/. Embryol. exp. Morph. 83, Supplement, 261-287 (1984)
261
Printed in Great Britain © The Company of Biologists Limited 1984
A mathematically modelled cytogel cortex exhibits
periodic Ca++-modulated contraction cycles seen in
Physarum shuttle streaming
By GARRETT M. ODELL
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy,
New York, 12181, U.S.A.
TABLE OF CONTENTS
1. Introduction
2. Shuttle streaming in Physarum polycephalum
3. Very simple gene-coded cytogen properties yield very complex, globally coherent
mechanical behaviour
A. How can mathematics contribute to understanding developmental biology?
B. The stress in a strand of cytogel should depend both on its chemical state and
on how much it is stretched
C. The mechanochemical constitution of cytogel must be determined experimentally
D. Coupling trigger chemical concentration kinetics to mechanical deformation
leads to self-regulating cytogel
E. Understanding the pattern formation capabilities of the model cytogel requires
mathematical analysis
4. Two opposing cortices provide a stylized model of shuttle streaming in Physarum
polycephalum
A. A Physarum vein consists of a continuum of coupled contractile cortex segments. We simplify to two
B. Two soap bubbles connected by a pipe yield a simple metaphor containing
most crucial concepts
5. How and why the two-cortex model oscillates
A. Phase portrait display of numerical solutions exhibits the dynamic system's
behaviour
B. The saddle-shaped stress response surface is responsible for oscillatory
behaviour
C. If c stimulates contractions, how can maximal c correlate with minimal tension?
6. Gradual parameter variations cause abrupt qualitative changes of oscillation
behaviour
7. References
8. Appendix
SUMMARY
If each of many cells of an embryo (or different zones in a single cell) possess identical active
cytogel machinery, having the 'right' mechanochemical response properties, then the collective
262
G. M. ODELL
interaction among those identical participants leads automatically to the globally coherent
tissue deformations seen in embryogenesis, and to shuttle streaming in the plasmodial slime
mould Physarum polycephalum.
Biologically plausible, and experimentally verifiable hypotheses are proposed concerning
how the tension generated by a strand of cytogel is determined by the deformation it suffers
and by the concentration of a contraction trigger chemical, Ca2+, whose kinetics involve
coupling to mechanical strain. The consequences of these hypotheses, deduced by solving the
appropriate differential equation systems numerically, and displayed in computer-animated
films, closely imitate diverse tissue deformation events seen in developing embryos.
The same hypotheses on cytogel behaviour are used to model a thick-walled Physarum vein
segment, and two such segments are set up to be able to pump endoplasm back and forth
between them. Under certain conditions, this model exhibits spontaneous rhythmic
mechanochemical oscillations, many features of which correlate well with shuttle streaming
in Physarum.
Small gradual variations of parameters, presumably under genetic control, are shown to
cause abrupt and biologically interesting bifurcations of the qualitative behaviour of the
model.
I . INTRODUCTION
This paper attempts a mathematical explanation of how rhythmic contractions
arise and are coordinated in the cortex of veins in the acellular slime mould
Physarum polycephalum. Section 2 gives a brief portrait of the biological
phenomenon.
The mathematical model involved is highly simplified and stylized to make it
easy to understand. My aim is not to fit data quantitatively, but to illuminate
qualitatively how certain mechanochemical properties of contractile cytogel lead
automatically to sustained rhythmic contractions, thence shuttle streaming.
Section 3 is an overview; it attempts to explain the kind of contribution
mathematical modelling studies can make in understanding biological phenomena. While gyrations of Physarum polycephalum perse would seem to have little
to do with developmental biology, I believe the connection is tight, and intend
part of section 3 to defend this claim. This section contains mathematized
declarations of the fundamental hypotheses I make about the mechanochemical
behaviour of cytogel.
In section 4 I build an intentionally simplified mathematical model of a
Physarum polycephalum shuttle streaming system. This involves two thick cortical shells, each made of the contractile cytogel hypothesized in section 3, and
filled with viscous endoplasm which can be pumped between the cortices
(through a connecting tube) whenever one of them contracts more strongly than
its opponent.
