be derived from Eq. 1. The resulting expression for the steady-state
solution for the concentration just below the surface is similar to that of Eq.
4, but more complex.
RESULTS
Second-messenger gradient formation under
point-source production
When a second messenger is produced at a certain cellular
location, how far and how fast will it diffuse from this point
source, given a specific diffusion coefficient and degradation
rate? To illustrate the main factors determining the fate of a
signal, we first consider the idealized case of a cylindrically
shaped cell. We assume that second-messenger production
takes place at only one end face and that degradation occurs
throughout the cell (Fig. 1, inset). When the production of
second messenger at the cylinder front face starts at t ⫽ 0, the
space–time dependence of the second-messenger concentration is described by a two-dimensional reaction– diffusion
equation, which has solution Eq. 2. In this equation, the space
constant ␭ ⫽ 公Dm/k⫺1 determines the dispersion range of the
second messenger. For long cells, about 95% of the molecules
are localized within a distance of 3␭ from the source.
In a cell with typical length L ⫽ 10 ␮m, the steady-state
concentration profiles were calculated for three values of
the diffusion coefficient: Dm ⫽ 1 ␮m2 s⫺1, a typical value
for membrane phospholipids or very large cytosolic proteins; Dm ⫽ 10 ␮m2 s⫺1, typical for cytosolic proteins; and
Dm ⫽ 100 ␮m2 s⫺1, characteristic for cAMP. The values
Biophysical Journal 81(3) 1314 –1323
1316
FIGURE 1 Diffusion of second messenger produced at one face of a
cylindrical cell. Concentration profiles of second messenger in the steady
state for different diffusion coefficients Dm; L ⫽ 10 ␮m and k⫺1 ⫽ 1.0 s⫺1.
Slow diffusion leads to a localized signal and fast diffusion leads to
dispersal of the gradient.
taken for the degradation rate and the second-messenger
production rate were k⫺1 ⫽ 1 s⫺1 and kP ⫽ 1 ␮Ms⫺1,
respectively (Fig. 1). A diffusion coefficient of Dm ⫽ 1
␮m2 s⫺1 yields a space constant of ␭ ⫽ 1 ␮m. The concentration of the second messenger is very high near the place
of production and falls off steeply to zero values at a few
micrometer away from the source. For Dm ⫽ 10 ␮m2 s⫺1 (␭
⫽ 3.3 ␮m), the second-messenger concentration in the
steady state still exhibits a noticeable gradient, but with a
diffusion coefficient of Dm ⫽ 100 ␮m2 s⫺1 (␭ ⫽ 10 ␮m),
the concentration profile is nearly flat. Because production
and degradation rates are kept constant, the total amount of
second-messenger molecules in the cell is the same in all
three cases.
The effect of second messenger degradation on signal
localization is complementary to that of diffusion speed,
due to the space constant ␭ ⫽ 公Dm/k⫺1. A lower degradation rate will lead to stronger dispersal of signals in the
cell, and also to a larger total concentration, i.e., stronger
signals. In contrast, high degradation rates will lead to more
localized signals, but unavoidably the total concentration
will decrease. To overcome a lower concentration, the production rate must be increased accordingly. The results
show that, to allow the formation of a steep gradient, the
space constant ␭ must be ⱕ1 ␮m. Thus, gradient transduction by a second messenger such as cAMP with a diffusion
coefficient over 100 ␮m2 s⫺1 requires a degradation rate
above 100 s⫺1, implying a half-life below 10 ms.
Receptor-coupled second-messenger production
in chemoattractant gradients
In Dictyostelium cells, receptor molecules are probably uniformly distributed around the cell’s periphery (Xiao et al.,
Biophysical Journal 81(3) 1314 –1323
Postma and Van Haastert
FIGURE 2 Diffusion of second messenger produced as a gradient in a
spherical cell. Steady-state second-messenger concentration profiles at or
just below the cell’s surface were calculated for different diffusion coefficients using a 60 – 40% gradient of receptor activity (dotted line), a
degradation rate of k⫺1 ⫽ 1.0 s⫺1 and radius r ⫽ 5 ␮m. The data are
normalized to 0.5 at the center of the cell. Diffusion leads to dissipation of
the gradient. The gradient is almost completely lost with a fast-diffusing
molecule (Dm ⫽ 100 ␮m2 s⫺1), while the intracellular gradient becomes
increasingly proportional to the receptor activity gradient at Dm ⫽
1 ␮m2 s⫺1.
1997). Because chemoattractant molecules can reversibly
bind to the receptors, the fraction of active receptors depends on the local chemoattractant concentration. We assume that the activated receptors locally couple to effector
enzymes that then produce second-messenger molecules at
the inner face of the plasma membrane. The subsequent
diffusion/degradation of second messenger will take place
at the cell’s inner surface (i.e., the plasma membrane in the
case of phospholipid second messenger) or in the cell’s
volume (i.e., the cytosol for a soluble second messenger like
cAMP).
We consider a spherically shaped cell with radius r ⫽ 5
␮m, placed in a chemoattractant concentration gradient
causing a 60 – 40% linear gradient in receptor activity; i.e.,
60% and 40% of the receptors are active at the front and
back, respectively, or formally (see Methods): R៮ *f ⫽ 0.6, R៮ *b
⫽ 0.4, the mean fraction of active receptors R៮ * ⫽ 0.5 and
the difference from the mean ⌬R* ⫽ 0.1.
The second-messenger gradient profile was analyzed for
the three values of the diffusion coefficient Dm: 1, 10, and
100 ␮m2 s⫺1, assuming a degradation rate of k⫺1 ⫽ 1.0 s⫺1,
a maximum production rate kR ⫽ 1.0 ␮Ms⫺1. The linear
receptor gradient of 60 – 40% (Fig. 2, dotted line) then
results in a linear second-messenger gradient of 59 – 41%,
55– 45%, and 51– 49% for the three values of Dm. In Eq 4,
the slope of the second-messenger gradient relative to the
gradient of active receptors is given by the factor F. Because
the factor F is always smaller than 1, the resulting secondmessenger gradient cannot be steeper than the gradient of
receptor activity. As encountered already in the case of the
Translocation Model for Gradient Sensing
1317
cylindrical model cell, only with a small space constant, i.e.,
a short dispersion range ␭ ⱕ 1 ␮m, the gradient in the
external signal is transduced into an almost identical internal gradient of second messenger; fast diffusion or slow
degradation will lead to a strong dispersion of the intracellular signal gradient. We conclude that slowly diffusing
second messengers with Dm ⬍ 1.0 ␮m2 s⫺1 can preserve the
external gradient, whereas fast-diffusing second messengers
with Dm ⬎ 100 ␮m2 s⫺1 effectively generate an average
global signal for realistic production and degradation rates.
A model for signal amplification with
effector translocation
In the previous section, we have demonstrated that the
second-messenger gradient will be always less steep than
the gradient in receptor activity. Dictyostelium cells and
neutrophils can respond to shallow chemoattractant gradients that are expected to induce a 12–10% gradient of
receptor activation (Tomchik and Devreotes, 1981), indicating that amplification of the signal is essential. We develop
a gradient amplification model that is based on translocation
experiments of PH domain-containing proteins in Dictyostelium (Parent et al., 1998; Firtel and Chung, 2000), neutrophils (Servant et al., 2000), and fibroblasts (Haugh 2000).
PH domains are known to bind to phosphatidyl inositol
phosphates (see, for example, Rebecchi and Scarlata, 1998)
that are generated upon receptor activation and possibly
mediate the formation of transduction complexes (Parent
and Devreotes, 1999). The diffusion coefficient of phosphatidyl inositol phosphates in living cells (about 1
␮m2 s⫺1) allows gradient preservation at realistic degradation rates. To introduce a nonlinearity in the system, we
assume that one essential component in the signal transduction cascade is a cytosolic PH-domain-containing protein
that translocates between the cytosol and the second messenger in the plasma membrane. The principal is very general and can be achieved in two ways, both in respect to the
identity of the second messenger in the membrane as the
identity of the translocating molecule (see Discussion).
Haugh et al. (2000) have proposed that PI3⬘-kinase may
play such a role in signal localization. The model is depicted
in four steps (Fig. 3): (A), before receptor stimulation only
a small number of effector molecules is bound to the membrane and is inactive; (B), after receptor activation the
membrane-bound effector molecules will be stimulated
leading to production of small amounts of phospholipid
second-messenger molecules; (C), because the phospholipid
concentration increases, more effector molecules will translocate from the cytosol to the membrane; and (D), because
the receptor now can signal to more effector molecules, this
rapidly leads to stronger phospholipid production and further depletion of cytosolic effector molecules. The amplification can be considered as a positive feedback mechanism,
where the phospholipid itself enhances its own production
FIGURE 3 The diffusion–translocation model for amplification of signal
transduction. The figure depicts four steps in the model: (A) receptor
activation, (B) second-messenger production, (C) effector translocation,
and (D) amplification.
by recruiting more effector molecules producing the phospholipid.
The production rate of second messenger, P, is directly
coupled to the local receptor activity and the local effector
enzyme concentration,
P共x兲 ⫽ k0 ⫹ kER*共x兲Em,
(5)
where k0 is the basal phospholipid production rate, kE is the
maximum production rate per effector molecule, and Em is
the local effector concentration in the membrane. We further assume that effector molecules in the cytosol (Ec) are
able to diffuse with diffusion coefficient DEc, and can
reversibly bind to the phospholipids with forward bindingrate constant kb and release-rate constant k⫺b. The total
amount of effector molecules in the membrane and the
cytosol, Etot, is held constant.
Amplification of a 60 – 40% external gradient
The concentration of effector molecule and second messenger were calculated with the translocation model using a
linear gradient of receptor activity of 60 – 40%. The left
panels of Fig. 4 show the concentration profiles of membrane-bound effector molecules, Em, and cytosolic effector
molecules, Ec, whereas the right panels show the crosssection of the cell; the gray scale indicates the local concentration of effector molecules. At 2 s after the application
of the gradient, the cytosolic concentration of effector molecules has decreased to ⬃25 nM (Fig. 4 B), after 4 s to about
Biophysical Journal 81(3) 1314 –1323
1318
FIGURE 4 Translocation of effector from cytosol to membrane during
gradient sensing with the diffusion–translocation model. Sequence of snapshots at four time points after application of a 60 – 40% gradient of receptor
activity. The cytosolic and membrane-bound effector are depicted in two
ways. The left-hand panels show the calculated values, where Em is the
density of effector molecules bound to the membrane and Ec the concentration of effector molecules in the cytosol. The panels at the right show the
concentration of effector molecules at a cross section of the cell using a
gray scale. The values for kinetic parameters used in the calculations are:
k0 ⫽ 10 molecules䡠␮m⫺2 s⫺1, kE ⫽ 20 molecules䡠s⫺1, k⫺1 ⫽ 1.0 s⫺1, kb ⫽
10 ␮M⫺1 s⫺1, k⫺b ⫽ 1.0 s⫺1, DEc ⫽ 50 ␮m2 s⫺1, and Dm ⫽ 1.0 ␮m2 s⫺1.
The total concentration of effector molecules was taken to be 50 nM, and,
before stimulation, ⬃10% of these effector molecules are bound to the
membrane.
10 nM (Fig. 4 C), which is followed by a slight further
decrease to 8 nM, reached in the steady state (Fig. 4 D).
This decrease is due to translocation of cytosolic molecules
to phospholipid binding sites in the membrane that have
been generated by effector molecules activated by receptors.
Whereas the effector molecules in the cytosol are spread
evenly, those in the membrane are progressively concenBiophysical Journal 81(3) 1314 –1323
Postma and Van Haastert
FIGURE 5 Second-messenger formation during gradient sensing with
the diffusion–translocation model. (A) At t ⫽ 0, a gradient of 60 – 40% is
applied to the cell. The dotted line depicts the steady-state second-messenger gradient if effector molecules would have been located at the
membrane from the beginning (cf. Fig. 2). Black lines show the gradient
calculated with the translocation model at three time points after application of the gradient. (B) Time courses of the concentration of second
messenger at the front and at the back of the cell. See Fig. 4 for the
parameter values.
trated at the front. Initially, during the first 2 s, the effector
concentration in the membrane increases at all sides of the
cell, and this increase is proportional to the 60 – 40% gradient of receptor activity. Later on, the membrane concentration of effector molecules increases further, however, this
increase occurs more strongly at the front of the cell than at
the back. In the steady state, reached after ⬃60 s, the
effector concentration in the front is about 16-fold higher
than in the back.
The concentration profiles of the phospholipid secondmessenger molecules are presented in Fig. 5. The dotted line
in Fig. 5 A represents the steady-state phospholipid concentration profile for a model without translocation (assuming
that all effector molecules are distributed homogeneously in
the membrane). In the translocation model, the secondmessenger concentration gradient that is formed after 2 s
Translocation Model for Gradient Sensing
has a slope nearly identical to that of the gradient in receptor
activity. At 4 s after application of the gradient, the concentration of second messenger has increased, which is
stronger at the front of the cell than at the back. In the steady
state, the second-messenger concentration has further increased at the front, but it has declined at the back of the
cell.
The time course of the concentration at the front and the
back of the cell is presented in Fig. 5 B. During the first
second, the phospholipid concentration increases only
slightly faster in the front than in the back of the cell, the
difference being proportional to the difference in receptor
activity at the front and back. Between about t ⫽ 1– 4 s, the
autocatalytic translocation process leads to an enhanced
phospholipid production, but starts to saturate at t ⫽ 4 s, due
to partial depletion of effector molecules in the cytosol. The
enhanced amplification then levels off. In the second amplification phase, taking place between t ⫽ 5– 60 s, the
membrane-bound effector molecules gradually translocate
from the back of the cell to the front. This leads to a decline
of phospholipids in the back of the cell and a further
increase frontally. The finally resulting 16-fold higher concentration of second messenger at the front is induced by
only a 1.5-fold higher fraction of active receptors (i.e., a
60 – 40% gradient), and thus leads to an ⬃10-fold amplification, if compared to a model without translocation. The
amplification process causes the second messenger production to be almost completely located at the front of the cell.
Amplification of signals at different receptor
activity gradients
The receptor-stimulated production of phospholipids was
investigated for different gradients in receptor activity (Fig.
6); i.e., gradients in receptor activity having different slopes
or different mean activity, while applying the same settings
of the kinetic parameters and diffusion coefficients as those
used before. The case of the 60 – 40% gradient in the previous section is indicated by point 1 in Fig. 6. Point 1 in Fig.
6, corresponding to a 52.5– 47.5% gradient, results in a
smaller difference in phospholipid concentration between
the front and the back of the cell, yielding an amplification
of 2.2 (point 2). For stronger gradients such as 75–25%
receptor activity (point 3), this results in a strong amplification of ⬃34-fold, leading to a more than 100-fold gradient
of phospholipid concentration over the cell length. All these
gradients lead to a considerable depletion of the cytosolic
effector concentration Ec from the initial 45 nM to ⬃10 nM.
With a 1.5-fold signal gradient at a lower mean receptor
activity (15–10% gradient; point 4), the cytosolic effector
concentration stays high, Ec ⫽ 31.8 nM, and the amplification is only 2.1-fold. For a 30 –20% gradient (point 5), we
obtain Ec ⫽ 17.7 nM and an amplification of 6.2, and for a
75–50% gradient (point 6) Ec ⫽ 7.26 nM and the amplification is 13.2. In general, it is observed that, at lower
1319
FIGURE 6 Contour-plot of gradient sensing. Ratio of second-messenger
concentration, after 60 s, at the front and the back of the cell mf/mb, was
calculated for different receptor activities at the front (R*f) and the back (R*b)
of the cell. See Fig. 4 for the parameter values and see text for explanation
of the marked points.
average receptor activity, amplification of the signal is
smaller than at higher average receptor activity with the
same receptor gradient. This is due to the limited depletion
of the cytosolic effector molecules at lower receptor activity, which reduces the second amplification phase. Translocation of membrane bound effector from the back to front
of the cell then hardly takes place.
Improved gradient amplification at low
receptor occupancy
At reduced average receptor occupancy, gradient amplification is less strong, mainly due to limited depletion of the
cytosolic effector molecule. To improve gradient detection
at low receptor occupance, the cell could increase secondmessenger production per activated receptor. Biochemically, this can be achieved by increasing the amount of
effector enzymes or by increasing the amount of second
messengers produced per effector molecule (kE). The second-messenger spatial concentration ratio (mf/mb) was calculated for different values of kE: 20 (previous value), 100,
and 200 molecules per effector. In all three cases, the
receptor occupancy at the front of the cell is 1.5-fold higher
than at the back of the cell. In Fig. 7 mf/mb is plotted against
different average receptor occupancy (R៮ *), demonstrating
that amplification at low receptor occupancy is improved
considerably by increasing the production rate of second
messengers per activated receptor. Higher second-messenger production rates lead to higher second-messenger conBiophysical Journal 81(3) 1314 –1323
1320
Postma and Van Haastert
FIGURE 7 Amplification at low receptor occupancy. The steady-state
ratio of second-messenger concentration at the front and the back of the
cell mf/mb, was calculated for different average receptor activities. (R*f) and
the back (R*b) of the cell. Enhancing depletion of the cytosolic effector
molecule through higher production rates of second messenger improves
amplification at low receptor occupancies.
centrations and hence more depletion. Figure 7 also reveals
that, for kE ⫽ 100 and kE ⫽ 200, amplification declines at
high receptor occupancy. Thus gradient detection can be
improved at low receptor occupancy by increasing the expression of effector enzymes at the cost of reduced sensitivity at high signal concentrations.
Improved detection of very shallow gradients
Some cells are able to detect very shallow gradients of
signal molecules that induce only a 1% gradient of receptor
occupancy. To allow detection of such shallow gradients, a
stronger amplification of the signal is essential. Here we
assumed that, in addition to translocation of effector molecule E, also a cytosolic activator molecule A translocates to
the membrane. Upon binding to the membrane, the activator
stimulates production of second-messenger molecules.
