Chapter 3
Generalised chemotaxis: the
chemotactic response to multiple
chemical stimuli
The chemotactic response of a cell population to a single chemical species has been extensively studied from a theoretical basis. However, as mentioned in Chapter 1, there are
well documented cases in biology of cells responding chemotactically to more than one
chemical simultaneously. In this chapter we derive continuous partial dierential equation models for cell movement in response to multiple chemotactic cues. Our derivation
generalises the approach of Othmer and Stevens (1997), who have recently developed
a modelling framework for studying dierent chemotactic responses to a single chemical species. We demonstrate the eect of multiple chemical cues on the form of the
sensitivity response function.
3.1 Introduction
In Chapter 1 we highlighted the importance of chemotaxis as a cell signalling mechanism, and have given examples of the broad diversity of biological systems where
chemotactic response is a key factor in determining population movement.
One of the more widely studied systems of chemical signalling is the cellular response
of bacterial populations (e.g. E. coli and S. typhimurium). Bacteria swim by means
of agella, long whip-like protrusions that drive cells through a uid. Flagella can be
rotated either clockwise or anti-clockwise. Clockwise rotation of the agella causes them
45
to ay outwards, and subsequently the bacterium tumbles randomly. An anti-clockwise
rotation, however, draws the agella into a coherent bundle and the bacteria swim
uniformly. In the absence of chemical cues, movement is characterised by swimming
followed by tumbling. This results in a random walk type model. In the presence
of a chemotactic cue, tumbling occurs less frequently when the bacterium is moving
in a \favourable" direction and more frequently when moving in an \unfavourable"
direction. The overall movement of the bacterium is towards regions of high (low)
concentrations of the chemoattractant (chemorepellent).
The transduction process occurring between detection of a signal and the motor mechanism resulting in directed rotation of the agella has been thoroughly investigated.
Briey, binding of an attractant to a cell surface receptor initiates an intracellular signalling pathway which couples the chemotactic receptor to the agella motors. For
a more detailed account of the pathways involved, see Parkinson (1993) or the book
Molecular Biology of the Cell (Alberts et al. (1994)).
Bacteria are also known to respond to not one, but multiple chemical signals. This includes the attractant response to a number of nutrients, such as sugars and amino acids,
and the repellent response to a variety of noxious chemicals. Mathematical modelling
of the chemotactic response focuses primarily on the response to a single attractant
(or repellent) species. In the following section we shall consider a generalisation of this
approach which encompasses cell response to multiple cues.
3.1.1 Mathematical modelling of chemotactic response
Here, we concentrate on the recent work of Othmer and Stevens (1997), motivated by
the response of myxobacteria which are known to respond to a number of signals of
both a diusible and non-diusible nature. Using a master equation for a continuous
time, discrete space random walk process, Othmer and Stevens have derived continuum
descriptions to model cell motion on a one-dimensional domain under a variety of forms
for the cellular response to the control (chemoattractant) species.
To model the chemotactic response to multiple chemical cues we adopt the approach
of Othmer and Stevens (1997) and derive a set of continuum models to describe cell
movement. We then analyse the model under a number of response laws.
46
3.2 Dynamics of movement
Othmer and Stevens (1997) have postulated a continuous time, discrete space random
walk, where walkers undergo instantaneous changes of position at random times. This
description of motion was labelled a space-jump process by Othmer et al. (1988).
The random walk model includes a transition rate (determining probability of a jump)
dependent on the concentration of a control species. In the following generalisation we
use the same master equation, but make the additional assumption that the transition
rate is dependent on the concentration of multiple chemical cues as opposed to a single
chemical cue.
Suppose that the probability pn(t) of a walker, initially at 0, is at n 2
evolves according to the equation
@pn
=
@t
Z at time t,
Tn+,1(U )pn,1 + Tn,+1 (U )pn+1 , (Tn+(U ) + Tn,(U ))pn:
(3.1)
Tn are the conditional probabilities per unit time of a 1{step jump to n1, and
(Tn+(U ) + Tn,(U )),1 is the mean waiting time at the nth site.
U represents density of the control species, and is given by
0 .
...
BB ..
BB u1n,1=2 u2n,1=2
u2n
U =B
BB u1n
B@ u1n+1=2 u2n+1=2
.