Section 5 shows samples of the dynamical behaviour this model can exhibit and
attempts to explain how the oscillations come about. The model's behaviour is
compared to experimental observations. Sections 4 and 5 give intuitive explanations of how oscillatory behaviour arises inevitably and is coordinated in the
model system. Pictures are used instead of mathematical formulae. Mathematical details underlying sections 4 and 5 are given in an appendix. I assume that my
Mechanochemical oscillations in mathematically modelled cytogel
263
mathematical details will put biologically oriented readers into the same deep
coma that the display and explanation of photographic plates of SDS gel electrophoresis data puts me.
Caveat: Intuitive discussions, however compelling the prose may be, are
flimsy and should never be taken seriously unless rigorous mathematical
solutions have actually been computed. The kind of verbal reasoning given in
section 5 to explain why the system oscillates could be subverted to explain
instead why oscillations are impossible. The mathematical model presented in
this paper comprises a dynamical system, i.e. a system of differential equations
asserting that the time rate of change of the state of the system is a function of the
present state of the system. Once initial conditions are set, the future behaviour
of the system evolves with ballistic certainty. Put another way, the kind of
mathematical model presented in this paper has a (stubborn) mind of its own and
cannot be made to conform to prose forecasts of its behaviour or the most fervent
hopes of its inventors. This perverse feature, at least, dynamical system models
share with biological cells.
In section 6, we see how a gradual variation in one of the parameters of the
model (a diffusivity) brings about gradual quantitative changes in the details of
the oscillations. But then, as a certain threshold value is crossed, a dramatic
bifurcation occurs abruptly involving a qualitative change in behaviour. Regarding the parameter as under genetic control, this provides one example of a very
important concept in understanding how minor genotype variation can give rise
to sudden and major changes in phenotype performance.
2.
SHUTTLE STREAMING IN PHYSARUM
POLYCEPHALUM
The acellular plasmodial slime mould Physarum polycephalum consists of a
single huge multinucleate cell which can arrange itself in the form of a gigantic
flat 'puddle' of cytoplasm from which ramifies a branching manifold of veins (see
Wohlfarth-Botterman, 1977). The whole organism can extend throughout one
square metre! The veins are long tubes with thick cytogel (ectoplasm) walls
through which a viscoelastic fluid (endoplasm) flows back and forth. Endoplasm
and ectoplasm are interconvertible. Space limitations here preclude any but the
briefest sketch of the extensive literature on this organism. For this I defer to N.
Kamiya's summary in his recent review article (Kamiya, 1981), and record eight
items, numbered below, (selected from his 13) most relevant to the simplified
model in this paper. In this list, quotation marks surround Kamiya's summary
assertions while {} brackets enclose additional details paraphrased from the
same review.
[2.1]"The streaming is caused by a local difference in internal pressure of the
plasmodium. This pressure difference, or the motive force responsible for the
streaming, is measurable by the double-chamber method."
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G. M. ODELL
[2.2]"The internal pressure is brought about by the active contraction of the
ectoplasmic gel."
[2.3]"Contractile force of the ectoplasmic gel can be measured as a unidirectional force in a segment of plasmodial strand both under isometric and isotonic conditions." {The current highest stress measured is 180 dyne/cm 2 .}
[2.4]" Tension oscillation is augmented conspicuously by stretching or by loading.'1'' {A 10 % strain triggers dramatic contractile oscillations.}
[2.5]"The structural basis of contraction is the membrane-bound fibrils in the
ectoplasm." {and} "The fibrils consist mainly of bundles of F-actin and of
smaller amounts of myosin and regulatory proteins."
[2.6]"Tension development is regulated by Ca 2+ ." {Calcium sequestering
vacuoles permeate the plasmodium serving as a sarcoplasmic reticulum
analogue.}
[2.7]"Actin filaments undergo cyclic changes in their aggregation pattern
and/or G-F transformation in each contraction-relaxation cycle." {Maximal
elongation coincides with maximal parallelism of filament bundles.}
[2.8]"The aggregation pattern of actin filaments and concentrations of free
calcium and free ATP oscillate with the same period as cyclic tension generation. In addition, electric potential difference, pH immediately outside the
surface membranes... and heat production are also known to change with the
same period as shuttle streaming. How these physiological parameters are
causally related, and what the origin of the oscillation is, still are unknown."