The local production of second messengers then becomes
P共x兲 ⫽ k0 ⫹ kER*共x兲Em
Am
,
Am ⫹ KA
(6)
where Am is the local activator concentration, and KA is the
concentration of activator molecules that gives half maximal active transduction complexes.
The receptor-stimulated second-messenger production
with this two-component model was investigated for different gradients in receptor activity (Fig. 8 A); i.e., gradients in
receptor activity having different slopes or different mean
activity. The contour lines correspond to second-messenger
concentration ratios between front and back (mf/mb) equal to
1, 10, and 100, where the number 1 means no amplification.
In the plot, we can distinguish two very different areas
Biophysical Journal 81(3) 1314 –1323
FIGURE 8 Contour-plot of gradient sensing using the effector–activator
model. (A) The steady-state ratio of second-messenger concentration at the
front and the back of the cell mf/mb, was calculated for different receptor
activities at the front (R*f) and the back (R*b) of the cell. Parameter values
used are: k0 ⫽ 10 molecules䡠␮m⫺2 s⫺1, kE ⫽ 20 molecules䡠s⫺1, k⫺1 ⫽ 1.0
s⫺1, for both effector and activator kb ⫽ 10 ␮M⫺1 s⫺1, k⫺b ⫽ 1.0 s⫺1,
Km ⫽ 25 molecules䡠␮m⫺2, DEc ⫽ 50 ␮m2 s⫺1 and Dm ⫽ 1.0 ␮m2 s⫺1. The
total concentration of effector and activator molecules was taken to be 50
nM. The model gives rise to a treshold concentration. Below this concentration no amplification occurs; above this concentration a strong activation
at the front and strong inhibition at the back takes place. (B) Secondmessengers gradients for the points indicated in panel A (1, 2, and 3).
Amplification is very large and also occurs at small differences of receptor
activity. Curve 4 was calculated with the same parameters as for curve 1,
except that the diffusion coefficient was ten-fold larger (Dm ⫽ 10.0
␮m2 s⫺1); the absence of a localized response indicates that the dispersion
range of the second messenger must be very small.
of amplification. Up to ⬃25% receptor occupancy at the
front of the cell, no strong amplification is observed. However, at higher receptor occupancy, a very strong all-ornothing amplification of the signal takes place.
The second-messenger concentration profiles for three
gradients of receptor occupancy are plotted in Fig. 8 B: (1)
Translocation Model for Gradient Sensing
40 – 60%, (2) 45–55%, and (3) 49 –51%. The results demonstrate that, at the back of the cell, receptor-coupled second-messenger production is strongly inhibited, whereas at
the front of the cell, second-messenger production is proportional to local receptor activity. A space–time analysis
reveals how the two-component model leads to high sensitivity: Upon gradient stimulation of receptor activity, the
second-messenger concentration increases faster at the front
than at the back off the cell, by which it will reach earlier a
critical threshold concentration. When this happens, the
front will recruit very fast more activator and effector molecules, thereby strongly increasing second-messenger production, whereas at the back of the cell, second-messenger
production is inhibited because the activator concentration
in the cytosol will be below the threshold level.
The importance of the second-messenger diffusion constant, also for the two-component translocation model, is
stressed by the second-messenger concentration profile indicated with (4) in Fig. 8 B. This curve corresponds to a
calculation using a 60 – 40% receptor activity and a 10-fold
higher second-messenger diffusion coefficient (Dm ⫽ 10
␮m2 s⫺1), leaving all other parameter values the same. The
response obtained is similar to the gradient of receptor
activity. Thus, the dispersion range (␭) appears to be a very
critical factor in the amplification mechanism and must be
much smaller than the cell’s length to allow effective transduction of chemical gradients.
DISCUSSION
Chemotactic gradient sensing requires transduction of the
signal to the locomotion machinery of the cell. The intracellular messengers might be proteins and lipids in the
membrane, or small molecules such as cAMP and proteins
in the cytosol. In all cases, the diffusion of the secondmessenger molecules has two pronounced effects, integration of receptor signals and communication to the locomotion system, but also dissipation of the spatial information.
This dissipation can be reduced if the second-messenger
molecule has a very short lifetime relative to its diffusion
rate. If we consider the diffusion coefficients of various
second-messenger molecules (see Table 1), it appears that
small soluble second messengers diffuse very fast, by which
they have a long range and a poor capacity to maintain
localized responses. Only with a very high turnover, such
second messengers are able to establish an intracellular
spatial gradient. Molecules that diffuse very slowly, such as
membrane proteins, have a very short range, even when
their degradation/inactivation is relatively slow. Although
these molecules can establish very steep gradients, diffusion
and communication to other parts of the cell is extremely
slow, taking several minutes. A 10-␮m cell that responds to
chemotactic signals after ⬃5 s therefore presumably relies
on second messengers diffusing a few micrometers in a few
seconds and having a turnover of a few seconds. Membrane
1321
TABLE 1
Dispersion of different second messengers
Second messenger
Diffusion
coefficient Dm
(␮m2 s⫺1)
Dispersion
Range*
␭ (␮m)
Dispersion
Time†
␶1 (s)
cAMP in cytosol
Protein in cytosol
Lipid in membrane
Protein in membrane
⬎100
10–50
1
⬃0.1
⬎10
⬃5
1
⬃0.3
⬃0.1
⬃0.25
⬃1
⬃1
All values were calculated with k⫺1 ⫽ 1.0 s⫺1.
*About 95% of the molecules are localized within a distance of 3␭ from the
source.
†
After ␭1 seconds, the concentration has reached 63% of the steady-state
value.
lipids have diffusion coefficients of ⬃1 ␮m2 s⫺1 and thus
fulfil that expectation. A similar conclusion, based on experimental data, was reached by Haugh et al. (2000).
Detection of a gradient with linear signal transduction can
never produce a second-messenger gradient that is steeper
than that of the applied signal. Because many organisms can
detect very small gradients at a low average receptor activity, a locally strong amplification is essential to generate an
all-or-nothing locomotion response. Meinhardt (1999) and
Meinhardt and Gierer (2000) have put forward a model that
explains gradient sensing by combining strong local autocatalytic activation with global inhibition, by which the net
activity at the front becomes much higher than in the back.
Calculations have shown that this model provides the required nonlinear amplification, but the model is biochemically rather complex and is sensitive to the parameter set
chosen. The model of Parent and Devreotes (1999) incorporates adaptation of the local stimulatory and global inhibitory responses to constant signals, by which the net amplification at the front becomes strongly enhanced. The present
diffusion–translocation model does not incorporate any of
these mechanisms except for a positive feedback on secondmessenger production, and still is very robust in generating
strong local responses. Nevertheless, incorporation of
global inhibition and adaptation expands the applicability of
the present model to more extreme situations such as rapid
reversal of the gradient (unpublished results).
The diffusion–translocation model
The heart of the diffusion–translocation model is that activation of the chemoattractant receptor results in the creation
of membrane-bound second-messenger molecules. The second messengers then act as docking sites for effector molecules in the cytosol that mediate (or facilitate) the transmission of the signal from the active receptor to the secondmessenger-producing enzyme. The transduction cascade
thus contains a positive feedback loop that strongly and
nonlinearly amplifies the signal. The amplification saturates
when the cytosolic pool of molecules becomes partially
depleted.
Biophysical Journal 81(3) 1314 –1323
1322
The membrane-bound second messenger could be any
lipid or protein that functions as (or induces) a docking sites
for cytosolic proteins. Thus, phosphatidyl inositol polyphosphates that are generated by receptor-stimulated kinases
may form binding sites for proteins with specific PH domains. Alternatively, receptor-stimulated protein tyrosin
phosphorylation can form binding sites for proteins with
SH2 domains. Possible candidates for the effector enzymes
that mediate second-messenger formation are heterotrimeric
G-protein subunits, small G-proteins, or a protein kinase
that activates a membrane-bound effector enzyme through
phosphorylation.
In its most simplified form, the second messenger is a
phospholipid, and the translocating component is the phospholipid-forming enzyme. We will discuss the model with
this simplified form in mind. The combined data of Figs. 4
and 5 reveal that a small spatial difference in receptor
activity is transduced into a phospholipid second-messenger
concentration gradient that initially has the same slope as
the gradient of receptor activity. Because the produced
phospholipid molecules are the binding sites for the effector
enzyme, slightly more effector enzymes translocate to the
front than to the back of the cell. The increased effector
concentration at the front of the cell leads to a more pronounced phospholipid production, and, subsequently, to a
depletion of the effector enzyme in the cytosol. In the
second phase of amplification, the membrane-bound effector enzymes dissociate at the back of the cell and gradually
translocate to the front. As a consequence, the phospholipid
second-messenger concentration further increases at the
front and decreases appreciably at the back of the cell.
Gradient amplification is achieved because second-messenger production becomes almost completely restricted to the
front of the cell. Effectively, the production point source
situation as described in the results section is obtained in
this way, and hence the corresponding requirements for
gradient formations apply. The main factor is the dispersion
range of the second messenger, which must be in the order
of 1 ␮m for small chemotactic cells.
Limitations and improvements of the model
The two phases of amplification in the diffusion–translocation model entail different mechanisms that cause both
limitations and enable improvements of the model. Amplification requires time, during which dispersion will take
place that reduces amplification. Therefore, amplification
should occur within the dispersion time. Because amplification is mainly determined by translocation of the cytosolic
effector enzyme, this implies that its translocation speed
must be faster than the dispersion of the second messenger.
With the diffusion speeds of a cytosolic protein as the
effector enzyme and a membrane phospholipid as the second messenger, this condition is easily fulfilled. Dispersion
of the second messenger can be reduced either by reducing
Biophysical Journal 81(3) 1314 –1323
Postma and Van Haastert
diffusion speed or by increasing the degradation rate of the
second messenger. Thus, simply by increasing the expression level of degrading enzymes, the second-messenger
gradient will become steeper. Slower second-messenger
diffusion can be achieved by either restricted diffusion
(corralled and percolation) or by protein-bound second messengers. It has been demonstrated that, in Dictyostelium,
signals adapt, meaning that, under persistent stimulation,
the intracellular signal returns to basal levels, which implies
that, after the initial activation, some form of inhibition
takes place. This phenomenon has not been included in the
model. However, it will enhance the localization process if
adaptation occurs faster at the side of lower activity, thereby
amplifying the localization of second messenger in areas
with higher activity (unpublished results).
The sensitivity of gradient sensing consists of two components, the detection of gradients at low concentrations and
the detection of shallow gradients. At low receptor activity,
amplification is predominantly determined by basal production activities and the second amplification phase is absent,
because there is no significant depletion of the cytosolic
effector enzyme. Thus, at receptor activities below ⬃10%,
amplification of the gradient becomes small. This has the
advantage that the system does not become unstable at low
receptor activities, which is a general problem of highly
nonlinear systems at low activity. Our calculations demonstrate that sensitivity at low average receptor activity can be
improved simply by increasing the amount of second messengers produced per activated receptor, which leads to
stronger depletion of the cytosolic effector. Biochemically,
this would mean that a cell can enhance the detection limit
of gradient sensing by increasing the expression of one of
the components of the signal transduction cascade. In the
basic model, only one component of the signaling transduction pathway translocates to the membrane, leading already
to considerable signal localization. Some cells are able to
respond to very shallow chemotactic gradients. When assuming that more components translocate to the second
messenger in the membrane, thereby forming activated
transduction complexes, second-messenger production can
virtually completely be restricted to one side of the cell,
even for minor gradients. Interestingly, such a two-component translocation model reveals a threshold of receptor
activity below which nonlinear amplification is very limited. This has the important advantage that, at low receptor
activity, the system is relatively silent and stable. As for the
one-component translocation model, the threshold can be
reduced to lower receptor activity by increasing the amount
of second-messenger molecules produced per activated receptor. Thus, cells become sensitive to small and shallow
gradients by inducing the expression of a co-activator and
by enhancing the expression of an existing component of
the signal transduction cascade. Such more complicated and
stronger nonlinear mechanisms can, however, lead to freezing of the intracellular signal, because multiple steady-state
Translocation Model for Gradient Sensing
concentration profiles may exist. In contrast to the onecomponent model, reversal of the external gradient signal
then may not lead to reversal of the internal gradient. A
similar very high sensitivity, but tendency for freezing of
the second-messenger gradient, was observed upon incorporating global inhibition and local activation into the translocation model. Preliminary work indicates that introducing
exact adaptation mechanisms, where internal signals are
only temporarily maintained, can solve this freezing problem.
The diffusion–translocation mechanism presented here
may very well form the central unit of signal transduction
processes in which spatial information has to be transduced.
On top of this unit, other biochemical modules, such as
adaptation or inhibition mechanisms, may fine-tune the
model such that it becomes increasingly sensitive and applicable for more complicated spatial signals such as gastrulation, and mesoderm induction.
We thank D. G. Stavenga and J. Roelofs for stimulating discussions and
critically reading of the manuscript.
REFERENCES
Allbritton, N. L., T. Meyer, and L. Stryer. 1992. Range of messenger action
of calcium ion and inositol 1,4,5-trisphosphate. Science. 258:
1812–1815.
Almeida, P. F. F., and W. L. C. Vaz. 1995. Lateral Diffusion in Membranes. Elsevier, Amsterdam. 305–357.
Arrio-Dupont, M., G. Foucault, M. Vacher, P. F. Devaux, and S. Cribier.
2000. Translational diffusion of globular proteins in the cytoplasm of
cultured muscle cells. Biophys. J. 78:901–907.
Carslaw, H. S., and J. C. Jaeger. 1959. Conduction of Heat in Solids.
Clarendon Press, Oxford, UK.
Chen, C., T. Nakamura, and Y. Koutalos. 1999. Cyclic AMP diffusion
coefficient in frog olfactory cilia. Biophys. J. 76:2861–2867.
Crank, J. 1979. The Mathematics of Diffusion. Clarendon Press, Oxford,
UK.
Devreotes, P. N., and S. H. Zigmond. 1988. Chemotaxis in eukaryotic
cells: a focus on leukocytes and Dictyostelium. Annu. Rev. Cell Biol.
4:649 – 686.
1323
Firtel, R. A., and C. Y. Chung. 2000. The molecular genetics of
chemotaxis: sensing and responding to chemoattractant gradients. Bioessays. 22:603– 615.
Haugh, J. M., F. Codazzi, M. Teruel, and T. Meyer. 2000. Spatial sensing
in fibroblasts mediated by 3⬘ phosphoinositides. J. Cell Biol. 151:
1269 –1280.
Haugh, J. M., and D. A. Lauffenburger. 1997. Physical modulation of
intracellular signaling processes by locational regulation. Biophys. J.
72:2014 –2031.
Heldin, C. H., and B. Westermark. 1999. Mechanism of action and in vivo
role of platelet-derived growth factor. Physiol. Rev. 79:1283–1316.
Korlach, J., P. Schwille, W. W. Webb, and G. W. Feigenson. 1999.
Characterization of lipid bilayer phases by confocal microscopy and
fluorescence correlation spectroscopy. Proc. Natl. Acad. Sci. U.S.A.
96:8461– 8466. [Published erratum. Proc. Natl. Acad. Sci. U.S.A. 1999.
96:9666].
Meinhardt, H. 1999. Orientation of chemotactic cells and growth cones:
models and mechanisms. J. Cell Sci. 112:2867–2874.
Meinhardt, H., and A. Gierer. 2000. Pattern formation by local selfactivation and lateral inhibition. Bioessays. 22:753–760.
Niv, H., O. Gutman, Y. I. Henis, and Y. Kloog. 1999. Membrane interactions of a constitutively active GFP-Ki-Ras 4B and their role in signaling. Evidence from lateral mobility studies. J. Biol. Chem. 274:
1606 –1613.
Parent, C. A., B. J. Blacklock, W. M. Froehlich, D. B. Murphy, and P. N.
Devreotes. 1998. G protein signaling events are activated at the leading
edge of chemotactic cells. Cell. 95:81–91.
Parent, C. A., and P. N. Devreotes. 1999. A cell’s sense of direction.
Science. 284:765–770.
Rebecchi, M. J., and S. Scarlata. 1998. Pleckstrin homology domains: a
common fold with diverse functions. Annu. Rev. Biophys. Biomol.
Struct. 27:503–528.
Servant, G., O. D. Weiner, P. Herzmark, T. Balla, J. W. Sedat, and H. R.
Bourne. 2000. Polarization of chemoattractant receptor signaling during
neutrophil chemotaxis [see Comments]. Science. 287:1037–1040.
Stock, A. M., and S. L. Mowbray. 1995. Bacterial chemotaxis: a field in
motion. Curr. Opin. Struct. Biol. 5:744 –751.
Stock, J. B., A. J. Ninfa, and A. M. Stock. 1989. Protein phosphorylation
and regulation of adaptive responses in bacteria. Microbiol. Rev. 53:
450 – 490.
Teruel, M. N., and T. Meyer. 2000. Translocation and reversible localization of signaling proteins: a dynamic future for signal transduction. Cell.
103:181–184
Tomchik, K. J., and P. N. Devreotes. 1981. Adenosine 3⬘,5⬘-monophosphate waves in Dictyostelium discoideum: a demonstration by isotope
dilution–fluorography. Science. 212:443– 446.
Xiao, Z., N. Zhang, D. B. Murphy, and P. N. Devreotes. 1997. Dynamic
distribution of chemoattractant receptors in living cells during chemotaxis and persistent stimulation. J. Cell Biol. 139:365–374.