.
...
ukn,1=2
ukn
ukn+1=2
...
1
CC
CC
CC ;
CC
A
(3.2)
..
..
for k control species. We have dened the control species on a lattice of half step size,
i.e. ujn,1=2 is the density of species uj at position n , 1=2. Thus, Tn : U ! R:
In the majority of biological cases processes take place on a nite domain and so boundary conditions need to be imposed. To model zero ux conditions, we consider a nite
segment of the lattice (,N; N ) and extend p and U to be even functions about ,N
and N . This ensures zero ux in the continuum limit and subsequent conservation of
mass.
As in Othmer and Stevens (1997), we consider three dierent types of models depending
on how the transition rates respond to U . These can be classied as strictly local models,
barrier models and gradient models.
47
3.2.1 Strictly local models
Under the strictly local model, the probability of a jump is dependent only upon the
concentration of the chemical species at the site of the walker. This is biologically
applicable to systems where the cell has only a very local mechanism for sensing its
environment.
We denote by the vector un the densities of chemical species u1; : : : ; uk at site n. Assuming a local response, we can write the transition rates at site n as Tn(U ) = T (un):
Then, Equation (3.1) becomes
@pn
=
@t
T +(un,1)pn,1 + T ,(un+1 )pn+1 , (T +(un) + T ,(un))pn:
(3.3)
Since we have assumed transitional probabilities independent of lattice site and dependent only on local densities it follows that there is no spatial bias and we write
T , = T +( T ). Hence, Equation (3.3) simplies to
@pn
=
@t
T (un,1)pn,1 + T (un+1)pn+1 , 2T (un)pn:
(3.4)
Dening x = nh, we can expand the right hand side to second order in h as a function
of x to give
@2
@p
2
=
h 2 (T (u)p) + O(h4 ):
(3.5)
@t
@x
We assume a scaling in the transition rates, T (u) = T (u), such that the limit
lim
h! 0;!1
h2 = constant ( D)
(3.6)
exists. Thus, in the diusion limit, Equation (3.5) becomes
@p
@2
= D @x2 (T (u)p);
@t
(3.7)
for x 2 , an isotropic and homogeneous medium. The derivation of this equation in
higher dimensions is similarly straightforward.
Equation (3.7) is subject to initial and, when a nite domain is required, boundary conditions. Furthermore p should always be nonnegative. If we assume to be a bounded
domain and impose zero ux conditions at the boundaries, the following conservation
equation must be satised
Z
p(x; t)dx =
Z
48
p(x; 0)dx = P0 :
(3.8)
In one dimension, the particle ux is given by
@ (T (u)p)
:
J = ,D
@x
(3.9)
This can be written in the equivalent form:
J =
@p
,D(T (u)) @x
, Dp @ (T@x(u)) :
(3.10)
Thus,
k
X
@ (T (u))
@ T @uj
=
:
@x
j =1 @uj @x
(3.11)
We dene the chemotactic sensitivity to species i, i, by:
i =
,DTui :
(3.12)
To conform to standard notation we write the ux,
J =
k
X
@p
+ p i (u) @ui :
,D(Ti (u)) @x
@x
i=1
(3.13)
If i > 0, the chemotactic component of the ux for species i is of an attractant nature.
Thus cells move up chemical gradients. The global movement of the cell however, is
determined by the overall response to the multiple chemical signals.
At a rst glance, it is not intuitively obvious how aggregation can occur with the strictly
local model: cells do not detect a chemical gradient, so how do they `know' to aggregate
at specic points of the domain? The key to understanding this is that it is not the
gradient that causes the cell to move, but the variation in frequency of jump. With
@T
@ uj > 0, the probability of a jump is higher at high chemical concentrations than at
low chemical concentrations. Subsequently, aggregation of cells is observed at positions
of low chemical concentration.
A solution can be obtained for the nonhomogeneous steady state of Equation (3.7) as
follows. Integrating (3.7) w.r.t x leads to
@
(T (u)p) = K;
@x
and, from Equation (3.9), zero ux boundary conditions imply K = 0.