{Minimal (ionophore induced) efflux from the outer boundary of the cortex
of free Ca2+ coincides with maximal tension production, and with maximal
efflux of ATP.} {See Kamiya, Yoshimoto & Matsumura, 1981, for the verdict
that electric potential depolarization plays little, if any, causal role in
coordinating contraction cycles.} {Yoshimoto & Kamiya (1978) showed experimentally that the endoplasm flow carries with it some factor which synchronizes the contraction-relaxation cycles at remote sites in the plasmodium,
but this factor does not affect the amplitude of the oscillations.}
3 . VERY SIMPLE GENE-CODED CYTOGEL PROPERTIES YIELD VERY COMPLEX, GLOBALLY COHERENT, MECHANICAL BEHAVIOUR
3A. How can mathematics contribute to understanding developmental biology?
This section begins with a short general essay on the role mathematical reasoning plays in the quest to understand developmental biology. The rest of the paper
gives a specific example of these general ideas.
'Genes infest embryonic cytoplasm and induce it to eat and manufacture itself
into a complex well-coordinated machine capable of squirting those genes into
the future, encapsulated in a packet of embryonic cytoplasm.' [This theme is
Mechanochemical oscillations in mathematically modelled cytogel
265
discussed at length in Dawkins, 1976.] The quest of developmental biology is to
understand the self organization process by which myriad identical proteins,
transcribed directly from genetic sequence code, interact with each other to ignite
an upward spiral of complexity amplification. This amplification process results
in the automatic assembly and behaviour of a machine [a cell, an organism, or
a social colony of organisms], a conventional blueprint for which would have to
contain orders of magnitude more (and very different) information than do the
genes.
Complexity amplification through collective interaction among a multitude of
nearly identical building block parts (proteins or cells) is responsible for amplifying linear amino acid sequence coding into the geometrical form specification
and the coordinated mechanical motion competence of the 'robot machines' that
genes trick cytoplasm to build.
Even very simple interaction rules iterated by many sub-units of an organism
(for example cells, proteins or organelles), can produce collective results of
astonishing complexity. Mathematics can be viewed as a language invented to
facilitate the precise expression of such rules, together with the logical deduction
of their consequences. In the quest to understand the dynamics of development,
mathematics as a language is much more potent than ordinary language.
The conceptual point of the mathematical work in this paper is to show that
if genes endow contractile cytogel, and the biochemical mechanisms controlling
it, with certain general, plausible, mechanochemical response properties, then
those properties, iterated by myriad interacting cells or domains of cytogel, lead
inexorably, without further genetic intervention, to complex globally coordinated patterns of tissue movement and behaviour.
Since essentially the same contractile protein molecules bring about diverse
biological events, this should be true, as well, of any worthy mathematical model
thereof. Indeed, the same mechanochemical response properties hypothesized
here to endow Physarum cytogel with its characteristic oscillatory behaviour can
also explain a variety of cell shape change phenomena crucial in embryogenesis
(see for example Oster, Odell & Alberch, 1980; Oster, Murray & Odell, 1984;
Segel, 1984, chapter 9). This builds a bridge to developmental biology - a fortunate one because the gigantic Physarum polycephalum can donate huge
strands of cytogel for the mechanical experimentation needed to measure and
characterize the general properties of the contractile cytogel, a topic to which we
now turn.
3B. The stress in a strand of cytogel should depend both on its chemical state and
on how much it is stretched
By 'cytogel' I mean the meshwork of protein filaments, including actin, myosin
aggregates, microtubules, intermediate filaments, various cross-linking proteins,
such as actin binding protein, and chemical regulation mechanisms that can
control the interaction of those proteins, all of which act in concert to endow
266
G. M. ODELL
cytoplasm with both passive viscoelastic mechanical strength as well as the capacity to contract actively.