Biophysical Journal 81(3) 1314 –1323
50
Biophysical Journal
Volume 82
January 2002
50 – 63
Models of Eukaryotic Gradient Sensing: Application to Chemotaxis of
Amoebae and Neutrophils
Andre Levchenko* and Pablo A. Iglesias†
*Divisions of Biology, California Institute of Technology, Pasadena, California 91125 and †Department of Electrical and Computer
Engineering, Johns Hopkins University, 105 Barton Hall, Baltimore, Maryland 21218 USA
ABSTRACT Eukaryotic cells can detect shallow gradients of chemoattractants with exquisite precision and respond quickly
to changes in the gradient steepness and direction. Here, we describe a set of models explaining both adaptation to uniform
increases in chemoattractant and persistent signaling in response to gradients. We demonstrate that one of these models can
be mapped directly onto the biochemical signal-transduction pathways underlying gradient sensing in amoebae and
neutrophils. According to this scheme, a locally acting activator (PI3-kinase) and a globally acting inactivator (PTEN or a
similar phosphatase) are coordinately controlled by the G-protein activation. This signaling system adapts perfectly to
spatially homogeneous changes in the chemoattractant. In chemoattractant gradients, an imbalance between the action of
the activator and the inactivator results in a spatially oriented persistent signaling, amplified by a substrate supply-based
positive feedback acting through small G-proteins. The amplification is activated only in a continuous presence of the external
signal gradient, thus providing the mechanism for sensitivity to gradient alterations. Finally, based on this mapping, we make
predictions concerning the dynamics of signaling. We propose that the underlying principles of perfect adaptation and
substrate supply-based positive feedback will be found in the sensory systems of other chemotactic cell types.
INTRODUCTION
Many biological systems have the ability to sense the direction of external chemical sources and respond by polarizing and migrating toward chemoattractants or away from
chemorepellants. This phenomenon, referred to as chemotaxis, is crucial for proper functioning of single-cell organisms, such as bacteria and amoebae, and multi-cellular
systems as complex as the immune and nervous systems.
Chemotaxis also appears to be important in wound healing
and tumor metastasis. A common feature of most chemotactic signaling systems is the ability to adapt to different
levels of external stimuli, so that it is the gradient of
signaling molecule rather than the average signal value that
determines the response. Chemotactic cells exhibiting perfect adaptation respond to spatially homogeneous increases
in external stimulus by transient activation of specific intracellular signaling pathways. The same signaling pathways, however, can be activated persistently if the signal is
presented in a spatially inhomogeneous, graded manner.
The goal of this analysis is to extend our understanding of
these processes by creating a single model explaining both
adaptation and gradient sensing.
The need for chemotaxis presents cells with a daunting
problem of detecting often exceedingly shallow and changing gradients of extracellular substances and regulating a
Received for publication 23 April 2001 and in final form 11 October 2001.
Dr. Levchenko’s present address is Dept. of Biomedical Engineering,
Johns Hopkins University, Baltimore, MD 21218.
Address reprint requests to Pablo Iglesias, Dept. of Electrical and Computer Engineering, Johns Hopkins University, 105 Barton Hall, 3400 N.
Charles St., Baltimore, MD 21218. Tel.: 410-516-6026; Fax: 410-5165566; E-mail [email protected].
© 2002 by the Biophysical Society
0006-3495/02/01/50/14 $2.00
complex locomotion apparatus to move in accordance with
the direction and the value of these gradients. The mechanism for attaining a highly complex and integrated response
such as this calls for an explanation and modeling in quantitative rather than qualitative terms. Mathematical and
computational modeling can provide a translation of seemingly logical biochemically-based arguments into a set of
predictions of dynamical and steady-state properties of the
system. Using mathematical formalism, alternative hypotheses can be contrasted more easily and criteria found for
discarding a hypothesis-contradicting experimental observations, sometimes in a subtle and counterintuitive way. In
addition, mathematical models can provide insight into
some general design principles that biochemically-based
cell control systems can use to perform a particular function.
All these considerations are especially true for the intricate
regulation of eukaryotic gradient sensing and chemotaxis,
leading to a long history of a quantitative and modeling
research.
Our model attempts first to derive principles that must be
true for any chemotactic cell capable of displaying both
perfect adaptation and persistent signaling with nonlinear
signal amplification, and then, to investigate whether and
how these principles are effected in a particular cell system.
As a result, we will obtain a model related to a simple
mechanism for gradient sensing, qualitatively outlined recently (Parent and Devreotes, 1999), thus providing support
for some of its premises and conclusions, but also making
the model more realistic and quantitative. We then complement it with a new mechanism for signal amplification and
show how this model can be experimentally verified. We
also argue that an alternative hypothesis for a mechanism of
gradient sensing outlined in the next paragraph is probably
invalid, at least in amoebae and neutrophils. The dynamical
Models of Eukaryotic Gradient Sensing
model presented in our study can thus provide a mathematical formalism that can be used more effectively as a paradigm of eukaryotic gradient sensing.
Unlike most explanations of eukaryotic gradient sensing
presented to date, the phenomenological model put forward
in Meinhardt (1999) is formulated mathematically and results in semiquantitative predictions. This model is based on
the principle, formulated by Turing and adapted in Gierer
and Meinhardt (1972) to explain biological pattern formation. The Turing principle postulates that stable patterns
may arise if there is an autocatalytic local production of an
activator that also causes production of an inhibitor. Unlike
the activator, the inhibitor is assumed to be capable of
long-range diffusion. This model is based on local positive
and global negative feedback that can lead to a substantial
amplification of a locally applied signal, thus making it
attractive in trying to account for substantial signal amplification observed in eukaryotic gradient sensing. However,
in addition to predicting signal amplification, the Turing
principle also predicts that the activation pattern becomes
stable. This presents a problem, because it is known that the
gradient-sensing signaling systems need to readjust themselves continuously to be able to sense changes in the
environment. In Meinhardt (1999), this difficulty is overcome by proposing a second inactivating enzyme with a
longer activation time, acting locally to “poison” the activity peak and “unlock” the local activation in the system.
This assumption however makes the model far less parsimonious and, thus, more difficult to accept.
Meinhardt also points out that the cytoskeleton rearrangements are integral to biochemical interpretation of his
model. Recently, it has been realized, however, that eukaryotic gradient sensing can be decoupled from cytoskeletondependent processes. In particular, Dictyostelium discoideum cells, in which actin polymerization is inhibited by
latrunculin A, can still activate a variety of signaling pathways in a gradient-dependent, spatially polarized manner
(Parent and Devreotes, 1999). These rounded cells, lacking
mobility and polarization imposed by actin polymerization,
clearly exhibit both adaptability and persistence of signaling
without substantial dynamic fluctuations observed in cells
with intact actin polymers. It can be demonstrated that
Meinhardt’s model fails to account for this behavior. In
particular, the model predicts no perfect adaptation, whereas
action of the two inhibitors make persistent activation at a
particular membrane location impossible. Finally, it should
be pointed out that the phenomenon of the Ca2⫹-induced
Ca2⫹ release from the intracellular stores postulated as the
mechanism of the positive feedback is doubtful, because
inhibition of Ca2⫹ concentration changes by cell permealization and other means does not affect gradient sensing in
D. discoideum, a common model system used in studies of
eukaryotic chemotaxis (Van Duijn and Van Haastert, 1992;
Traynor et al., 2000). Several studies suggested that the only
aspect of chemotaxis for which upregulation of Ca2⫹ is re-
51
quired is efficient detachment of the uropode in migrating
amoebae and neutrophils (Eddy et al., 2000). These considerations call into question the general applicability of the approach in Meinhardt (1999) to modeling of gradient sensing.
Decoupling chemoattractant gradient sensing from cell
movement and cytoskeletal rearrangements can greatly facilitate consideration of the underlying biochemical regulation. Indeed, the actin cytoskeleton remodeling and its connection to various intracellular and extracellular cues
mediated by a multitude of regulatory proteins may present
an intimidating if not impossible task for a modeler. The
matter is further complicated by the need to account for cell
adhesion properties, loss and synthesis of the cell membrane, variable cell shape, etc. It is therefore of importance
that gradient sensing and cytoskeleton regulation represent
separable components of chemotaxis. In this work, we concentrate on the analysis of the cytoskeleton-independent
gradient sensing, creating a model that may be integrated
with the model of cytoskeletal regulation at a later point.
We present, here, necessary conditions for the organization or topology of a gradient-sensing biochemical network
and a plausible biochemical scheme that may embody these
principles in D. discoideum and neutrophils. Whereas the
models by Meinhardt and others (D. Lauffenburger and A.
Arkin, personal communications) were driven primarily by
the need to explain high gain in gradient sensing, we use a
different strategy. It consists in accounting first for perfect
adaptation (because it does not involve spatial consideration, and the model is simpler), then in seeing how this
model needs to be modified to account for persistent signaling, as opposed to adaptation, in the presence of gradients. Finally, we explore possible mechanisms of signal
amplification consistent with adaptation and persistent signaling.
MODEL AND RESULTS
Perfect adaptation to spatially uniform changes
in ligand concentrations
Perfect adaptation is commonly observed in gradient-sensing systems (Alon et al., 1999; Van Haastert, 1983). A
simple analysis shows that perfect adaptation allows a sensory system to respond to the gradient itself rather than to
the absolute value of the signal. Any deviation from the
precise character of adaptation can result in persistent activation in the absence of signal gradients, a situation that can
severely limit the range of inputs, over which a system can
operate efficiently. We thus need to account for perfect
adaptation in our model of gradient-sensing signal transduction.
A mechanism for generating robust perfect adaptation
based on receptor modification has been proposed previously for bacterial chemotaxis (Alon et al., 1999; Yi et al.,
2000). However, receptor modification has been shown to
Biophysical Journal 82(1) 50 – 63
52
Levchenko and Iglesias
compared to the activation of the two enzymes, the concentration of R* is a function of the ratio of the concentrations
of A and I. The steady-state concentration of R* is independent of both the external signal concentration and the ratio
of inactivation rates. The peak value of the transient response is inversely proportional to the adaptation time.
Possible mechanisms of signal amplification
(gain in signaling)
FIGURE 1 Scheme for achieving perfect adaptation. (A) The response
element can be found in one of two states: active (R*) or inactive (R).
Transitions between the two states are catalyzed by the activation (A) and
inactivation (I) enzymes. The activity levels of the two enzymes are both
regulated by the signal (S). (B) Sample predictions of the adaptation
scheme depicted in (A). Normalized concentration of the response element
is plotted as a function of normalized time for a 10% increase at time 0 in
the concentration of the signaling molecule S. The ratio ␣ of inactivation
rates of I and A determines the transients. For 0 ⬍ ␣ ⬍ 1, activation of A
is faster than that of I, resulting in a transient increase in the concentration
of R*. Smaller values of ␣ lead to larger transients and longer adaptation
times. Plots corresponding to values of ␣ of 0.1 (largest transient) 0.3, 0.5,
0.7 and 0.9 (smallest transient) are shown.
be unessential for G-protein-mediated adaptation in eukaryotes (Kim et al., 1997), the main focus of this study.
Therefore, we consider here two different mechanisms allowing the achievement of precise adaptation downstream
of the receptor in a signaling pathway.
The scheme leading to precise adaptation proposed here
is based on the assumption that a signal S increases the
concentration of an activator A, whose action is to convert
some response element R into the activated form R* (Fig.
1). For adaptation to take place, an inactivator I, mediating
the reverse conversion of R* to R, needs to be introduced. S
activates both A and I in fixed proportion. As shown in the
Appendix, this scheme can lead to perfect adaptation because the corresponding equations have solutions for the
concentration of R* that are not a function of S. Formulation
of these equations requires further assumptions, namely that
the reactions of activation (production) of A and of I have
the same form of dependence on S. A similar scheme
achieving perfect adaptation (not shown) can be formulated
relying on the inactivation of I being a saturated process.
Below, we show that these two mechanisms represent plausible descriptions of the biochemical processes underlying
gradient sensing in amoebae and neutrophils. In Fig. 1 B we
demonstrate numerically that the scheme in Fig. 1 A leads to
perfect adaptation to changes in the external signal S. The
model equations for this and the other simulations are found
in the Appendix. The amplitude and duration of the response is a function of the ratio of inactivation rates of the
activator A and inactivator I. If the activation of R is fast
Biophysical Journal 82(1) 50 – 63
Significant nonlinear signal amplification has been postulated to occur in the biochemical pathways mediating eukaryotic gradient sensing. As discussed above, this phenomenon appears to be so dramatic that previous modeling
efforts have been focused primarily on description of the
high gain rather than adaptation aspects of the sensory
signal transduction. Moreover, the presence of a Turing-like
positive feedback mechanism is often assumed. However,
as mentioned above, the Turing mechanism is not particularly appealing as a means for amplification in gradient
sensing. Here, we propose a different amplification scheme,
also based on a positive feedback loop.
The overall rate of an enzymatic reaction far from saturation with a given reaction efficiency (measured as the
ratio of the maximum rate to the Michaelis constant, vmax/
KM) can be increased if the concentration of the active
enzyme or that of the substrate is augmented. In the feedback scheme we propose (Fig. 2 B), the reaction product R*
affects the supply of the substrate R, but not the activation
of the enzyme. The supply here can mean regulation of a
reaction resulting in production of R or a transport process
leading to increased local concentrations of R. The concentration of the active enzyme is taken to be proportional to
the external signal S. If the signal is absent, the reaction
cannot proceed and the positive feedback, if present, is
discontinued. If the signal is present, the reaction proceeds,
and the substrate-supply feedback gets activated. This sort
of positive feedback is inherently dependent on the presence
of the external signal. It is important to emphasize that the
positive feedback-containing scheme presented in Fig. 2 B,
unlike other mechanisms involving positive feedback (such
as that in Meinhardt, 1999), allows the sensory system to be
sensitive to variations of extracellular signal strength. Indeed, this scheme includes a reaction that can only proceed
in the presence of external signal activation. Therefore, the
feedback loop in Fig. 2 B contains a sort of a circuit breaker
that allows the system to avoid going into signal-independent auto-activation cycling mode. As soon as the external
signal is removed (or the system has adapted to it) the
positive feedback is inactivated.
In principle, signal amplification does not need to involve
a positive feedback as a mechanism. For instance, amplification may occur due to high cooperativity of the activation
process or because not only the concentration of the product
(R*) but also that of the substrate (R) are positively regu-
Models of Eukaryotic Gradient Sensing
53
3 B, adaptation occurs on the level of R*, whereas amplification takes place one step downstream, on the level of R*1.
Placing adaptation upstream of the gain-producing processes may be advantageous if the same upstream signal
activates several downstream pathways. Each of those pathways may have different amplification mechanisms and
characteristics, but all cease activation as soon as adaptation
in a single upstream reaction takes place. We will discuss
below a possible relevance of this argument to various
G-protein-mediated signaling processes in eukaryotic gradient sensing.
Persistent signaling in the presence of ligand
gradients: gradient sensing
FIGURE 2 Possible mechanisms for signal amplification. (A) Nonlinear
signal amplification can be achieved by having the activation process act at
multiple levels as in A, where the enzyme catalyzes the activation of both
the response element R* and its substrate R. (B) Stronger amplification is
achieved through the “substrate supply” mechanism of positive feedback in
which the response element acts to increase the concentration of its
precursor. (C) Input– output response of the systems depicted in (A)
(dashed line) and (B) (solid line). For comparison, the case when the signal
A acts only on the conversion from R to R* with no positive feedback is
shown (dotted line). As discussed in the text, for small and large concentrations of A, the two systems in (A) and (B) exhibit the same slope and thus
have the same gain. However, for intermediate concentrations of A, the
positive feedback scheme exhibits much higher slopes. Constants used are
␣ ⫽ 100, ␤ ⫽ 100, ␥ ⫽ 1, ␦ ⫽ 10, ⑀ ⫽ 0 (dashed) and ␣ ⫽ 100, ␤ ⫽ 100,
␥ ⫽ 1, ␦ ⫽ 0, ⑀ ⫽ 10 (solid).
lated by the signal S (Fig. 2 A). Although simulations
corresponding to both these schemes show a nonlinear
response to changes in the input levels, the positive feedback-mediated scheme shown in Fig. 2 B can provide far
higher gains. We can expect, therefore, that, in systems
exhibiting high-gain behavior, as many gradient sensing
systems seem to do, the positive feedback outlined in Fig. 2
B will be present in one way or another.
Provided that the inactivator acts to reverse the processes
mediated by A, the possible amplification schemes in Fig. 2
can be combined with the adaptation scheme proposed in
Fig. 1 to obtain high-gain signaling with perfect adaptation
(Fig. 3). It is thus possible to achieve both adaptation and
amplification properties on the same level in a sensory
signaling pathway (conversion of R into R* and the reverse
process).
Adaptation and amplification may also occur on different
levels. For example, in the two-step pathway shown in Fig.
In the previous sections, we analyzed how G-protein-activated signaling systems can adapt perfectly to spatially
homogeneous changes in ligand concentration. The same
systems become activated persistently in the presence of
ligand gradients. It is of interest, then, to see whether the
simple schemes presented above are sufficient to explain
both adaptability and persistent signaling depending on the
spatial organization of the input signal. If we assume all
intracellular signaling to occur locally (e.g., within a close
neighborhood of the receptor–G-protein complex), perfect
adaptation would mean that signaling activity would tend to
the same steady-state value independent of the local extracellular ligand concentration. Therefore, the assumption of
local signal transduction in combination with perfect adaptation does not allow formation of a gradient of intracellular
signaling activity even in the presence of an external ligand
gradient. It follows that some aspect of signaling has to be
global (diffusible) within the cytosol. Additional analysis
reveals (data not shown) that the highest activity gradient
results if the inactivator is allowed to diffuse while the
activator A is assumed to be immobile or slowly diffusing
(see also Postma and van Haastert, 2001). This assumption
is sufficient to predict both adaptability and persistent signaling, as illustrated in Fig. 3. In Fig. 3, C and D, we
illustrate the gain amplification of the positive feedbackcontaining scheme depicted in Fig. 3 B and its gradientsensing capabilities. The only signaling molecule that is
allowed to diffuse is the inactivator. At first, the system is
excited by a homogeneous change in external source—all
parts of the cell experience the same activation levels. At a
later point, the levels of R* (Fig. 3 C) and R*1 (Fig. 3 D)
adapt to the signal. Note, however, that the increase in
concentration for R*1 is considerably larger. The system is
then excited by a spatially inhomogeneous signal. The concentration of S at the “front” is increased by 5% and that of
the “rear” is decreased by 5% with corresponding changes
linearly along the length of the cell. The system now exhibits a corresponding graded response. Note that, once
again, the increase in activity of R*1 is considerably larger
Biophysical Journal 82(1) 50 – 63
54
Levchenko and Iglesias
and displays nonlinearity of response. The equilibrium responses in both R* and R*1 along the length of the cell are
contrasted in Fig. 3 E.