49
(3.14)
Integrating (3.14) w.r.t. x gives:
p=
c
(3.15)
T (u)
where c is a constant, obtained by integrating further w.r.t. x
Z L 1 !,1
c = P0
dx :
(3.16)
0 T (u)
We thus obtain the following solution for the non-constant steady state:
Z L 1 !,1
P0
p(x) =
(3.17)
T (u(x)) 0 T (u) dx :
From a mathematical viewpoint, one can categorise dierent types of behaviour depending on the form of the solution to Equation (3.17). In simple terms, we dene
Aggregation to occur if Equation (3.17) has a nonconstant bounded solution; Blow-up
is the evolution to a nonconstant and unbounded solution; Collapse is the reduction to
the uniform solution.
3.2.2 Barrier models
The second type of model considered in Othmer and Stevens (1997) is described as a
barrier model. In this scenario, the u-dependence of the transition rate at site n is at
n 1=2. We therefore write the transitional rates, Tn (U ) = T (un1=2 ). Equation (3.1)
becomes
@pn
=
@t
T (un+1=2)pn+1 + T (un,1=2 )pn,1
,(T (un+1=2 ) + T (un,1=2))pn:
(3.18)
Under scaling of the transition rates, T (u) = T (u), we obtain the following evolution
equation
@p
@
=
D
@t
@x
T
!
@p
(u) @x
:
(3.19)
@p
In this case, particle ux is given by J = ,DT (u) @x
which is equivalent to a diusive
ux process. Clearly aggregation is not possible in this model.
Alternatively, we suppose that the decision when to jump is independent of the decision
where to jump. Under this assumption, the mean waiting time across the domain is
constant, and we model this by dening renormalised transition rates, N , where
Nn+(U ) + Nn,(U ) = constant:
50
(3.20)
Without loss of generality, we take the constant as 1. The above equation is satised
by choosing the following forms for our new transition rates:
(3.21)
Nn(U ) = N (un,1=2 ; un+1=2 ) = T (u T ()u+nT1=2()u ) ;
n+1=2
n,1=2
and N : Rk Rk ! R.
Equation (3.21) demonstrates non-local dependence on the transition rates as follows:
N +(un,1=2 ; un,3=2 ) = T (u T ()u+n,T1=2()u ) ;
(3.22)
n,1=2
n,3=2
N ,(un+1=2 ; un+3=2 ) = T (u T ()u+n+1T=2()u ) :
(3.23)
n+3=2
n+1=2
The master equation is now given by:
@pn
=
@t
N +(un,1=2 ; un,3=2)pn,1 + N ,(un+1=2 ; un+3=2 )pn+1
,(N +(un+1=2 ; un,1=2) + N ,(un,1=2 ; un+1=2))pn:
(3.24)
From the denition of N , N +(x; y) = 1 ,N ,(y; x). We can dene N + = N and write
Equation (3.24) as:
@pn
= pn+1 , pn , N (un+3=2 ; un+1=2)pn+1 + N (un,1=2; un,3=2 )pn,1;
@t
= pn+1 , pn
, (pn+1 , pn)(N (un+3=2 ; un+1=2) , N (un+1=2; un+3=2 ))
+ (pn , pn,1)(N (un,3=2; un,1=2 ) , N (un,1=2 ; un,3=2))
, pn(N (un+3=2; un+1=2) , N (un+1=2 ; un+3=2))
, pn(N (un,3=2; un,1=2) , N (un,1=2 ; un,3=2))
, pn+1N (un+1=2; un+3=2 ) + pn,1N (un,3=2; un,1=2 ):
(3.25)
In the diusion limit, transition rates are evaluated at (x; x). Rescaling the renormalised
transition rates, N = N , and assuming
D = n!1
limh!0
equation (3.25) becomes:
0
h2
(3.26)
2 ;
1
k @uj
X
@ @p
@p
A:
=
D @ , 2p
N
1j , N2j
@t
@x @x
@x
j =1
51
(3.27)
Here N1j represents the derivative with respect to the j th argument and N2j represents
derivative with respect to the (j + n)th argument, both evaluated at (un; un).
We evaluate N1j at (un; un) using Equation (3.21) to get
N1j = 4T 1(u) @ T@u(u) = 14 (ln T )uj :
j
Similarly one can show N2j = ,N1j .