What form should a description of the mechanical behaviour of such cytogel
take? One useful approach is to try to characterize only the simplest features of
its macroscopic rheology, and that will be the approach here. For a more detailed
exposition, see Oster & Odell (1984a and 19846). If we had a strand of cytogel
so large that we could attach hooks connected to force transducers to it and then
stretch it, we could measure the counter stress (force per unit area) that the
strand exerts as a function of imposed deformation (see item [2.3] above).
It is customary to use a dimensionless measure of deformation called strain,
which we denote by e, and define as follows. When a strand of cytogel, whose
initial relaxed length is Lo, is deformed to a new length L, then the strain
associated with this deformation is e= (L—Lo)/Lo. Thus, L = Lo(H-e) so that
e = 0 means no deformation has taken place, while e—> (—1) means L—»0, a limit
that never can be reached. We will use O to denote the static force in one
'contractile fibre', many of which constitute a cytogel strand.
The measurements just described would tell us a great deal about the mechanical constitution of a passive material. Cytogel is not passive, however, because
the chemical environment surrounding the contractile filaments determines the
degree of cross linking and the ability of the actomyosin machinery to generate
active contractile stresses.
The concentration of free calcium ions is known to play a significant role in
modulating the mechanical state of contractile protein machinery in general [see,
for example, Taylor, Hellewell, Virgin & Heiple (1979) and Taylor & Fechheimer (1982)], and, in particular, in Physarum (see item [2.6] above). Other
chemicals certainly play a role (cf. [2.8]), but for simplicity, I will consider only
a single 'trigger chemical' to be important, and denote its concentration by c. In
what follows, this single 'symbolic' trigger chemical, c, could, and for quantitative accuracy, must, be replaced by an n-vector of chemical concentrations.
The mathematical model would survive, but its intuitive understanding might
not.
Assuming that a single chemical's concentration, c, serves to modulate the
action of contractile cytogel, we could perform a sequence of experiments upon
a strand of cytogel, stretched between hooks, and immersed in a solution
medium which provides ATP fuel, etc, for the actomyosin components and in
which c is 'clamped' at a sequence of known concentrations. Measuring the force
exerted upon the hooks by the strand we could obtain the kind of data suggested
in Fig. 1, which shows how <E> varies as a function of c and e. The measured data
could as well resemble that in Fig. 2. If such data turn out to yield a saddle shaped
surface, as depicted in Fig. 1, then this endows cytogel with quite remarkable
abilities to form geometrically interesting patterns, and abilities to generate
sustained mechanochemical oscillations. This latter claim will be established in
the rest of this paper.
Mechanochemical oscillations in mathematically modelled cytogel
267
Fig. 1. Thisfiguredepicts the main hypothesis on the constitution of cytogel. Stress
in an isolated strand of cytogel, <&(e,c) is plotted on the vertical axis as a function of
trigger chemical concentration, c = Ca2+, and strain, e. The taller peak, at moderate
c and high strain, is supposed due to passive stiffening when cross-linked filaments
are stretched into parallel alignment, after which further extension is possible only
by breaking filaments or cross links. The lower peak at high c and negative strain is
supposed due to active, c-triggered, actomyosin, sliding filament force generation.
When the strand contracts (e<0) increased filament overlap results in tension amplification. This lower peak causes the saddle shape of the surface whose significance
is explained in the text. Regardless of e, O —> 0 as c becomes large because sufficiently high Ca2+ causes solation [see Taylor et al. (1979)]. The level curve is the stressfree <E> = 0 locus.
\
Fig. 2. An alternative version of Fig. 1, in which the effect of increased active tension generation does not result from strand contraction.
The level curve is the stress-free <J> = 0 locus.
W
D
o
p
ON
OO
Mechanochemical oscillations in mathematically modelled cytogel
269
Figs 1 and 2 summarize only a static relationship between stress, <£, c, and
strain. When cytogel deforms dynamically not only the amount of deformation
(the strain) but also the rate of deformation affects the stress. Thus the total
stress, o, will be:
{ | ^ } o
(3.1)
where /i is a viscosity coefficient, and No is the fibre density in the unstrained
state. O(e,c) is the function graphed in Fig. 1. In this paper, I assume, for
simplicity, that //(c) = jU= constant.