G-protein-mediated gradient sensing: relationship
between the model and biochemistry
In this section, we consider the known biochemistry of
eukaryotic gradient sensing based on activation of G-protein-associated chemokine receptors (Gi family of G-proteins), as studied in D. discoideum and neutrophils. The
principal signaling pathways, shown to be essential in a
variety of experiments, are illustrated in Fig. 4 A. Receptor
occupation by a chemoattractant leads to G-protein activation followed by activation of various downstream effectors, most notably phosphatidylinositol 3-kinase-␥ (PI3K)
(Rickert et al., 2000). Activation of PI3K leads to phosphorylation of various phosphoinositides at the D-3 position of
the inositol ring. Phosphoinositide phosphates PI(4,5)P2 and
PI(3,4,5)P3 (henceforth denoted P2 and P3, respectively) can
further transduce the signal by providing binding sites for
various downstream PH-domain-containing signaling components, such as PLC␦, AKT, and CRAC (Parent and Devreotes, 1999; Servant et al., 2000). A variety of PH-domaincontaining markers has been developed allowing for
straightforward monitoring of phosphoinositide formation
by observing changes in membrane-associated fluorescence. Thus, phosphoinositide concentrations are commonly used as readouts of chemoattractant-stimulated signal transduction. Concentrations of these molecules will
also be used as signaling outputs in our analysis.
Another signal-transduction molecule activated by Gi,
and capable of affecting phosphoinositide levels in response
to chemoattractants, is PLC (specifically ␤2 and ␤3 isoforms) (see Wu et al., 2000 for a summary of recent results).
FIGURE 3 Integrated adaptation and signal amplification schemes. (A)
The adaptation scheme of Fig. 1 can be combined with the amplification
schemes of Fig. 2 at the same level. This scheme is similar to that of
Biophysical Journal 82(1) 50 – 63
Fig. 1 A, but, in this case, the activation enzyme also increases the
concentration of the substrate R. Provided that the activation of the two
enzymes A and I by the external signal S is proportional, the system will
adapt, but the transients will be amplified compared to Fig. 1 A. (B) The
schemes can also be combined in series. Here the adaptation scheme of Fig.
1 A is followed by the signal amplification scheme of Fig. 2 B. The
effectiveness of the latter scheme is illustrated by plotting the corresponding concentrations of R* (C) and R*1 (D). In these figures, a one-dimensional model of the cell’s response is simulated. At 0 s, the concentration
of S is increased uniformly by 10%. As can be seen, the peak increase in
the concentration of R*1 is significantly higher than that of R*. When the
external signal becomes nonhomogeneous, this has significant implications
to the cell’s polarization. At 120 s, the external concentration is increased
5% at the front and decreased 5% at the rear. In Fig. 3 C and D, we show
the concentrations at the front (dotted), center (solid), and rear (dashed).
(E) The concentrations at steady state (240 s) across the normalized length
of the cell. The normalized concentrations of R* (dotted) and R*1 (solid) are
plotted. As expected from Eq. A8 in the text, the ⫾5% external gradient in
S manifests itself in a smaller gradient for R*. However, because this
concentration is near the point of highest gain in the second subsystem, a
large concentration gradient in R*1 ensues.
Models of Eukaryotic Gradient Sensing
55
FIGURE 4 Spatial sensing mechanism of Dictyostelium and neutrophils. (A) The essential biochemical pathways implicated in gradient sensing
(described in detail in the text). The output of the pathway is the concentrations of P3. The nonlinear character of response is assumed to be mediated by
one or more small G-proteins (SGP). This feedback is of the substrate supply kind, illustrated in Fig. 2 B. (B) Simulation of the model corresponding to
(A). Concentration of P3 is shown both as a function of time and position along the cell. The signal is increased homogeneously by 20% at 0 s and removed
at 100 s. It is seen that the system adapts perfectly both these changes. A graded input is applied (⫹5% at the front, ⫺5% at the rear and varying linearly
along the length of the cell) at 200 s. The graded response in P3 concentration, which disappears after removal of the graded input at 300 s, is shown.
PLC acts by hydrolyzing P2 to insosytol-3-phosphaste and
diacylglycerol, both of which may affect downstream signaling events, including Ca2⫹ upregulation and activation
of protein kinase C (PKC). It may appear that chemotactic
events can be influenced negatively by PLC activation,
because the P2, the substrate for PI3K, is depleted. Experimental evidence shows that depletion of PLC can indeed
affect chemotaxis positively for some chemoattractants, but
has no significant effect on chemotaxis toward other chemoattractants. The effect of PLC is thus not uniform
throughout chemoattractant-receptor families and is probably receptor specific. In particular, it has been suggested
that PLC activation leads to downregulation of signaling by
certain receptor classes by receptor desensitization through
activation of PKC (Ali et al., 1999). In this report, we
assume that, although PLC may affect P2 concentration to
some degree, this effect is insignificant for the signaling
pathways involved in gradient sensing.
TABLE 1
Numerous studies indicate that plasma-membrane concentrations of P3 is relatively low in the absence of stimulation and return precisely to the baseline values following
spatially uniform changes in ligand concentration (perfect
adaptation). The total cellular and possibly cell membrane
concentration of P2, in contrast, is relatively high in the
absence of the signal and does not seem to change significantly following exposure to chemoattractant (Stephens et
al., 2000). However, a wealth of data indicates that P2
induces uncapping and subsequent elongation of actin filaments by modulating interaction of profilin, ␣-actinin, vinculin, talin, and various actin-capping proteins with actin
(Czech, 2000). These findings suggest a possibility of significant local changes in concentration of this phosphoinositide that might be masked when compared to its total
cellular concentration. A recent study provides support for
this view (Tall et al., 2000). Finally, the baseline concentration of PI(4)P (denoted here as PIP) is relatively high,
Kinetic rate constants assumed in simulation in Fig. 4 B
k⫺K ⫽ 5.0 s⫺1
k⫺P ⫽ 0.57 s⫺1
k⫺S ⫽ 2.1 s⫺1
k⫺1 ⫽ 2.1 s⫺1
k⫺2 ⫽ 1.4 s⫺1
k⫺3 ⫽ 5.0 ␮M⫺1 s⫺1
kK ⫽ 5.0 s⫺1
kP ⫽ 0.49 s⫺1
kS ⫽ 8.0 s⫺1
k1 ⫽ 1.0 ⫻ 103 ␮M s⫺1
k2 ⫽ 6.2 ⫻ 10⫺4 s⫺1
k3 ⫽ 8.0 ⫻ 10⫺2 ␮M⫺1 s⫺1
D ⫽ 5 ␮m2 s⫺1
k11 ⫽ 1.4 s⫺1
k21 ⫽ 4.3 ␮M⫺1 s⫺1
k31 ⫽ 2.9 ⫻ 10⫺3 ␮M⫺1
Biophysical Journal 82(1) 50 – 63
56
whereas its regulation during signal transduction is not well
understood (Stephens et al., 2000).
As indicated in Fig. 4 A, phosphoinositide regulation may
involve a number of positive feedback mechanisms. It is
well documented, for example, that activity of PIP5Ks, the
enzymes acting to form P2 from PIP, is positively regulated
by small G-proteins Rac, Rho, and Arf (Czech, 2000).
These proteins, in turn, can be upregulated through formation of P3 (Missy et al., 1998). Significantly, formation of
PIP in the Golgi complex can be positively regulated by Arf
through recruitment of phosphatidylinositol-4-OH kinase
(PI4K) (Czech, 2000). It is conceivable that similar upregulation of PIP in the plasma membrane may occur through
the action of Rho and Rac. The resultant high concentrations
of PIP will provide more substrate for formation of P3
(through increased P2 formation). Finally, a significant role
of Cdc42 activation by phosphoinositides in response to a
chemoattractant has been suggested by recent experiments
(Glogauer et al., 2000). Again, this small G-protein may, by
analogy with Rac, Rho, and Arf be involved in the biochemical positive feedback proposed here.
The perfect adaptation of P3 is of central importance to
our model. Initially, it was suggested that adaptation took
place primarily at the level of receptor by feedback phosphorylation of its C-terminal cytoplasmic domain following
signal propagation (Knox et al., 1986). However, as mentioned above, receptor modification is not necessary for
adaptation, and additional mechanisms underlying this
property must exist (Kim et al., 1997). In the following
discussion, we assume that the chemoattractant receptor is
modified to prevent phosphorylation. From Fig. 4 A, it can
be seen that the only remaining possibilities are to assume
that either adaptation of the receptor-associated G-protein,
or that the formation of P3 adapts to changes in signaling
input. Although perfect adaptation at the level of G-protein
seems attractive for explanation of a variety of G-proteinactivated processes (both phosphoinositide-dependent and
not) exhibiting precise adaptation, there are experimental
indications that no such adaptation takes place. First, preincubation of D. discoideum cells with the chemoattractant
cAMP decreases the ability of the G-protein to be activated
by a GTP analog in a receptor-independent manner (Pupillo
et al., 1992). Second, FRET studies of dissociation of ␣ and
␤␥ subunits of G-protein occurring in G-protein activation
reveal that these subunits remain dissociated as long as the
receptor is occupied (Janetopoulos et al., 2001). These findings indicate that G-protein remains activated continuously
in the presence of the signal, making it likely that adaptation
occurs downstream of G-protein activation.
Assuming that adaptation takes place downstream of
G-protein activation, this implies that activation of the Gprotein and its effectors including PI3K remains elevated as
long as the ligand is present. Because the level of P3 adapts
perfectly, a phosphatase opposing PI3K, most likely PTEN,
has to be upregulated in response to G-protein activation.
Biophysical Journal 82(1) 50 – 63
Levchenko and Iglesias
Currently, we do not know precisely the mechanism of
PTEN activation and thus cannot assert the identity of the P3
inhibitor. However, we can predict, on the basis of the
above considerations, that this inactivator is either regulated
directly by the receptor or G-protein in a manner similar to
activation of PI3K, or is activated by its substrates, e.g., P3.
In the latter case, we also predict that inactivation of this
phosphatase is a saturated process. Regulation of the inactivator enzyme by the substrates appears to be less likely,
because no adaptation occurs when P3 is produced by means
other than G-protein-mediated signaling, e.g., by insulin or
PDGF receptor activation (Oatey et al., 1999). Adaptation is
thus likely to be dependent on the events upstream of
phosphoinositide metabolism.
As discussed above, to account for persistent signaling in
the presence of chemoattractant gradients, candidates for
inactivator molecules (PTEN or similar molecular species)
have to possess an additional property. Namely, the inactivator molecule has to be diffusible in its active state. Members of PTEN families have this property, and can be
considered as relevant contenders for the role of inhibitor.
The issue of signal amplification in the biochemical pathways defined above can now be addressed. We mentioned
that there are feedback mechanisms mediated by the small
GTPases of the Rho family that can increase the production
of P2 and PIP following upregulation of P3. Because P2 and
PIP are substrates required to produce P3, the amplification
can proceed according to the gain mechanisms described
above (Fig. 2). Indeed, production of P3 is amplified by the
positive feedback from P3 directly onto formation of its
substrate P2. Here the activator A is PI3K, whereas the
substrate R is P2 and the product R* is P3. In both cases, the
activation ceases as soon as the activator (the active PI3K)
is removed. These gain schemes are therefore sensitive to
signal variations (including changes in the ligand gradients)
and can operate successfully if adaptation occurs at the level
upstream of phosphoinositides. As illustrated above, these
schemes are also compatible with the adaptation mechanisms operating at the level of phosphoinositides.
From the above consideration, the following likely
scheme of G-protein-mediated signal transduction events in
D. discoideum and neutrophils in response to chemoattractants emerges. Following G-protein activation of PI3K, this
enzyme increases the concentration of P3. Following this
initial increase, one or more members of the Rho family of
small G-proteins is activated, leading to upregulation of
membrane-localized PIP5K and PI4K. These kinases, in
turn, increase production of PI3K substrates: P2 and PIP,
leading to signal amplification. The G-protein activation
also leads to activation of PTEN or a similar PI3K-counteracting phosphatase. As a result, in spatially uniform
concentrations of the ligand, the action of PI3K is balanced
to achieve the baseline levels of D-3-position inositol phosphorylation. The feedback loops are interrupted and the
signaling mechanism adapts perfectly. If the system is faced
Models of Eukaryotic Gradient Sensing
57
TABLE 2
150
Sensitivity analysis
Gradient
100
k⫺K
kK
k⫺P
kP
D
k⫺S
kS
k⫺1
50
0
-50
-100
-5
-4
-3
-2
-1
0
1
2
3
4
5
FIGURE 5 Spatial sensing mechanism: response to various external
gradients. The system of Fig. 4 was subjected to varying gradients, ⫾2%,
⫾5%, ⫾10%, and ⫾25%, relative to that of the center, in the concentration
of the external source. The spatial response along the length of the cell is
plotted.
with a chemoattractant gradient, diffusion of PTEN or a
similar inactivator leads to an incomplete balance of PI3K
action and results in persistent signaling oriented in the
direction of the gradient.
The mathematical description of our model of the adaptation mechanism of D. discoideum and neutrophils is found
in the Appendix, and the results of corresponding simulations are shown in Fig. 4 B. The kinetic constants used are
given in Table 1. The homogeneous concentration of Gprotein signal was first increased by 20% from its baseline
level at t ⫽ 0 s. This results in a perfectly adapting spatially
homogeneous response. This response is duplicated when
the G-protein signal returns to basal level (100 s). Finally, a
graded input is applied (200 s) by a graded level of Gprotein signal varying from the basal level ⫹5% at the front
and ⫺5% at the rear and linearly throughout the cell. Our
model exhibits a graded response, with the ratio activity
between front and rear nearing ⫾25% that disappears when
the G-protein signal once again returns to the basal level
(300 s).
Although the responses seen in Fig. 4 B are highly
nonlinear, their amplitudes are not as great as may be
expected on the basis of experimental data, often showing
what apparently are more substantial increases. However,
the experiments are often performed with gradients not as
shallow as the ⫾5% gradient assumed in Fig. 4 B. To
explore the response characteristics of the system depicted
in Fig. 4 further, we subjected the cell model to different
gradients of the G-protein, varying from ⫾2% to 25% (Fig.
5). It can be easily seen that the intracellular activity gradient is a sensitive nonlinear function of the gradient value.
In particular, there is a more than two-fold increase in
activity at the cell front in the ⫾25% gradient. This result is
important, because it shows that the system in Fig. 4 can
Decrease
(%)
Increase
(%)
5.1
5.4
9.9
5.1
24.5
5.1
5.4
5.1
5.4
5.1
2.2
5.4
154.9
5.4
5.1
5.4
Gradient
k1
k11
k⫺2
k2
k21
k⫺3
k3
k31
Decrease
(%)
Increase
(%)
5.4
65.1
5.1
66.2
5.4
5.1
5.0
67.0
5.2
65.0
5.4
60.4
5.1
5.4
5.0
60.8
Nominal parameter values were either increased or decreased five-fold.
The observed gradient, defined as the ratio P3(front)/P3(rear) ⫺ 1, was
computed. The nominal value is 65.1%. From the numbers obtained, it can
be seen that deviations away from the nominal parameter values cause a
loss in the ultrasensitivity of the system (see text).
respond to the value of the gradient, not just the presence
versus absence of gradient. The relatively low response seen
for small gradient values is consistent with low precision of
gradient detection seen in cells migrating in shallow chemoattractant gradients. The precision goes up as the gradient values increase.
To measure the system’s sensitivity to parameter variations, each kinetic constant was increased/decreased fivefold. The response was checked for two qualitative properties. First, does the system adapt? To determine this, we
calculated the concentration just before 100 s to just before
0 s. We found that, in all cases, the difference between these
two concentrations amounted to ⬍0.2%. Thus, perfect adaptation is a structural property of the model, not unlike that
in Barkai and Leibler (1997). Second, we determined
whether the system can detect spatial gradients. For this, we
compute the activity ratio between front and rear of the cells
at 300 s. The respective changes are seen in Table 2. From
these data, it is clear that large deviations from most of the
nominal kinetic parameters can cause a loss of the ultrasensitivity obtained. This can be explained by analogy to the
gain mechanism depicted in Fig. 2 B and analyzed in the
Appendix. In particular, a relative-large increase in internal
gradient is achieved if the input concentration is centered at
the “transition” regime of the system in Fig. 2 B. In Fig. 2
C, this amounts to log A ⫽ 0. This means that a small
relative difference between front and rear in the concentrations of this stimulus will cause a large relative difference in
the response.
However, if the kinetic parameters deviate from their
nominal values, the regime of operation will shift either to
the right or left on Fig. 2 C, where the slope (log R* versus
log A) is ⬃1. Thus, the internal gradient will mirror the
external gradient. We thus conclude that the property of
spatial sensing, like that of adaptation, is robust, but that the
ultrasensitivity seen is not.