Equation (3.27) therefore reads
@p
=
@t
@2p
D 2
@x
and particle ux is described by:
(3.28)
0 k
1
X
@ @
@u
, D @x
p (ln T )uj j A ;
@x
j =1
0
1
k
X
@u
@p
J = ,D + D @p (ln T )uj j A :
@x
@x
j =1
(3.29)
(3.30)
Therefore, for the renormalised barrier model, we have the following denition for
chemotactic sensitivity to species i:
i = D(ln T )ui :
(3.31)
Hence the chemotactic eect of species i will be positive if D(ln T )ui is positive.
3.2.3 Non-local models
The last type of model we consider here examines transition rates based on a gradient.
A number of dierent forms can be considered, and we look here at the simplest, with
nearest neighbour dierences taken into account.
We dene Tn1 : U ! R by,
Tn+,1 = + ( (un) , (un,1));
Tn,+1 = + ( (un) , (un+1));
(3.32)
(3.33)
and : Rk ! R with ( 0) and constants.
This gives us the master equation,
@pn
= (pn+1 , 2pn + pn,1) , ( (un+1 ) , (un ))(pn+1 + pn)
@t
, ( (un ) , (un,1))(pn + pn,1):
52
(3.34)
Assuming a scaling in the transition rates such that
D = h!lim
h2 ;
0;!1
we obtain in the diusion limit
(3.35)
!
k @
X
@2p
@
@uk
@p
=
D 2 , 2D
p
:
@t
@x
@x i=1 @uk @x
(3.36)
Thus, for the individual chemotactic sensitivities, we have
i = 2Dui
(3.37)
This last form is the most widely applied of the models and is of the same form as
the models initially developed by Patlak (1953a) and subsequently by Keller and Segel
(1970) to explain bacterial cell population movement.
3.3 Chemotactic responses under dierent transition rates
The local, barrier and gradient models considered above describe dierent cellular responses to chemical gradients. We now consider dierent forms of the transition rates
and compare subsequent response laws in the dierent models.
3.3.1 Chemotactic sensitivity laws
The equation for a cell population responding to a single chemical cue, under a gradient
model, is given by:
!
@2p @
@u
@p
= D @x2 , @x p(u) @x ;
(3.38)
@t
where u is the chemoattractant and (u) the chemotactic sensitivity. A variety of
forms of have been considered. These include a constant law obtained by setting the
transitional response proportional to u (e.g. Keller and Segel (1970)) and the logistic
law, (u) = 1=u (Keller and Segel (1971)). The latter is a phenomenological law to
feature decreased chemotactic response at higher chemical concentrations.
A more realistic law can be derived by considering signal binding of the chemical to cell
surface receptors. We assume binding of a single molecule of the attractant switches a
53
receptor from an inactive form (Ri ) to an active form (Ra ):
+
k
Ri + u *
), Ra :
(3.39)
k
This is then relayed to the movement machinery. Assuming binding equilibrates on a
faster time scale than the subsequent transduction and movement response, we determine the number of activated receptors by Ra = R0 u=(K + u), where K = k,=k+ and
R0 = Ra + Ri is the total number of cell surface receptors. We further assume that
the transitional response is proportional to the total number of occupied receptors and
write = R0 u=(K + u) in Equations (3.32) and (3.33). Equation (3.37) then gives the
following chemotactic response:
0
(3.40)
=
(K + u)2
where 0 = 2DR0K . The above form of response is commonly called the receptor
law and was rst derived by Segel (1977). More complex forms have been derived
to describe the chemotactic sensitivity response, and we refer the reader to Ford and
Lauenburger (1991) for further details.
3.3.2 Sensitivities in the generalised chemotaxis equations
Example 1: Linear, T = a + b u:
The transitional rate in this case is determined by a linear combination of the chemical concentrations at a site. Under the local, barrier and gradient models we obtain,
respectively, the following terms for the chemotactic sensitivities,
j = ,Dbj ;
j = D a+bbj u ;
j = 2Dbj :
(3.41)
This illustrates dierences in behaviour between the three models. The local model
results in aggregation at low chemical concentrations, whereas the barrier and gradient
models display aggregation at high concentrations. The aggregation of the local model is
through the increased frequency of a jump at higher chemical concentration levels. The
form of the velocities of movement for the barrier and gradient model is dierent. The
barrier model has a chemotactic velocity depending on the total chemical concentration,
whereas the gradient model has constant sensitivity terms.