3C. The mechanochemical constitution of cytogel must be determined experimentally
Cytogel is so complex that only data obtained from measurement can certify
the 'correct' version of Fig. 1. In the absence of such data, we substitute in Fig.
1 the conjectured outcome of experiments yet to be performed. Its caption
explains some reasons for postulating the qualitative shape shown. My hope is
that the mathematical demonstration of consequences of the shape of this surface
on pattern formation will stimulate experimental biologists to make the crucial
measurements.
Indeed, a fully adequate experimental determination of the endoplasm in the
Physarum plasmodium has been made recently by Sato, Wong & Allen (1983),
showing it to be an inhomogeneous anisotropic viscoelastic fluid with a yield
stress. In section 4, I will grossly oversimplify this complex rheology and let a
simple Newtonian viscous fluid play the role of the endoplasm.
3D. Coupling trigger chemical concentration kinetics to mechanical deformation
leads to self-regulating cytogel
Among other things, Fig. 1 indicates that the force exerted by a strand of
cytogel held at a fixed length depends upon the c concentration. From item [2.8],
we know variations in c are synchronized with mechanical deformations. Items
[2.4] and [2.6] constitute a near proof that strain, stress, and c must be coupled
causally and not just correlated. In our model we couple the trigger chemistry
kinetics to the mechanics so that each affects the other. We do this by asserting
that;
— = ac - - r c + ye + A +
flux
(3.2)
Y
J
y
dt l + j3c2
'
The first term in (3.2) represents a c-stimulated autocatalytic release of trigger
chemical from vesicles into which it is resequestered at the rate re (cf. [2.6]). The
third term couples the control chemistry to mechanical deformation: increased
strain leads to increased production or release of c. This term will turn out to
yield the behaviour cited as [2.4] above. A represents a constant leakage from
EMB 83S
270
G. M. ODELL
vesicles. The last term, flux, represents leakage of trigger chemical from neighbouring domains. (3.2) is more thoroughly justified in Oster & Odell (1984a and
1984ft).
The meaning of (3.2) is most easily explained graphically by Fig. 3, which
shows how (3.2) can have one, two, or three equilibrium solutions depending
upon the value of [ye + X+flux]. Thus, in an assembly of many segments of
cytogel, mechanically linked, the contraction of some elements can, by stretching others, induce dramatic alterations of their trigger chemical concentrations.
Changes in the lengths and trigger chemical concentrations move each segment's
'operating state' around on the surface graphed in Fig. 1, and thus change the
contractile force generated by each segment. In this way, an ensemble of identical segments of cytogel, coupled together, can exhibit very complex dynamic
behaviour.
3E. Understanding the pattern formation capabilities of the model cytogel
requires mathematical analysis
Consider two identical cylinders of cytogel chained together and arranged so
the sum of their lengths is constrained to remain constant. Thus, one can contract
only if it stretches the other by an equal amount. Suppose that, initially, each
strand has the same length and the trigger chemical concentration in each is at
the upper stable equilibrium shown as c = Ch in Fig. 3B. Is this initial configuration stable to perturbations?
The answer depends upon the nature of the function (j>(e,c) graphed in Fig. 1,
in a small neighbourhood of the point e = 0 and c = Q,. If ®(£,c) is the kind of
saddle-shaped surface shown in Fig. 1, and this initial point happens to lie just
at the saddle point, or wherever d<&/de^ 0, then the initial configuration will be
unstable. When these circumstances obtain, if one of the two cytogel strands
contracts a bit, then, by becoming shorter, it gets stronger. Simultaneously, its
opponent becomes weaker by being stretched. Thus the strand that first shortened will out pull its opponent and shorten even more. The equilibrium is
unstable. The changed lengths of the two strands will induce changes in their
trigger chemical concentrations [via. equation (3.2)], and these changes will
influence the eventual outcome.
What would happen? Would the two opposing cytogel strands oscillate back
and forth forever, or would the system approach a different stable equilibrium,
possibly involving one short and one long strand? Either outcome is biologically
interesting. To emphasize the 'caveat' in the introduction, I will omit the answer
to this question. In section 4, we set up a much more complicated 'contest' between opposing cytogel cortices, then give actual solutions in sections 5 and 6.