Biophysical Journal 82(1) 50 – 63
58
DISCUSSION
In this study, we argue that substantial insights into the
problem of eukaryotic gradient sensing can be gained from
mathematical analysis of the necessary conditions for some
of the biochemical properties of this process. Namely, any
proposed biochemically-based model has to be able to account for both perfect adaptation of signaling to spatially
homogeneous variations of chemoattractant concentration
and also for high-gain persistent polarized signaling in
response to chemoattractant gradients. The model also
needs to be able to predict a high degree of sensitivity of
polarized signaling to changes in the ligand gradient, e.g.,
due to gradual changes in the value or direction of the
gradients. As demonstrated here, these necessary conditions
substantially limit the number of possible ways a gradientsensing biochemical network can be organized. The mathematically motivated limitations, coupled with information
on various aspects of known biochemistry, allowed us to
suggest a plausible scheme of gradient sensing in D. discoideum and neutrophils.
First, we considered the property of perfect adaptation.
Perfect adaptation is commonly observed in various Gprotein or growth-factor receptor-mediated signaling pathways implicated in gradient sensing. Several models of
perfect adaptation in these pathways have been suggested
before. For example, in Tang and Othmer (1994), it is
proposed that adaptation occurs due to activation of both
activating and inhibitory G-proteins. Other investigators
ascribed adaptation mechanism to receptor regulation properties (Knox et al., 1986). We also should note that a
receptor modification-based model has been successfully
advocated for bacterial chemotaxis (Alon et al., 1999).
Here, we investigate two models distinct from those proposed previously primarily because recent experimental observations add new restrictions on how a biochemical
scheme underlying adaptation can operate. In particular, it
has been determined that receptor phosphorylation is not
essential in adaptation and that no redistribution of receptor
molecules occurs in migrating Dictyostelium cells (Kim et
al., 1997). In addition, only one G-protein species has been
shown to mediate adaptation and chemotaxis (Neptune and
Bourne, 1997), which argues against a previously proposed
model of G-protein-based adaptation (Tang and Othmer,
1994).
The adaptation schemes proposed here postulate, in general, the existence of a process, the regulation of which
results in adaptation of the activity of this process with
respect to an external stimulus. The possible adaptation in
activity of the G-protein to activation of the associated
receptor or adaptation of phosphoinositide phosphorylation
can be considered as examples of this process. As downregulation of activation is presumed to be mediated by an
inhibitory molecule, assumptions on how the inactivator
itself is activated need to be made. Two possibilities can
Biophysical Journal 82(1) 50 – 63
Levchenko and Iglesias
then be considered: inactivator regulation by the activity of
the process itself or by events upstream of the process. The
fact that adaptation is perfect leads to further limitations on
the mechanisms of inactivator regulation. In particular, if
the inactivator is regulated downstream of the adaptation
process, the inactivator downregulation has to be saturated.
If, however, an upstream process regulates the inactivator,
this regulation needs to be proportional to the activation of
the corresponding activator. These conditions for perfect
adaptation can be tested experimentally. In particular, on the
basis of the plausible biochemical scheme proposed in the
text, one can predict that the steady-state activation of
PTEN or a similar phosphatase is proportional to activation
of PI3K. Further analysis of PTEN regulation is needed to
verify this prediction.
Another prediction concerning the inactivator is that it
has to be freely diffusible in the cytosol. This prediction
reflects an added necessary condition needed to explain
persistent signaling polarization in chemoattractant gradients in addition to perfect adaptation to spatially homogeneous changes in chemoattractant. Again, for PTEN or other
inactivator candidate, this prediction can be verified experimentally. For instance, any PTEN modification preventing
its diffusion is likely to limit the accuracy of gradient
detection in chemotaxis. It is important to emphasize that
the predictions as to the nature of the inactivator regulation
and its diffusivity are made on the basis of consideration of
the basic properties of perfect adaptation and persistent
signaling and, thus, are expected to hold for any candidate
for the role of the inhibitor.
A major challenge in modeling gradient sensing is to
reconcile the strongly nonlinear signal response with high
sensitivity to the presence and the amplitude of the signal.
So far, it has been common to assume that some sort of
Turing-like positive feedback acting from the activator of
signaling onto its own production is needed to explain the
high gain characteristics of signal amplification. This hypothesis, however, invariably leads to formation of stable
signaling patterns independent of the external ligand gradient. We argue here that several alternative schemes, with
only one relying on a positive feedback, can be proposed to
explain nonlinear signal amplification. Furthermore, there is
a significant difference between the nature of the positive
feedback mechanism proposed here and the modification of
the Turing mechanism proposed by Meinhardt and others.
The mechanism suggested here postulates that gradient
sensing can be mediated by a positive feedback from the
activator onto the supply of its precursor (or its inactivated
form) rather than on the activator production itself. This
kind of “substrate supply”-driven feedback scheme provides
an opportunity to amplify the response significantly, while
retaining the sensitivity to the presence of the promoteractivating enzyme. No requirement for a second inhibitor,
as needed in the Meinhardt model, is any longer present.
This approach can be used to modify not only the gradient-
Models of Eukaryotic Gradient Sensing
sensing signaling schemes suggested before, but also perhaps other models, in which a Turing-like pattern-generating schemes are utilized but not justified.
Consideration of the biochemical mechanisms implicated
in gradient sensing in D. discoideum can serve to verify the
plausibility of a signal gain scheme. Indeed, although Ca2⫹induced Ca2⫹ release proposed by Meinhardt has been
shown to be dispensable for gradient sensing in D. discoideum, phosphorylation of various phosphoinositides is
thought to be at the core of the corresponding signal transduction. Here, we propose the existence of a small Gprotein-mediated positive feedback scheme leading to production the phosphoinositides P3. Its concentration, very
low in quiescent cells, undergoes sharp transient (in adaptive response) or persistent (in gradient-sensing response)
increases following exposure to a chemoattractant. A quick
analysis of the available biochemical information on the
underlying signal transduction reveals that the mathematical
mechanisms suggested for the substrate supply-mediated
positive feedback are likely to be embodied in the biochemical mechanisms. The positive feedback can be mediated by
upregulation of PIP and P2 concentrations through the action of one or more small G-proteins (Cdc42, Rac, or Rho),
which are, in turn, activated by P3. Evidence for the importance of small G-protein-mediated feedback in cell orientation has been obtained recently (Rickert et al., 2000).
Although, in this study, we concentrated on modeling of
gradient sensing that can occur independently of cell locomotion, in unperturbed chemotactic systems, the influence
of various factors omitted from this analysis, such as cytoskeleton-mediated polarization of signaling components,
can be of major importance. Indeed, even in the absence of
an external gradient, a variety of cells, including D. discoideum, can migrate in random directions. These cells are
polarized with various signaling molecules, including Gproteins, localized to what can be regarded as the front and
the back of the cell (Jin et al., 2000). Polarization of signaling apparatus can create intracellular signaling gradient
even in the absence of external chemoattractant gradients.
Cytoskeleton-mediated extension of filopodia can further
influence the gradient detection by providing the opportunity for temporal gradient sensing, in which gradients are
measured by subtracting ligand concentrations detected at
different times in the same subcellular location. Numerous
questions are still open in this aspect of chemotaxis, such as
what causes symmetry breaking in the cytoskeleton architecture leading to cell polarization and how commonly
observed oscillations in actin cytoskeleton structure
(Vicker, 2000) can influence chemotaxis. This paper provides a “stepping stone” for addressing these questions
through an analysis of cytoskeleton-independent gradientdetection mechanisms that can regulate actin polymerization.
We anticipate that, in the further chemotaxis models, the
relative roles of P2 and small G-proteins (Cdc42, Rac, and
59
Rho) in regulation of actin polymerization will be further
accounted for. It is becoming clear that actin is polymerized
at the leading edge of migrating cells according to the
dendritic nucleation model, whereby a regulatory protein
complex, Arp2/3, both creates new filaments and crosslinks them into a branching meshwork. Recently, it has been
shown that costimulation of Arp2/3 by both P2 and small
G-proteins (Cdc42 in particular) through accessory WASP
protein family is essential for its activation (Blanchoin et al.,
2000). It is important to have congruent activation of both
these regulators in formation of cell membrane protrusions
at the front of the cell. The nature of the positive feedback
signal amplification suggested here guarantees that both P2
and small G-proteins become activated only if PI3K signaling is present. Unrelated regulatory events leading to increasing concentrations of just P2 or small G-proteins can
mediate other important processes, such as stabilization of
potassium channels (Kobrinsky et al., 2000) or membrane
reshaping (Loyet et al., 1998) but not formation of spikes or
philopodia characteristic of migrating cells.
In this study, we illustrated how the mathematical model
of the processes underlying perfect adaptation and reversible signal amplification can be mapped to biochemical
signaling networks that became known through experimental studies in amoebae and neutrophils, probably the most
common model systems in studies of chemotaxis. It will be
of interest to see whether these general mathematical principles will hold for biochemical signaling networks found in
other chemotaxing systems. Studies of chemotropism in
yeast revealed important differences in the identity of the
sensory pathways involved in gradient sensing in D. discoideum and Saccharomyces cerevisiae (Arkowitz, 1999). For
instance, PI3K does not seem to be a major player in yeast
gradient sensing, whereas the MAPK Fus3 seems to have an
essential role. In addition, in migration of fibroblasts or
neuronal growth cones, reception of the signal is not mediated by G-proteins. Despite these biochemical differences,
we suggest that the major underlying principles proposed in
this study, such as the combination of a mechanism for
perfect adaptation with substrate-supply-mediated reversible signal amplification will be at the core of most eukaryotic gradient-sensing systems.
It is of interest to compare our gradient-sensing model
with qualitative descriptions suggested before. One of the
popular ones, proposed in Parent and Devreotes (1999), is
conceptually very similar to the one proposed here in that it
assumes that perfect adaptation is mediated by a broadly
defined balance of the actions of the activator and inactivator of signaling, whereas the persistent response to gradients
is generated by an imbalance of these components due to the
inactivator diffusion. However, no particular mechanisms
(either mathematical or biochemical) have been proposed
and analyzed by the authors, which limited the predictive
power of the model. In addition, no clear explanation for the
sources of nonlinearity in response has been suggested. The
Biophysical Journal 82(1) 50 – 63
60
Levchenko and Iglesias
modeling framework proposed here, both in its general
theoretical and its plausible biochemical embodiments, provides more opportunities for direct experimental test and
further refinement.
trations of these two enzymes are Atot and Itot, respectively. A third
enzyme, the external signal S, mediates activation of two enzymes, whereas
inactivation is assumed to be constitutive. Again, using the form Eq. A1,
the differential equations describing the active states can be written as
dA
⫽ ⫺k⫺AA ⫹ k⬘AS共Atot ⫺ A兲,
dt
APPENDIX
Description of model from Fig. 1
We assume that the response element is found in both an active, R*, and an
inactive, R, state. Conversion from the inactive to the active state is through
the activator enzyme A, whereas inactivation is through the enzyme I.
In Fig. 1 A, we presented three reactions, in which a molecule is
converted by an activator enzyme into the active state and by an inactivator
enzyme into the inactive state. These reaction cycles are assumed for the
activator A, the inactivator I, and the response element R. The relationship
among these components is as follows. For the reaction of R, the activator
is A and the inactivator is I. For the reaction of A, the activator is S (the
external signal) and the inactivator is a constitutively active molecule (not
denoted). By analogy, I is activated by S, and inhibited constitutively. The
consideration of all of these reactions will follow the general treatment in
Goldbeter and Koshland (1981).
First, we consider a general set of reactions for the interconversion
between the active form W* and inactive form W of a signaling molecule.
With the activation mediated by an enzyme E1, the enzyme–substrate
complex designated as U1 and association, dissociation, and reaction
(catalysis) constants denoted, respectively, as kc1, ku1, and ka1, and with the
inactivation mediated by E2 and the corresponding parameters (kc2, ku2, and
ka2), we have,
dI
⫽ ⫺k⫺II ⫹ k⬘IS共Itot ⫺ I兲.
dt
Moreover, we assume that the inactivating reactions for both A and I are far
more efficient (as measured by the value of the ratio of the catalytic rate
constant and the Michaelis–Menten constant) than the activating reaction
mediated by S. A result of this assumption is that the available substrate for
S far exceeds the concentration of the two enzymes; i.e., Atot ⬎⬎ A and
Itot ⬎⬎ I, so that these equations can be simplified as
dA
⫽ ⫺k⫺AA ⫹ kAS,
dt
(A3a)
dI
⫽ ⫺k⫺II ⫹ k⬘IS,
dt
(A3b)
where kA ⫽ k⬘AAtot and kI ⫽ k⬘I Itot. We can rewrite these equations using
the dimensionless fraction of active molecules r ⫽ R*/Rtot, dimensionless
time ␶ ⫽ k⫺at, and similarly, dimensionless concentrations: a ⫽ (kR/k⫺A)A,
2
2
i ⫽ (kRkAk⫺I/kIkA
) I, and s ⫽ (kAkR/k⫺A
)S. The new dimensionless
equations are
dW
⫽ ⫺kc1WE1 ⫹ ku1U1 ⫹ ku2U2,
dt
da
⫽ ⫺共a ⫺ s兲,
d␶
(A4a)
dU1
⫽ kc1WE1 ⫺ 共ku1 ⫹ ka1兲U1,
dt
di
⫽ ⫺␣共i ⫺ s兲,
d␶
(A4b)
dr
⫽ ⫺␤ir ⫹ a共1 ⫺ r兲,
d␶
(A4c)
dW*
⫽ ⫺kc2W*E2 ⫹ ku2U2 ⫹ ka1U1,
dt
dU2
⫽ kc2W*E2 ⫺ 共ku2 ⫹ ka2兲U2.
dt
(A1)
It is usual to derive the Michaelis–Menten kinetics from the quasi-steadystate approximation, in which it is assumed that a fast steady state for
intermediate enzyme–substrate complexes is achieved (dU1/dt ⫽ dU2/dt ⫽
0). Using these conditions, the equation for the rate of change in W* given
above can be rewritten as
dW*
W*E2
WE1
⫹ ka1
⫽ ⫺kc2W*E2 ⫹ ku2
dt
KM2
KM1
⫽ ⫺␦W*E2 ⫹ ␭WE1.
(A2)
If both enzymes E1 and E2 operate far from saturation, we can assume that
the values for these enzymes given in Eq. A2 correspond to the total rather
than free concentrations. We will use this form for describing the particular
reactions for R, A, and I. The reaction for R is thus
dR*
⫽ ⫺k⫺RIR* ⫹ kRAR.
dt
As mentioned above, we assume that the activating and inhibitory enzymes
also occur in two states: active (A and I) and inactive. The total concenBiophysical Journal 82(1) 50 – 63
where ␣ ⫽ (k⫺I/k⫺A) and ␤ ⫽ [(k⫺R/kR)(k⫺A/kA)]/(k⫺I/kI). At the steady
state, the concentrations of normalized activator and inactivator are both
equal to that of the signaling molecule s. For any value of s ⬎ 0, the
normalized concentration of the active response element is
rss ⫽
a/i
,
a/i ⫹ ␤
(A5)
and we note that it is the ratio of enzyme concentrations that determines the
steady-state value of the activity, and that rss is independent of the concentration of the signal s. Moreover, it is the ratio of inactivation constants
␣ that determines the magnitude of the transient. For ␣ ⬍ 1 (resp. ␣ ⬎ 1)
activation of A is faster (resp. slower) than that of I and, hence a transient
increase (resp. decrease) in the concentration of r results to a positive
increase in the concentration of s. When the concentration of the signaling
molecule, s, is zero, Eqs. A4 have an arbitrary equilibrium value for the
concentration of the active response element rss. However, to maintain
continuity in the steady-state value of rss, we postulate that the only
sensible value is that given by Eq. A5.
To contrast the scheme proposed here with that of Goldbeter and
Koshland (1981), we note that, in their study, the equations governing the
activation of the two enzymes occurs in the saturating region of Michaelis–
Menten kinetics. Based on this assumption, they obtain “ultrasensitivity” in
the steady-state response element R* to changes in the external concentration of S. We consider the opposite regime, in which the enzyme activation
Models of Eukaryotic Gradient Sensing
61
occurs in the linear region of Michaelis–Menten kinetics. Interestingly, the
Goldbeter–Koshland model also results in the degree of activation being a
function of the ratio of activating and inhibitory enzymes. Therefore, even
if we assume a Goldbeter–Koshland-style zero-order-sensitivity regime for
R activation, we can still predict perfect adaptation, provided that the
concentrations of active A and I depend on S linearly. In this case, there are
both adaptation of R* and amplification of the signal S occurring without
any extra mechanisms. Although this scheme may seem attractive for a full
description of gradient sensing, a different, positive feedback-based amplification scheme described in the next section agrees better with experimental data (see text).
Description of models from Fig. 2
Both systems can be described by the pair of differential equations;
dR*
k2AR
⫽ ⫺k⫺2R* ⫹
,
dt
kM ⫹ R
dR
⫽ ⫺k⫺1R ⫹ 共k1 ⫹ k1aA ⫹ k1*R*兲
dt
⫹ k⫺2R* ⫺
k2AR
.
kM ⫹ R
(A6)
The first equation describes an enzymatic conversion from R to R* using
Michaelis–Menten kinetics, as well as the reverse reaction. In the second
equation, we assume that there is a basal level of production of R, with
kinetic constant k1. Two other possibilities exist. If k1a ⫽ 0, then the
enzyme A catalyzes the production of substrate as in Fig. 2 A. If k1* ⫽ 0,
then R* provides a positive feedback loop as in Fig. 2 B.
We look for possible equilibria of these two equations. Setting the first
equation to zero, one obtains
R* ⫽
␣AR
k2 AR
⫽
,
k⫺2 kM ⫹ R ␤ ⫹ R
Finally, the third case—that of Fig. 2 B— has ␦ ⫽ 0. Two solutions
exist, but only one is non-negative,
R* ⫽
⫺共␤ ⫹ ␥ ⫺ A␣⑀兲 ⫹ 冑共␤ ⫹ ␥ ⫺ A␣⑀兲2 ⫹ 4␣␭A
.