54
Example 2: Receptor based response laws
We considered above a function for the transition rate derived from the total number of occupied receptors on the cell surface. We now examine the eect of multiple
chemotactic cues on the transition rate.
We can consider a variety of interactions at the local level, however we concentrate here
on two fairly simple cases. In the rst case we assume the chemical species occupy the
same receptor whereas in the second case we assume the chemicals occupy dierent
receptors. Experimental evidence suggests both cases apply: the cell surface contains
several types of chemotactic receptors, each receptive to a small group of chemicals.
We refer the reader to the excellent text, Molecular Biology of the Cell (Alberts et al.
(1994)), for fuller details.
In the latter case we consider two types of chemoattractant molecules, u and v, which
bind to two dierent receptor types, Ru and Rv . This is a straightforward extension of
the one species case, and we derive the following expressions for the number of activated
receptors, Rau and Rav :
Rau =
R0u u
;
Ku + u
Rav =
R0v v
:
Kv + v
(3.42)
R0u and R0v are the total number of receptors of type Ru and Rv , respectively. Ku =
k1, =k1+ and Kv = k2, =k2+ , where k1, ; k1+; k2, and k2+ are the rate constants due to rate
equations similar to that given by (3.39). It is simple to extend the above to n chem-
ical species, and we obtain the following expression for the total number of activated
receptors:
n R0 uj
X
X
uj
;
(3.43)
Rauj =
K +u
j =1
j
j
where Rauj is the number of activated receptors of type Ruj , R0uj is the total number of
receptors of type Ruj and Kj = kj,=kj+. Using the above expression for the transitional
rate, for the barrier model we have an individual chemotactic sensitivity response given
by:
DKi
i =
;
(3.44)
u (K + u )
whereas for the gradient model we have
i =
i
i
i
2DR0ui Ki
(Ki + ui)2 :
55
(3.45)
The above form is a generalisation of the receptor law mechanism for cellular response
to multiple chemical species.
When considering dierent molecules occupying the same type of chemotactic receptor,
we can suppose an inactive receptor Ri is activated by binding to one of two molecules,
u or v , into the activated state Rau or Rav ,
k+
Ri + u *
)1, Rau ;
k1
k+
Ri + v *
)2, Rav :
k2
(3.46)
(3.47)
We take the total number of receptors to be constant, R0 = Rau + Rav + Ri. Assuming
binding of either of the molecules occurs on the same time scale and faster than the
subsequent transduction and movement response, we have:
[R0 , (Rau + Rav )] uk1+ = k1,Rau ;
[R0 , (Rau + Rav )] vk2+ = k2,Rav :
(3.48)
(3.49)
Assuming the transitional response is proportional to the number of activated receptors,
Rau + Rav , we manipulate Equations (3.48) and (3.49) to get
R (k+ k, u + k+ k, v )
(3.50)
Rau + Rav = , ,0 1 2+ , 2 1+ , :
k1 k2 + (k1 k2 u + k2 k1 v )
We dene K1 = k1,=k1+, K2 = k2, =k2+, to obtain:
R0 (K2 u + K1 v )
Rau + Rav =
:
(3.51)
K1 K2 + K2 u + K1 v
More generally, for the case of n species, u1; : : : ; un, we dene Raui to be the number of
receptors activated by chemical species ui and R0 to be the total number of receptors.
Following the above procedure we derive
P
n
X
R0 [ ni=1 ui Kp =Ki ]
;
(3.52)
Raui =
P
Kp + [ ni=1 uiKp =Ki ]
i=1
where Kj = kj,=kj+ and Kp = Qnj=1 Kj .
Using the above expression for the transition rate and assuming the gradient description
of response, we obtain the following expression for the chemotactic response to the j th
species:
2DKpKj
:
(3.53)
j =
(Kp + [Pni=1 uiKp=Ki])2
56
The only variation in the sensitivity for each individual species results from the dierent
values of Kj . Therefore, the total chemotactic ux is
X
@u
2Dp
K p Kj j :
(3.54)
P
2
n
@x
(Kp + [ i=1 uiKp=Ki]) j
The global direction of movement is determined by a linear function of the individual
chemical gradients.