2⑀
We first consider the instance where the concentration of A is small, that
is, ␤ ⫹ ␥ ⬎⬎ ␣⑀A. In this instance, the solution matches that of the
previous Section. In the other extreme case, where A is very large, the
solution approaches R* ⫽ ␣A ⫹ ␥/⑀. Thus, asymptotically, this solution
matches that of the previous section for large values of concentration in A.
The third regime of interest is the “transition,” which we describe below.
For the two schemes depicted in Fig. 2 there are three regimes, depending on the concentration of A. When A is small, both schemes give rise to
concentrations of R* that vary linearly with respect to A, and having the
same slope. This is the regime when the contributions of either the k1a or
k1* terms are small compared to the k1 term. Thus, neither positive
feedback nor the contribution of A on the substrate R is playing a significant role. When R ⬎⬎ kM, the system saturates. This can only happen when
either k1a ⫽ 0 or k1* ⫽ 0. In this regime, more positive feedback or
contribution of A on R will not be beneficial, and, therefore, the two
schemes have the same slope and, hence, gain.
Where the systems can differ significantly is in the transition area, and
we will compare the gains there. To do this, we assume that
␥ ⬍⬍ R ⫽ ␥ ⫹ ␦A ⫹ ⑀R* ⬍⬍ kM ⫽ ␤.
It is straightforward to check that when ⑀ ⫽ 0, this leads to R* ⫽ (␣␦/␤)A2.
For the positive feedback scheme, in the region far from saturation, R* ⬵
AR/␤ ⫽ ␣A(⑀R* ⫹ ␥)/␤. Thus,
R* ⬇
␣␥A
,
␤共1 ⫺ ␣⑀A兲
provided that A ⬍ 1/(␣⑀). This scheme can provide arbitrarily larger gains
near A ⫽ 1/(␣⑀). However, as the concentration of A approaches this
threshold, we quickly reach saturation. A comparison of the gains for the
two schemes is shown in Fig. 2 C.
Description of models in Fig. 3
where ␣ ⫽ k2/k⫺2 and ␤ ⫽ kM. Similarly,
R⫽
1
共k ⫹ k1aA ⫹ k1*R*兲 ⫽: ␥ ⫹ ␦A ⫹ ⑀R*.
k⫺1 1
Together, these equations lead to the following quadratic equation for the
steady-state concentration of R*:
⑀共R*兲2 ⫹ 共␤ ⫹ ␥ ⫹ A共␦ ⫺ ␣⑀兲兲R* ⫺ ␣A共␥ ⫹ ␦A兲 ⫽ 0.
We consider some special cases in detail. In the base-line level, we
assume that k1a ⫽ k1* ⫽ 0; equivalently, ␦ ⫽ ⑀ ⫽ 0. In this case, there is
only one solution, R* ⫽ [␣␥/(␤⫹␥)]A, showing that the concentration of
R* increases linearly with that of A.
The situation depicted by Fig. 2 A, where there is no feedback, amounts
to setting ⑀ ⫽ 0. Once again, there is only one solution: R* ⫽ ␣A[(␥ ⫹
␦A)/(␤ ⫹ ␥ ⫹ ␦A)]. Notice that, in the two extreme regimes (small and
large concentrations of A), the resultant concentration for R* varies linearly
with that of A. For small A, the slope (␣␥/(␤ ⫹ ␥)) is the same as in the
previous case. For large A, the slope (␣) is strictly larger. In the transition
regime, the concentration of R* varies as the square of A. This is the region
when the production of R due to A is now significantly more than the basal
level (k1aA ⬎⬎ k1) but R has not saturated (kM ⬎⬎ R).
We first analyze the model of Fig. 3 A and consider the analysis of the
system, assuming that the concentration of the external signaling molecule
is homogeneous. This system then combines the schemes of Figs. 1 A and
2 A. The equations governing the evolution of the system are those of Eqs.
A3, together with
dR*
⫽ ⫺k⫺2IR* ⫹ k2AR,
dt
dR
⫽ ⫺k⫺1IR ⫹ k1aA ⫹ k⫺2R* ⫺ k2AR,
dt
(A7)
which is Eq. A2 in the transition regime discussed in the previous section.
Analysis of this pair of equations leads to a steady-state concentration
value for R* as in the previous section,
R*ss ⫽
k2k1a
共A/I兲2,
k⫺2k⫺1
so that the steady-state concentration of R* depends on the square of the
ratio of concentrations of A and I.
The system described by Fig. 3 B consists of a cascade of the systems
described in Fig. 1 A and the system in Fig. 2 B. Thus, if we once again
Biophysical Journal 82(1) 50 – 63
62
Levchenko and Iglesias
concentrate on the transition regime of the previous section, the steadystate concentration of R*1 is given by
R*1⫺ss ⫽
␣␥共A/I兲
.
␤共1 ⫺ ␣⑀共A/I兲兲
We note that it is, once again, the ratio of enzyme concentrations that
governs the behavior of the system. When the activity of these enzymes is
regulated as in Eqs. A3, constant concentrations of S will lead to steadystate concentrations of the response element that are independent of the
level S. This analysis is valid when the system is stimulated by spatially
homogeneous source signal concentrations. To account for graded inputs,
the system is modified slightly. We assume that the concentrations of all
system species are localized except for the inhibitor, which is allowed to
diffuse. To account for this diffusion, we modify one term to the differential equation describing the inactivator dynamics, Eq. A3b,
⭸I共t, x兲
⫽ ⫺k⫺II共t, x兲 ⫹ kIS共t, x兲 ⫹ Dⵜ2I共t, x兲.
⭸t
For boundary conditions, we assume no flux at either end. Note that the
concentration of the inactivator is now indexed according to the spatial
dimension.
We model the cell as a one-dimensional system where the concentration
of the external signal varies linearly as the spatial parameter x. Thus S(x) ⫽
s0 ⫹ s1x. By solving the above equation, the steady-state concentration of
the inactivator enzyme can then be calculated as
I共x兲 ⫽
冉
冉
kI
sinh ␴x cosh ␴x关cosh ␴ ⫺ 1兴
s0 ⫹ s1 x ⫺
⫹
k⫺I
␴
␴ sinh ␴
冊冊
,
where ␴ ⫽ 公k⫺I/D. Because the activator enzyme cannot diffuse, its
steady-state concentration profile mirrors that of the external source,
A(x) ⫽ (kA/kA)S(x). Recall that the concentration of the response element
is governed by the ratio of these two enzymes, which is now A(x)/I(x), and
this equals
冉
冉
cosh ␴x关cosh ␴ ⫺ 1兴 sinh ␴x
⫺
␴ sinh ␴
␴
kAk⫺I
⬍
.
k⫺AkI
冊冊
⫺1
(A8)
It is the parameter ␴ that determines the spatial response of the system.
For small values of ␴, the concentration of I(x) is approximately constant
across the length of the cell. Hence, the ratio A(x)/I(x) mirrors that of the
external source. When ␴ is large, the concentration I(x) is linear across the
length of the cell, and hence the ratio of A(x)/I(x) is independent of the
spatial parameter x.
The output of a simulation involving the system in Fig. 3 B is shown in
Fig. 3, C–E. Parameters used are kA ⫽ 1 s⫺1, k⫺A ⫽ 1 s⫺1, kI ⫽ 0.05 s⫺1,
k⫺I ⫽ 0.05 s⫺1, k1 ⫽ 1 ␮M s⫺1, k*1 ⫽ 100 s⫺1, k⫺1 ⫽ 1 s⫺1, k2 ⫽ 1 ␮M⫺1
s⫺1, k⫺2 ⫽ 1 s⫺1, kM ⫽ 100 ␮M, and D ⫽ 0.8 ␮m2 s⫺1. As discussed with
regards to the system of Fig. 1 A, the property of perfect adaptation does
not depend on the value of the parameters chosen. The concentration of the
response element R*, which serves as the output of the first subsystem and
input to the second subsystem, has a steady-state value of 1 ␮M. The other
parameter values (k1, k1, k⫺1, k2, k⫺2, kM, and D) where chosen to provide
large amplification near this operating point. This is achieved by selecting
parameters so that the point of highest slope in Fig. 2 C matches the
concentration of the input of the second subsystem, R*.
Biophysical Journal 82(1) 50 – 63
Description of Model from Fig. 4 A
We assume a one-dimensional model of a cell, 10-␮m long. The differential equations describing our model are given below. Except for the
following two cases, the reactions are assumed to follow first-order kinetics. First, in the conversion of PI to PIP, and PIP to P2, the rate constants
k1 and k2 are augmented by terms proportional to the concentration of the
small G-protein; these are meant to express the increase in conversion that
is mediated by these proteins. Second, the conversion of P2 to P3 follows
Michaelis–Mentin quasi-steady-state dynamics. These assumptions lead to
the following set of differential equations:
⭸PI3K
⫽ ⫺k⫺KPI3K ⫹ kKG
⭸t
⭸PTEN
⫽ ⫺k⫺PPTEN ⫹ kPG ⫹ Dⵜ2PTEN
⭸t
⭸SGP
⫽ ⫺k⫺SSGP ⫹ kSPIP3
⭸t
⭸PIP
⫽ ⫺k⫺1PIP ⫹ 共k1 ⫹ k11SGP兲 ⫹ k⫺2P2
⭸t
⫺ 共k2 ⫹ k21SGP兲PIP
⭸P2
⫽ ⫺k⫺2P2 ⫹ 共k2 ⫹ k2SGP兲PIP ⫹ k⫺3PTEN 䡠 P3
⭸t
A共x兲 kAk⫺I
s1
⫽
1⫹
I共x兲 k⫺AkI
s 0 ⫹ s 1x
䡠
It is seen that the system adapts perfectly to homogeneous changes in
the concentration of the external signal S. Graded changes in S lead to
graded changes in both the concentrations of R* and R*1. However, as
expected from Eq. A8, the relative concentration gradient in R* is smaller
than that of S. However, the amplification subsystem involving R*1 leads to
large gradients in this molecule’s concentration, as seen in Fig. 3, D and E.
Varying parameters from these nominal values can have significant effects
on the magnitude of the response element R*1 (not shown).
⫺ k3PI3K
P2
1 ⫹ k31P2
⭸P3
P2
⫽ ⫺k⫺3PTEN 䡠 P3 ⫹ k3PI3K
⭸t
1 ⫹ k31P2
The analysis of this set of equations is similar to that of the previous
section. The only difference is that there are now two positive feedback
loops. Nevertheless, it is possible to show that, for spatially homogeneous
levels of G-protein signaling, the concentration of the signal P3 is independent on the level of concentration of G. For spatially graded inputs, the
spatial distribution of P3 depends on the spatially local ratio of concentrations of PI3K and PTEN. These spatial distributions are governed as in the
previous section. Note, however, that, for these spatially inhomogeneous
differences in concentrations to have the greatest effect on the distribution
of concentrations of P3, this ratio must lie in the transition region described
in Description of Models from Fig. 2. Kinetic constants used are found in
Table 1. These constants were chosen to give basal levels of PIP, P2, and
P3 around 50, 30, and 0.05 ␮M, which is in the range reported (Stenmark,
2000). The diffusion coefficient used corresponds to that predicted in
Postma and van Haastert (2001).
The authors gratefully acknowledge P. N. Devreotes, D. Bray, M. D. Levin
and D. A. Lauffenburger as well as the anonymous reviewers for their
Models of Eukaryotic Gradient Sensing
careful reading of the manuscript and for their helpful insights and suggestions.
Partial support for this work was provided by Burroughs Wellcome Fund’s
Fellowship in Computational Biology (Interfaces program at Caltech) to
A.L., the Whitaker Foundation (P.I.) and the National Science Foundation
(P.I.) under grant No. DMS-0083500.
REFERENCES
Ali, H., R. M. Richardson, B. Haribabu, and R. Snyderman. 1999. Chemoattractant receptor cross-desensitization. J. Biol. Chem. 274:
6027– 6030.
Alon, U., M. G. Surette, N. Barkai, and S. Leibler. 1999. Robustness in
bacterial chemotaxis. Nature. 397:168 –171.
Arkowitz, R. A. 1999. Responding to attraction: chemotaxis and chemotropism in Dictyostelium and yeast. Trends. Cell Biol. 9:20 –27.
Barkai, N., and S. Leibler. 1997. Robustness in simple biochemical networks. Nature. 387:913–917.
Blanchoin, L., T. D. Pollard, and R. D. Mullins. 2000. Interactions of
ADF/cofilin, Arp2/3 complex, capping protein and profilin in remodeling of branched actin filament networks. Curr. Biol. 10:1273–1282.
Czech, M. P. 2000. PIP2 and PIP3: complex roles at the cell surface. Cell.
100:603– 606.
Eddy, R. J., L. M. Pierini, F. Matsumura, and F. R. Maxfield. 2000.
Ca2⫹-dependent myosin II activation is required for uropod retraction
during neutrophil migration. J. Cell Sci. 113:1287–1298.
Gierer, A., and H. Meinhardt. 1972. A theory of biological pattern formation. Kybernetik. 12:30 –39.
Goldbeter, A., and D. E. Koshland. 1981. An amplified sensitivity arising
from covalent modification in biological systems. Proc. Natl. Acad. Sci.
U.S.A. 78:6840 – 6844.
Glogauer, M., J. Hartwig, and T. Stossel. 2000. Two pathways through
Cdc42 couple the N-formyl receptor to actin nucleation in permeabilized
human neutrophils. J. Cell Biol. 150:785–796.
Janetopoulos, C., T. Jin, and P. N. Devreotes. 2001. Receptor-mediated
activation of heterotrimeric G-proteins in living cells. Science. 291:
2408 –2411.
Jin, T., N. Zhang, Y. Long, C. A. Parent, and P. N. Devreotes. 2000.
Localization of the G protein ␤␥ complex in living cells during chemotaxis. Science. 287:1034 –1036.
Kim, J. Y., R. D. Soede, P. Schaap, R. Valkema, J. A. Borleis, P. J. Van
Haastert, P. N. Devreotes, and D. Hereld. 1997. Phosphorylation of
chemoattractant receptors is not essential for chemotaxis or termination
of G-protein-mediated responses. J. Biol. Chem. 272:27313–27318.
Knox, B. E., P. N. Devreotes, A. Goldbeter, and L. A. Segel. 1986. A
molecular mechanism for sensory adaptation based on ligand-induced
receptor modification. Proc. Natl. Acad. Sci. U.S.A. 83:2345–2349.
Kobrinsky, E., T. Mirshahi, H. Zhang, T. Jin, and D. E. Logothetis. 2000.
Receptor-mediated hydrolysis of plasma membrane messenger PIP2
leads to K⫹-current desensitization. Nat. Cell Biol. 2:507–514.
Loyet, K. M., J. A. Kowalchyk, A. Chaudhary, J. Chen, G. D. Prestwich,
and T. F. Martin. 1998. Specific binding of phosphatidylinositol 4,5bisphosphate to calcium-dependent activator protein for secretion
(CAPS), a potential phosphoinositide effector protein for regulated exocytosis. J. Biol. Chem. 273:8337– 8343.
63
Meinhardt, H. 1999. Orientation of chemotactic cells and growth cones:
models and mechanisms. J. Cell Sci. 112:2867–2874.
Missy, K., V. Van Poucke, P. Raynal, C. Viala, G. Mauco, M. Plantavid,
H. Chap, and B. Payrastre. 1998. Lipid products of phosphoinositide
3-kinase interact with Rac1 GTPase and stimulate GDP dissociation.
J. Biol. Chem. 273:30279 –30286.
Neptune, E. R., and H. R. Bourne. 1997. Receptors induce chemotaxis by
releasing the ␤␥ subunit of Gi, not by activating Gq or Gs. Proc. Natl.
Acad. Sci. U.S.A. 94:14489 –14494.
Oatey, P. B., K. Venkateswarlu, A. G. Williams, L. M. Fletcher, E. J.
Foulstone, P. J. Cullen, and J. M. Tavare. 1999. Confocal imaging of the
subcellular distribution of phosphatidylinositol 3,4,5-trisphosphate in
insulin- and PDGF-stimulated 3T3–L1 adipocytes. Biochem. J. 344:
511–518.
Parent, C. A., and P. N. Devreotes. 1999. A cell’s sense of direction.
Science. 284:765–770.
Postma, M., and P. J. van Haastert. 2001. A diffusion-translocation model
for gradient sensing by chemotactic cells. Biophys. J. 81:1314 –1323.
Pupillo, M., R. Insall, G. S. Pitt, and P. N. Devreotes. 1992. Multiple cyclic
AMP receptors are linked to adenylyl cyclase in Dictyostelium. Mol.
Biol. Cell. 3:1229 –1234.
Rickert, P., O. D. Weiner, F. Wang, H. R. Bourne, and G. Servant. 2000.
Leukocytes navigate by compass: roles of PI3K␥ and its lipid products.
Trends. Cell Biol. 10:466 – 473.
Servant, G., O. D. Weiner, P. Herzmark, T. Balla, J. W. Sedat, and H. R.
Bourne. 2000. Polarization of chemoattractant receptor signaling during
neutrophil chemotaxis. Science. 287:1037–1040.
Stenmark, H. 2000. Phosphatidylinositol 3-kinase and membrane trafficking. In Biology of Phosphoinositides. S. Cockroft, editor. Oxford University Press, Oxford, UK. 32–108.
Stephens, L., A. McGregor, and P. Hawkins. 2000. Phosphoinositide
3-kinases: regulation by cell surface receptors and function of 3-phosphorylated lipids. In Biology of Phosphoinositides. S. Cockroft, editor.
Oxford University Press, Oxford, UK. 32–108.
Tall, E. G., I. Spector, S. N. Pentyala, I. Bitter, and M. J. Rebecchi. 2000.
Dynamics of phosphatidylinositol 4,5-bisphosphate in actin-rich structures. Curr. Biol. 10:743–746.
Tang, Y., and H. G. Othmer. 1994. A G protein-based model of adaptation
in Dictyostelium discoideum. Math. Biosci. 120:25–76.
Traynor, D., J. L. Milne, R. H. Insall, and R. R. Kay. 2000. Ca(2⫹)
signalling is not required for chemotaxis in Dictyostelium. EMBO J.
19:4846 – 4854.
Van Duijn, B. and P. J. Van Haastert. 1992. Independent control of
locomotion and orientation during Dictyostelium discoideum chemotaxis. J. Cell Sci. 102:763–768.
Van Haastert, P. J. 1983. Sensory adaptation of Dictyostelium discoideum
cells to chemotactic signals. J. Cell Biol. 96:1559 –1565.
Vicker, M. G. 2000. Reaction-diffusion waves of actin filament
polymerization/depolymerization in Dictyostelium pseudopodium extension and cell locomotion. Biophys. Chem. 84:87–98.
Wu, D., C. K. Huang, and H. Jiang. 2000. Roles of phospholipid signaling
in chemoattractant-induced responses. J. Cell Sci. 113:2935–2940.
Yi, T. M., Y. Huang, M. I. Simon, and J. Doyle. 2000. Robust perfect
adaptation in bacterial chemotaxis through integral feedback control.
Proc. Natl. Acad. Sci. U.S.A. 97:4649 – 4653.
Biophysical Journal 82(1) 50 – 63
REPORTS
that Cdc25C mutations that impair PBD binding result in incomplete mitotic activation in
vivo (Fig. 4E).
Plk1 localizes to centrosomes and kinetochores in prophase and to the spindle midzone during late stages of mitosis through
its COOH-terminus (14, 15). Centrosomal
localization requires both the PB1 and PB2
regions but not kinase activity (13). We
therefore arrested U2OS cells with nocodazole, permeabilized them with streptolysinO, and incubated them with GST-Plk1 PBD
in the absence or presence of peptide competitors (6 ). Plk1 PBD localized to the
centrosomes of nocodazole-arrested cells
(Fig. 5), as verified by costaining with an
antibody to ␥-tubulin. This localization was
disrupted by incubation of Plk1 PBD with
its optimal phosphopeptide ligand but was
unaffected by the nonphosphorylated analog (Fig. 5; fig. S4).
Our identification of Plk1 PBD as a pSer
or pThr binding domain that mediates Plk1
localization to substrates and centrosomes (6)
(i) provides a molecular mechanism for the
essential function of the Plk1 COOH-terminus (13–15), (ii) adds another member to a
growing collection of pSer/pThr-binding
modules, and (iii) demonstrates the general
utility of our phospho-motif–based affinity
screen for identifying binding modules interacting with substrates of any kinase whose
phosphorylation motif is known. Parallels between the PBD in Plk1 and SH2 domains in
Src family kinases are apparent. Like the Src
SH2 domain, the PBD binds to the kinase
domain, inhibiting its phosphotransferase activity in the basal state (18). Thus, both the
PBD and SH2 domains target their respective
kinases toward phosphorylated substrates
upon activation but inhibit their catalytic activity in the basal state (19, 20). A requirement for priming phosphorylation of Plk1
substrates is similar to that observed for the
kinase GSK3 (21). In the case of mitotic
regulation by Plk1, small amounts of Cdc2cyclin B activity during prophase are insufficient to fully activate Cdc25C but could provide priming phosphorylation of Cdc25C for
interaction with the PBD. Activation of Plk1
later in mitosis might then lead to an initial
wave of Cdc25C phosphorylation and activation (17), generating more Cdc2-cyclin B
activity, priming additional Cdc25C molecules for phosphorylation by Plk1, and resulting in a positive feedback loop (12) (fig. S5).
This model may explain the observation that
maximal intracellular targeting and activation
of Cdc25C by constitutively active Plk1 requires coexpression of cyclin B1 (22). In this
regard, the PBD may be an attractive target
for the design of anticancer chemotherapeutics because its compact tripeptide binding
motif may be particularly amenable to the
design of small molecule mimetics.
References and Notes
1. T. Pawson, P. Nash, Genes Dev. 14, 1027 (2000).
2. M. B. Yaffe, A. E. Elia, Curr. Opin. Cell Biol. 13, 131
(2001).
3. T. Obata et al., J. Biol. Chem. 275, 36108 (2000).
4. M. B. Yaffe et al., Cell 91, 961 (1997).
5. K. D. Lustig et al., Methods Enzymol. 283, 83 (1997).
6. Materials and methods are available as supporting
material on Science Online.
7. M. B. Yaffe, L. C. Cantley, Methods Enzymol. 328, 157
(2000).
8. F. M. Davis, T. Y. Tsao, S. K. Fowler, P. N. Rao, Proc.
Natl. Acad. Sci. U.S.A. 80, 2926 (1983).
9. J. M. Westendorf, P. N. Rao, L. Gerace, Proc. Natl.
Acad. Sci. U.S.A. 91, 714 (1994).
10. M. B. Yaffe et al., Science 278, 1957 (1997).
11. E. A. Nigg, Curr. Opin. Cell Biol. 10, 776 (1998).
12. D. M. Glover, I. M. Hagan, A. A. Tavares, Genes Dev.
12, 3777 (1998).
13. Y. S. Seong et al., J. Biol. Chem. 277, 32282 (2002).
14. K. S. Lee, T. Z. Grenfell, F. R. Yarm, R. L. Erikson, Proc.
Natl. Acad. Sci. U.S.A. 95, 9301 (1998).
15. K. S. Lee, S. Song, R. L. Erikson, Proc. Natl. Acad. Sci.
U.S.A. 96, 14360 (1999).
16. A. Kumagai, W. G. Dunphy, Cell 70, 139 (1992).
, Science 273, 1377 (1996).
17.
18. Y. J. Jang, C. Y. Lin, S. Ma, R. L. Erikson, Proc. Natl.
Acad. Sci. U.S.A. 99, 1984 (2002).
㛬㛬㛬㛬
19. W. Xu, S. C. Harrison, M. J. Eck, Nature 385, 595
(1997).
20. F. Sicheri, I. Moarefi, J. Kuriyan, Nature 385, 602
(1997).
21. P. Cohen, S. Frame, Nature Rev. Mol. Cell Biol. 10,
769 (2001).
22. F. Toyoshima-Morimoto, E. Taniguchi, E. Nishida,
EMBO Rep. 3, 341 (2002).
23. A. Bateman et al., Nucleic Acids Res. 27, 260 (1999).
24. Single-letter abbreviations for the amino acid residues are as follows: A, Ala; C, Cys; D, Asp; E, Glu; F,
Phe; G, Gly; H, His; I, Ile; K, Lys; L, Leu; M, Met; N, Asn;
P, Pro; Q, Gln; R, Arg; S, Ser; T, Thr; V, Val; W, Trp; and
Y, Tyr.
25. We thank F. Kanai and P. A. Marignani for providing
cDNA plasmid pools used in the screen, F. McKeon for
providing the Cdc25 construct, and members of the
M.B.Y. and L.C.C. laboratories for technical assistance
and helpful discussions. Supported by National Institutes of Health grants GM52981 (M.B.Y.) and
GM56203 (L.C.C.) and a Burroughs-Wellcome Career
Development Award (M.B.Y.).
Supporting Online Material
www.sciencemag.org/cgi/content/full/299/5610/1228/
DC1
Materials and Methods
Figs. S1 to S5
Table S1
4 October 2002; accepted 23 December 2002
Spontaneous Cell Polarization
Through Actomyosin-Based
Delivery of the Cdc42 GTPase
Roland Wedlich-Soldner,1 Steve Altschuler,2 Lani Wu,2 Rong Li1*
Cell polarization can occur in the absence of any spatial cues. To investigate the
mechanism of spontaneous cell polarization, we used an assay in yeast where
expression of an activated form of Cdc42, a Rho-type guanosine triphosphatase
(GTPase) required for cell polarization, could generate cell polarity without any
recourse to a preestablished physical cue. The polar distribution of Cdc42 in this
assay required targeted secretion directed by the actin cytoskeleton. A mathematical simulation showed that a stable polarity axis could be generated
through a positive feedback loop in which a stochastic increase in the local
concentration of activated Cdc42 on the plasma membrane enhanced the
probability of actin polymerization and increased the probability of further
Cdc42 accumulation to that site.
The ability to self-organize is a fundamental property of living systems (1, 2). Cell
polarity, or the generation of a vectorial
axis controlling cell organization and behavior, is an important example. Although
cell polarization is often directed by asymmetric cues from the environment or the
cell’s history (3), several cell types polarize
randomly in response to global temporal
signals, presumably through a self-organization mechanism. One view is that directed polarization in response to spatial cues
1
Department of Cell Biology, Harvard Medical School,
Boston, MA 02115, USA. 2Bauer Center for Genomics
Research, Harvard University, Cambridge, MA 02138,
USA.
*To whom correspondence should be addressed. Email: [email protected]
could occur by biasing an intrinsic selfpolarization system (4).
A simple and genetically tractable system
such as yeast is useful for studying spontaneous cell polarization at the molecular level.
However, in normal haploid yeast, cell polarization is guided by spatial cues such as the
bud scar or a pheromone gradient (5, 6).
Cue-dependent signaling generally involves
localization of Cdc24, a guanine nucleotide
exchange factor (GEF) for Cdc42 (7, 8). To
render cell polarization in yeast independent
of any specific physiological signals, we took
advantage of the observation that expression
of a constitutively activated form of Cdc42 is
sufficient to cause polarization in otherwise
nonpolarized cells, arrested in the G1 phase of
the cell cycle, where Cdc24 is inactive (9,
10). This polarization event presumably by-
www.sciencemag.org SCIENCE VOL 299 21 FEBRUARY 2003
1231
REPORTS
passes the GEF and the spatial signals from
the bud scar.
The original assays (11, 12) were modified (13) to improve the efficiency of polarization and to enable us to view live cells.
Cells were arrested in G1 with the use of a
strain with a methionine-repressible Cln2 as
the sole source of G1 cyclins (14). Expression
of a constitutively active form of Cdc42,
Cdc42Q61L (15), tagged with (myc)6-GFP
(green fluorescent protein) at the N-terminus
(referred to as MG-Cdc42Q61L), was induced
under the control of the Gal1/10 promoter.
The level of MG-Cdc42Q61L rapidly increased during the initial 3 hours of induction
and then reached a plateau estimated to be
higher than the endogenous level of Cdc42 by
a factor of 5 to 6 (fig. S1A) (16). Shortly after
induction, the constitutively active MG-
Cdc42Q61L exhibited a uniform distribution
in the plasma membrane (Fig. 1A). With
increasing time of induction, cells formed
one or two polar caps of MG-Cdc42Q61L on
the plasma membrane (Fig. 1, B and C, and
Table 1). MG-Cdc42Q61L was found on the
plasma membrane and on various internal
membranes including vacuolar, nuclear, and
endocytic membranes (17) (Fig. 1B) (fig.
S1B), as well as on a large number of highly
motile, dot-like structures that accumulated
below the MG-Cdc42Q61L caps (arrows in
fig. S1C).
The process of MG-Cdc42Q61L polar cap
formation was followed by time-lapse imaging using confocal microscopy. In all six cells
that developed a polar cap in the course of the
imaging experiments, the GFP signal started
to accumulate at a single site on the plasma
membrane and increased in intensity and
width in a time-dependent fashion, until a
steady state was reached (Fig. 1D). The position of the caps did not drift during or after
their formation (Fig. 1, D and E) (movie S1),
which suggests that stable polarity axes were
established. Besides MG-Cdc42Q61L, fluorescence staining showed that two other polarity
markers—F-actin and Sec4, a component of
secretory vesicles (18)—were also concentrated at the caps, which often became protrusions (Fig. 1, F and G). Additionally, when
the polarized Cdc42Q61L-expressing cells
were released from G1 arrest, buds were seen
to develop from polar caps (fig. S1E). Thus,
the sites of Cdc42Q61L accumulation were
true sites of polarized growth.
Next, we investigated whether polarization of MG-Cdc42Q61L occurred independently of any preexisting structural or
chemical asymmetry. First, staining with
calcofluor showed no correlation between
the localization of the MG-Cdc42Q61L polar caps and the bud scars, which normally
dictate the positions of the new buds (5)
(Fig. 2, A and B). Additionally, deletion of
the BUD1 gene, encoding a factor essential
for bud site selection (5), resulted in only a
small reduction in the fraction of cells that
formed MG-Cdc42Q61L polar caps (Table
1) (fig. S1F). Another potential spatial cue
might come from an asymmetric distribuTable 1. Polarization of GFP-Cdc42 in wild-type
and mutant strains.
Fig. 1. MG-Cdc42Q61L in G1 cells leads to cell polarization. (A) Distribution of MG-Cdc42Q61L after
1 hour of induction. Note that the image is overexposed compared to (B). (B) Formation of polar
MG-Cdc42Q61L caps after 3 hours of induction. V and N indicate vacuoles and the nucleus. (C)
Kinetics of MG-Cdc42Q61L polarization. Percentages of cells with one or two caps were plotted
against time of induction. (D) Time series showing the development of a polar cap (left cell, large
arrow) and stability of a polar cap (right cell, small arrow). Time in minutes is given on each frame
(time 0 ⫽ 90 min of induction). Linescans around the plasma membrane of the cell on the left are
shown on the right of each frame. (E) Chymographs around the periphery of the cells represented
in (D) in left-to-right order. Time is given in minutes. (F) MG-Cdc42Q61L localization and rhodamine-phalloidin staining of F-actin (arrow denotes an actin cable) in a cell after induction of
MG-Cdc42Q61L for 3 hours. (G) Sec4 staining in cells expressing MG-Cdc42Q61L. Scale bars, 3 ␮m.
1232
Strains and
conditions*
Cells with 1
cap (%)
Cells with 2
caps (%)
wt, 23°C†
wt, 30°C‡
wt, shift to 30°C§
⌬bud1, 30°C‡
wt, 30°C, Noc㛳
myo2-66, 23°C†
myo2-66, shift to
30°C‡
myo2-66, 30°C‡
sec3-2, 23°C†
sec3-2, shift to
30°C§
sec3-2, 30°C‡
⌬tpm2, tpm1-2,
23°C†
⌬tpm2, tpm1-2,
shift to 30°C§
⌬tpm2, tpm1-2,
30°C‡
sec6-4, 23°C†
sec6-4, shift to
30°C§
sec6-4, 30°C‡
53.3 ⫾ 3.4
71.2 ⫾ 1.0
52.0 ⫾ 1.4
55.0 ⫾ 0.8
73.6 ⫾ 2.4
62.0 ⫾ 12.0
2.0 ⫾ 1.6
34.0 ⫾ 3.3
6.6 ⫾ 2.5
32.2 ⫾ 2.9
8.3 ⫾ 2.6
3.0 ⫾ 0.8
4.7 ⫾ 2.9
0
0
55.0 ⫾ 6.2
9.3 ⫾ 2.5
0
0.3 ⫾ 0.5
0
0
58.6 ⫾ 5.3
0
0
2.7 ⫾ 1.7
0
4.0 ⫾ 2.2
0
70.3 ⫾ 4.2
11.0 ⫾ 3.3
7.0 ⫾ 1.6
1.0 ⫾ 0.8
0
0
*All strains are in the ⌬cln1,2,3, MET3-CLN2, GAL-MGCdc42Q61L background, which is designated as wild type
(wt).
†Cells were arrested for 4 hours and then
induced for 5 hours.
‡Cells were arrested for 3 hours
and then induced for 4 hours.
§After 4 hours of
induction at 23°C cells were shifted to 30°C for 30
min.
㛳Cells were treated with 10 ␮M nocodazole
during induction of MG-Cdc42Q61L.
21 FEBRUARY 2003 VOL 299 SCIENCE www.sciencemag.org
REPORTS
tion of cytoplasmic microtubules. However, treatment with the microtubule-destabilizing drug nocodazole had no effect on the
formation of MG-Cdc42Q61L caps (Table 1)
(fig. S1G). A third possible spatial cue
could come from the geometry of the cell.
To test this possibility, we induced MGCdc42Q61L cap formation in shmoo-shaped
cells that were the result of a prior incubation with ␣-factor (a mating pheromone). In
77.0 ⫾ 2.9% of the cells (n ⫽ 3 experiments with ⱖ100 cells each), MGCdc42Q61L caps did not overlap with, and
were distributed randomly relative to, the
original shmoo tip (Fig. 2C). Finally, we
found that phosphatidylinositol 4,5bisphosphate [PI(4,5)P2] and sterol-rich
lipids—implicated in actin assembly (19)
and cell polarization (20), respectively—
were uniformly distributed in the plasma
membrane of G1-arrested cells (Fig. 2, D
and E). Thus, the MG-Cdc42Q61L–induced
cell polarization was likely to reflect an
underlying self-organizing process that was
not directed by preexisting spatial cues.
Because actin filaments accumulated at
polar MG-Cdc42Q61L caps, we tested whether F-actin was required for polarization of
MG-Cdc42Q61L. When the actin polymerization inhibitor latrunculin A (LatA) was added
during induction of MG-Cdc42Q61L expression, cells completely failed to form MGCdc42Q61L caps (16). Furthermore, when
LatA was added after the polar caps had
formed, polarized distribution of MGCdc42Q61L was abolished in less than 15 min
(Fig. 2F). This effect was completely reversible after LatA washout (fig. S1H). Thus,
F-actin was required for both the establishment and maintenance of the Cdc42Q61L polar cap in G1-arrested cells.