Example 3: Interacting sensitivity laws
So far we have considered transitional rates as a linear combination of the total number
of activated receptors. We now consider the transitional rate to be proportional to the
product of the number of activated receptors of type Ru and Rv from Equation (3.42).
Hence,
T = (K R+0uuR)(0Kv uv+ v) :
(3.55)
u
v
Chemotactic sensitivity responses for the gradient model are therefore given by
2DKuR0u R0v v ; = 2DKv R0v R0u u :
u =
(3.56)
v
(Ku + u)2(Kv + v)
(Kv + v)2(Ku + u)
Notice that the above implies that for low v the chemotactic sensitivity response to
species u is much weaker than the response at high concentrations of v. Thus the
sensitivity response to a specic chemical concentration gradient is dependent on the
concentrations of both chemicals at that site.
To demonstrate further the diversity of sensitivity responses for the multiple species
case, we hypothesise the following situation for transduction of the signal. If we suppose
that activation of a cell surface receptor is dependent on the binding of two chemicals,
u and v , then we obtain the following equation for binding,
k+
Ri + u + v *
), Ra :
k
(3.57)
Once more we assume a fast time scale for the binding process and obtain the following
expression for the number of activated receptors,
Ra =
R0 uv
;
K + uv
57
(3.58)
where K is the ratio k,=k+. Thus, with the gradient model, we obtain the following
form for the chemotactic sensitivity response to species u,
u =
R0 Kv
;
(K + uv)2
(3.59)
and the equivalent equation for the chemotactic sensitivity to species v. Once again,
this demonstrates how sensitivity can depend on both chemical species.
With the barrier model we obtain the following form for the chemotactic response,
u =
K
u(K + uv )
:
(3.60)
Assuming K large with respect to the product of chemical concentrations, uv, this can
be approximated by a logistic-type sensitivity law at lower chemical concentrations.
3.4 Discussion
Othmer and Stevens (1997) have proposed a model for the motion of a bacterium
in the presence of a single chemoattractant. The model, based on the random walk
description of motion, gives rise to a continuum description based on local rules of the
population response to the chemoattractant. In particular, the commonly used forms
for chemotaxis in the literature have been shown to be special cases for specic local
rules.
In this chapter we have reviewed their work and generalised it to include the population response to multiple chemical signals. Of particular interest here, is the eect of
multiple chemotactic signals on the form of the chemotactic response laws. In Section
3.3 we have begun to explore the type of functions that can be derived for these sensitivities. Clearly, by considering only a few ways under which multiple cues can interact,
a wide range of forms can be derived, including generalised versions of many of the
standard forms used in the literature. The laws have been derived with two motives:
biological and illustrative. Consider, for example, the sensitivity response laws given
by Equations (3.45), (3.53) and (3.56). The former two have both been derived with
attention to biological details. The latter, however, has been derived with less attention to the biology. The purpose of this law has been to demonstrate how consideration
of multiple response cues can lead to a greater degree of complexity in the sensitivity
58
laws. Furthermore, the assumptions made in derivation of this third law are by no
means biologically unrealistic.
The long time behaviour of solutions for the continuous equations in response to a
single non-diusible control (or chemical) species has been examined by Othmer and
Stevens under dierent forms for growth of the control species and chemotactic sensitivity terms. In particular the above authors have begun to explore the conditions under
which equations of this form may lead to the dierent types of behaviour, aggregation,
blow up and collapse. Furthermore, Levine and Sleeman (1997) have derived a heuristic approach to obtaining analytical results for the dierent types of behaviour. The
development of singularities in chemotactic systems where the control species diuses
has previously been examined by a number of authors (see, for example, Childress and
Percus (1980), Jager and Luckhaus (1992)). Rascle and Ziti (1995) have studied these
aspects for a case where the control species does not diuse.
As would be expected, all three forms of behaviour (aggregation, blow-up and collapse)
can be obtained when the cell population is responding to multiple chemotactic cues.
However, while the type of behaviour that occurs appear to be the same, the form
of the solutions can be quite dierent. In the following chapter we consider this in
more detail. More specically, we examine the form of the cell density pattern at
the heterogeneous steady state solution when responding to more than one underlying
(and time independent) chemical eld. Under the dierent forms of the chemotactic
sensitivity derived in Section 2.3 we demonstrate dierent forms of patterning for the
cell density.
59