Actin cables in yeast are known to have
a crucial role in membrane transport toward
the sites of polarized growth (21, 22) and
Fig. 2. Role of spatial cues
and membrane transport in
MG-Cdc42Q61L–induced polarization. (A) Localization of
actin and bud scars (stained
with calcofluor, CF) in a
G1-arrested cell expressing
MG-Cdc42Q61L. The arrow
shows the site of polarization. (B) Quantification of
the position of the MGCdc42Q61L cap relative to
the closest bud scar. Bud
scars were categorized as
“adjacent” if the closest scar
was less than 45° away
from the polar cap on either
side, as “opposite” if less
than 45° away from the
pole opposite of the polar
cap, or as “side” if not in one
of the above categories. (C)
MG-Cdc42Q61L cap formation in a cell with a shmoo
shape due to prior ␣factor treatment. Arrow,
shmoo tip; arrowhead, MGCdc42Q61L cap. (D and E)
G1-arrested cell (before
MG-Cdc42Q61L expression),
showing a uniform distribution of PI(4,5)P2 [(D), visualized with GFP-2xPH-PLC␦
(31)] and sterol-rich lipids
[(E), stained with filipin (20)],
respectively. (F) G1-arrested
cells were allowed to express MG-Cdc42Q61L for 3
hours and were then treated
with 100 ␮M LatA for 15
min. (G to J) MG-Cdc42Q61L
cap formation in various
temperature-sensitive yeast strains. Cells were induced for 3 hours at 23°C and then incubated for 1 hour at
23° or 30°C (restrictive or semi-permissive temperature). (G) myo2-66; (H) ⌬tpm2 tpm1-2; (I) sec3-2; (J)
sec6-4. Scale bars, 3 ␮m. (K) Subcellular fractionation of MG-Cdc42Q61L and Sec4. Low-speed supernatant
(S2) and pellet (P2) and high-speed supernatant (S3) and pellet (P3) from wild-type and sec6-4 cells were
obtained as described (13, 26). The P3 fraction from sec6-4 cells was further extracted with vesicle buffer
containing no salt (LS), 0.5 M KCl (HS), or 1% Triton X-100 (TX).
might be required for MG-Cdc42Q61L polar
cap formation. To test this hypothesis, we
examined the effects of mutations in the
secretory pathway. myo2-66, a temperature-sensitive mutation affecting a type V
myosin known to be responsible for the
transport of secretory vesicles along actin
cables (22–24), prevented the formation
and maintenance of the MG-Cdc42Q61L polar cap at the restrictive but not the permissive temperature (Fig. 2G and Table 1). A
similar defect was also observed in a temperature-sensitive tropomyosin mutant defective specifically in actin cable formation
(Fig. 2H and Table 1). Additionally, mutations (sec3-2 and sec6-4) in components
of the exocyst complex required for fusion
of secretory vesicles at the plasma membrane (25) also blocked MG-Cdc42Q61L
polar cap formation (Fig. 2, I and J, and
Table 1). Thus, actomyosin-directed vesicle transport and fusion were essential for
MG-Cdc42Q61L polarization.
To investigate whether Cdc42Q61L localized to secretory vesicles, we fractionated MG-Cdc42Q61L– expressing cells by differential centrifugation (26). We found that
MG-Cdc42Q61L behaved similarly to the
secretory vesicle marker Sec4 (Fig. 2K). In
wild-type extracts, both proteins were
present in the P2 fraction (containing plasma membrane) and to a lesser extent in P3
(containing secretory vesicles). In extracts
from a sec6-4 strain at 37°C, both proteins
were enriched in P3 (relative to the wild
type, amounts of P3/P2 in sec6-4 extracts
were increased by a factor of 3.33 in Sec4
and by a factor of 3.18 in MG-Cdc42Q61L).
Both MG-Cdc42Q61L and Sec4 could be
extracted from P3 by Triton X-100 but not
by 0.5 M KCl (Fig. 2K), which suggests
that the association was mediated through
lipid interactions.
Thus, Cdc42Q61L polarization involved
transport of GTP-bound Cdc42 along actin
cables. The assembly of the actin cables is
in turn dependent on the formin-like proteins, which can be activated by GTPbound Cdc42 present on the plasma
membrane (27–29). Newly synthesized
Cdc42Q61L was originally (shortly after induction) deposited randomly in the plasma
membrane (Fig. 3A, time 1). If, at a certain
location, actin nucleation happened to occur, leading to formation of actin cables
toward this site (Fig. 3A, time 2), new
Cdc42Q61L would be selectively recruited
to this site through membrane transport
along actin cables (Fig. 3A, times 2 and 3).
Deposition of Cdc42 would further stimulate actin nucleation at this site, leading to
the formation of a Cdc42Q61L cap.
To assess whether the above positive
feedback circuit is sufficient to induce cell
polarization, we performed mathematical
www.sciencemag.org SCIENCE VOL 299 21 FEBRUARY 2003
1233
REPORTS
simulations (13) that demonstrate the generation of Cdc42Q61L peaks, similar in
shape and number to those observed in
living cells, from either a random initial
distribution (Fig. 3B) or a completely uniform distribution of Cdc42Q61L (fig. S2A).
The observed evolution (Fig. 3C) (fig.
S2A) strongly resembled the observed kinetics of polarization after induction of
MG-Cdc42Q61L (Fig. 1C). Variation of the
initial concentration of plasma membrane–
bound Cdc42Q61L or the transport rate (c)
affected the number of caps formed per
cell: The smaller the value of c, or the more
Cdc42Q61L initially deposited on the plas-
ma membrane, the more caps were predicted to form (Fig. 3D). Furthermore, the amplitude of individual peaks became smaller
as more peaks formed per cell (Fig. 3B).
We experimentally tested one prediction
of the simulation by varying the initial
amount of Cdc42Q61L on the plasma membrane. Expression of MG-Cdc42Q61L was
induced for various amounts of time in
G1-arrested cells. The cells were then depolarized by brief treatment with LatA and
allowed to repolarize for 3 hours after LatA
washout. In this way, cells could begin
polarization with different initial concentrations of MG-Cdc42Q61L on the plasma
membrane. Because the level of MGCdc42Q61L plateaus after 3 hours of induction by galactose (fig. S1A), the final levels
of MG-Cdc42Q61L in each of the treatments
should be the same. Consistent with the
prediction, we observed an increase in the
fraction of cells with more than two
MG-Cdc42Q61L caps with increasing initial
concentrations of MG-Cdc42Q61L on the
plasma membrane (Fig. 3, E and F). Furthermore, in cells with multiple polar caps,
the size of the caps was smaller relative to
those in cells with fewer caps (Fig. 3G).
We have demonstrated a cytoskeletondependent mechanism that could account
for the intrinsic ability of cells to polarize
in response to Cdc42 activation. This
mechanism involves a positive feedback
loop between Cdc42-dependent actin polymerization and F-actin– dependent delivery
of Cdc42 to the plasma membrane. It remains to be determined to what extent such
an intrinsic polarization mechanism contributes under physiological conditions
where cell polarization is controlled by spatial signals. In neutrophils, the actin cytoskeleton plays an important role in the
amplification of the spatial signal provided
by gradients of chemoattractants (30).
Thus, the cytoskeleton-dependent positive
feedback loop could also be used as a powerful signal amplification mechanism that
amplifies a small initial asymmetry in the
distribution of polarity inducers, thereby
establishing the polarity axis toward a
physiologically relevant orientation.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
Fig. 3. Mathematical simulation of MG-Cdc42Q61L polarization. (A) Schematic representation
of the model for the establishment of Cdc42Q61L polar cap. A horizontal black bar represents
an activated section of plasma membrane. Light gray circles, Cdc42Q61L; dark gray vertical bars,
actin cables. Direction of transport is indicated by arrows. GW indicates the Gaussian
distribution. Time is in arbitrary units. (B) Representative examples of simulations showing
initial distribution of Cdc42Q61L and final distribution in cells that formed one or two caps
(initial Cdc42Q61L on the plasma membrane ⫽ 1%, c ⫽ 5). (C) Kinetics of cap formation with
low initial Cdc42Q61L concentration on the membrane. The evolution of 100 cells was
simulated (c ⫽ 5; initial percentage of Cdc42Q61L on the plasma membrane ⫽ 0.1%). (D)
Histograms showing the distribution of cells with 0, 1, and ⱖ2 caps simulated under different
conditions (c ⫽ 1, 5, and 10; fractions of Cdc42Q61L initially on the plasma membrane ⫽ 0.001,
0.01, and 0.1). (E) Formation of Cdc42Q61L caps with various amounts of initial Cdc42Q61L on
the plasma membrane. MG-Cdc42Q61L expression was induced in arrested cells for 0, 30, and
180 min. Cells were then treated with LatA for 15 min, washed, and allowed to repolarize for
3 hours. (F) Examples of cells that formed multiple Cdc42Q61L caps (arrowheads) after LatA
washout. Scale bar, 3 ␮m. (G) Quantification of Cdc42Q61L distribution in the cells shown in
(F) using linescans around the cell periphery. Dashed line denotes background.
1234
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
References and Notes
T. Misteli, J. Cell Biol. 155, 181 (2001).
T. D. Seeley, Biol. Bull. 202, 314 (2002).
W. J. Nelson, Science 258, 948 (1992).
M. Kirschner, J. Gerhart, T. Mitchison, Cell 100, 79
(2000).
J. Chant, I. Herskowitz, Cell 65, 1203 (1991).
R. A. Arkowitz, Trends Cell Biol. 9, 20 (1999).
D. Pruyne, A. Bretscher, J. Cell Sci. 113, 365 (2000).
M. P. Gulli, M. Peter, Genes Dev. 15, 365 (2001).
A. Nern, R. A. Arkowitz, J. Cell Biol. 148, 1115
(2000).
Y. Shimada, M. P. Gulli, M. Peter, Nature Cell Biol. 2,
117 (2000).
T. Lechler, G. A. Jonsdottir, S. K. Klee, D. Pellman, R. Li,
J. Cell Biol. 155, 261 (2001).
M. P. Gulli et al., Mol. Cell 6, 1155 (2000).
See supporting data on Science Online.
A. Amon, S. Irniger, K. Nasmyth, Cell 77, 1037 (1994).
M. Ziman, J. M. O’Brien, L. A. Ouellette, W. R. Church,
D. I. Johnson, Mol. Cell. Biol. 11, 3537 (1991).
R. Wedlich-Soldner, S. Altschuler, L. Wu, R. Li, data
not shown.
T. J. Richman, M. M. Sawyer, D. I. Johnson, Eukaryot.
Cell 1, 458 (2002).
P. Novick et al., Ciba Found. Symp. 176, 218 (1993).
T. Takenawa, T. Itoh, Biochim. Biophys. Acta 1533,
190 (2001).
M. Bagnat, K. Simons, Proc. Natl. Acad. Sci. U.S.A. 99,
14183 (2002).
D. Pruyne, A. Bretscher, J. Cell Sci. 113, 571 (2000).
D. W. Pruyne, D. H. Schott, A. Bretscher, J. Cell Biol.
143, 1931 (1998).
B. Govindan, R. Bowser, P. Novick, J. Cell Biol. 128,
1055 (1995).
21 FEBRUARY 2003 VOL 299 SCIENCE www.sciencemag.org
REPORTS
24. G. C. Johnston, J. A. Prendergast, R. A. Singer, J. Cell
Biol. 113, 539 (1991).
25. D. R. TerBush, T. Maurice, D. Roth, P. Novick, EMBO J.
15, 6483 (1996).
26. B. Goud, A. Salminen, N. C. Walworth, P. J. Novick,
Cell 53, 753 (1988).
27. M. Evangelista et al., Science 276, 118 (1997).
28. I. Sagot, A. A. Rodal, J. Moseley, B. L. Goode, D.
Pellman, Nature Cell Biol. 4, 626 (2002).
29. D. Pruyne et al., Science 297, 612 (2002).
30. F. Wang et al., Nature Cell Biol. 4, 513 (2002).
31. C. J. Stefan, A. Audhya, S. D. Emr, Mol. Biol. Cell 13,
542 (2002).
32. We thank P. Crews for latrunculin A; A. Amon, A.
Bretscher, S. Emr, C. Kaiser, and P. Novick for reagents
and yeast strains; J. Waters-Shuler and the Nikon
Imaging Center for help on confocal microscopy; and
M. D. Altschuler, J. Gunawardena, M. Kirschner, T.
Michison, A. Murray, E. Conlon, B. Ward, and S.
Wedlich for advice and comments on the manuscript.
Supported by an EMBO fellowship (R.W.S.) and by
NIH grant GM057063 (R.L.).
Separate Evolutionary Origins of
Teeth from Evidence in Fossil
Jawed Vertebrates
Moya Meredith Smith1* and Zerina Johanson2
Placoderms are extinct jawed fishes of the class Placodermi and are basal among
jawed vertebrates. It is generally thought that teeth are absent in placoderms
and that the phylogenetic origin of teeth occurred after the evolution of jaws.
However, we now report the presence of tooth rows in more derived placoderms, the arthrodires. New teeth are composed of gnathostome-type dentine
and develop at specific locations. Hence, it appears that these placoderm teeth
develop and are regulated as in other jawed vertebrates. Because tooth development occurs only in derived forms of placoderms, we suggest that teeth
evolved at least twice, through a mechanism of convergent evolution.
The origins and evolution of the dentition of
jawed vertebrates have come under recent scrutiny. New research has questioned the classic
concept of evolution of teeth by co-option of
external skin denticles at the margins of the jaws
and the obligatory evolution of “teeth with jaws”
(1). The current consensus view (Fig. 1A) is that
placoderms are phylogenetically placed as the
most basal group of jawed vertebrates (2–4) and
that teeth evolved after jaws, within jawed vertebrates. “Teeth,” those structures produced at
controlled locations from specific toothproducing tissues (in a dental lamina) (5), is a
shared derived character (synapomorphy) of
chondrichthyans, acanthodians, and osteichthyans (Fig. 1A). Given this accepted phylogenetic
position of placoderms, whether or not they have
real teeth is a crucial issue. Therefore, we examined placoderm dentitions for structural evidence
of tooth addition (e.g., pattern and organization
of dental elements) (1) that would allow us to
identify real teeth in placoderms such as occur in
all other jawed gnathostomes. Further, we used
this evidence to infer developmental processes
(1, 6, 7)—specifically, whether new teeth were
added in a pattern indicative of development
from a tooth-specific organ during growth of the
dental plates. The placoderm dentition is mor1
Craniofacial Development, Dental Institute, King’s
College London, Guy’s Campus, London Bridge, London SE1 9RT, UK. 2Palaeontology, Australian Museum,
6 College Street, Sydney, NSW 2010, Australia.
*To whom correspondence should be addressed. Email: [email protected]
phologically variable, with the Antiarchi and
Ptyctodontida lacking teeth completely (the dentition is currently unknown in the Petalichthyida)
(8), whereas in more basal taxa, dentitions comprise denticles in various arrangements (9). If
teeth originate within the placoderm phylogeny,
in more derived taxa (Fig. 1), then a dual, independent origin is indicated and teeth are not
homologous among jawed vertebrates.
Our observations derive from dentitions on
the upper and lower dental plates (supragnathals
and infragnathals) operating as dorsoventrally
occluding feeding structures in placoderms (Fig.
2, A to F) (fig. S1) (9). We noted that in certain
placoderm taxa new teeth arose on the jaws in an
Supporting Online Material
www.sciencemag.org/cgi/content/full/1080944/DC1
Materials and Methods
References
Figs. S1 and S2
Movie S1
28 November 2002; accepted 21 January 2003
Published online 30 January 2003;
10.1126/science.1080944
Include this information when citing this paper.
organized way, added to recognizable rows on
each dental plate (Fig. 2, B to F, arrows) (fig. S1,
A to E), indicating that tooth development was
patterned and regulated. We infer that successive
teeth (both older retained functional teeth and
new primary teeth) develop from tooth primordia, regulated in time and space by tooth-specific
tissues, as occur on a dental lamina (5).
We observed from precise, vertical sections
through the new teeth (Fig. 2F) (fig. S2A) that
each was composed of regular dentine formed
from cells within a pulp cavity (Fig. 2, H and
I) (fig. S2, B and C) (9), contrary to accepted
opinion (4, 10). These criteria of histology
and structural arrangement are consistent
with those for the presence of teeth in other
jawed vertebrates. We conclude that individual placoderm teeth were generated in a predictable, ordered pattern and were made of
regular tubular dentine (9).
The pattern of adding teeth from precise
positions on the jaw is a principal component of
tooth addition to statodont dentitions of other
jawed vertebrates, and is said to be diagnostic of
a dental lamina (5). Examples are found in lungfish radial tooth rows (Fig. 2G) (6, 7), chondrichthyan tooth families (1, 5), and symphyseal tooth
whorls in acanthodians and osteichthyans (1, 5).
In the placoderm dentition, new teeth are added
out of the bite, at one end of a preexisting tooth
row, and are regulated in time and space with
others of the set (Fig. 2, B to F). These features
apply to all gnathostome dentitions, including
the fossil group Acanthodii (5), and, we would
Fig. 1. Phylogenetic relationships of
jawed vertebrates and the Placodermi.
(A) Phylogenetic relationships of the
“jawed vertebrate” clade, with the
taxon Placodermi branching from the
basal node. The character “teeth,” as
mapped on the cladograms, is defined
as those conical structures composed
of regular dentine and added at the
end of a tooth row, only at specific
patterned loci in a sequential, regulated manner. (B) Phylogenetic relationships within the Placodermi, with the
Arthrodira representing the most derived taxon. We have observed
“teeth” in one of the basal arthrodire
groups, the Actinolepida, and also in
the Eubrachythoracida, although between there are several unresolved
arthrodire clades in which teeth appear to be absent.
www.sciencemag.org SCIENCE VOL 299 21 FEBRUARY 2003
1235