Which outcome, if either, would happen depends on the exact details of the
function O(e,c) graphed in Fig. 1 and upon the right-hand side of equation (3.2).
When the details that characterize the cytogel's mechanochemical response
properties are under genetic control, we see how small quantitative changes in
Mechanochemical oscillations in mathematically modelled cytogel
271
rate constai ts can lead to large qualitative changes in behaviour. Prose forecasts
cannot resolve such subtleties, even for the trivial two-strand situation sketched
above; OL!\ mathematical analysis can.
ye + A • •
ye+A-*
Fig. 3. Equation (3.2) depicted graphically, dc/dt on the vertical axis is plotted as
a function of c. Supposing flux = 0, frame A shows the kinetics for a value of e
sufficiently negative (i.e. for a cytogel strand shorter than its original length) that
only one (low c) equilibrium value exists. It is stable. Frame B, in which e = 0, shows
the existence of three equilibrium values of c. The lowest, CL , and highest, q,, are
stable; the intermediate one is unstable. Frame C shows how stretching, until e is
sufficiently positive, eliminates all but the high Q, equilibrium.
272
4 . TWO
G. M. ODELL
OPPOSING CORTICES PROVIDE A
SHUTTLE STREAMING IN PHYSARUM
STYLIZED
MODEL
POLYCEPHALUM
OF
4A. A Physarum vein consists of a continuum of coupled contractile cortex segments. We simplify to two
We now try to make a caricature of a Physarum vein which we can schematize
as a long hollow tube whose thick walls are composed of ectoplasm (cytogel).
The tube is filled with endoplasm, a (very) viscous incompressible fluid which is,
in fact, ectoplasm in a solated state.
We will make the simplifying assumption, that interconversion between
endoplasm and ectoplasm does not occur. We attribute the observed shuttle
streaming of the endoplasm in the vein lumen to spatially heterogeneous
contraction-relaxation cycles in the cortex, and aim to show that a cytogel cortex
with the mechanochemical constitution described in section 3 automatically
initiates such cycles.
The next several paragraphs nominate some more severe simplifications as
virtues rather than flaws.
Even though I use only a single trigger chemical, c, in this study, there is of
course every possibility of a purely chemical oscillation occurring in Physarum
cytoplasm, involving cyclic AMP, calmodulin, and other enzymes, nearly independent of mechanical deformations. One consequence of this is a periodic
fluctuation in Ca+2. This fluctuation could then modulate the cortex contraction
activity (see, for example, Tyson, 1982). Item [2.4] above, however, suggests
strong coupling between the mechanical state and the control chemistry. A
major point of the present study is to show that control chemistry, incapable
alone of oscillating, can participate in globally coordinated oscillations when
cross coupled with the protein contraction machinery, which too cannot generate
oscillations by itself. Once this is established, it is obvious that additional control
chemical participants can greatly enrich the behavioural subtleties of the whole
system. This is especially true if the added chemical participants constitute a
chemical control system which oscillates autonomously, and can therefore 'beat
against' the mechanochemical oscillation established in this paper.
A more mechanically sophisticated approach than we will take here would be
to imagine slicing the vein tube, perpendicular to its length, into many serial
'ring' sections. We could model each ring as a circular strand of cytogel, and link
each ring to its immediate neighbours by cytogel strand tethers. This would give
us a finite-element model of a thick-walled tube. We could then use the
Navier-Stokes equations, characterizing the flow of incompressible viscous
fluid, to model endoplasm flow and equate the fluid and cytogel stresses on the
interior boundary of the lumen. This approach is now in progress, but no concrete results are available yet. It involves a large system of coupled non-linear
differential equations, in implicit form, which will require considerable time and
effort to solve. Even when solved, no intuitive insights are guaranteed.
Mechanochemical oscillations in mathematically modelled cytogel
273
To increase our chances of understanding the behaviour of our mathematical
model, we attempt a maximal reduction in its complexity commensurate with
retaining a stylized vestige of the main concept involved. That main concept is
the following:
If one zone of the vein cortex contracts, the endoplasm it displaces must
stretch some other cortex zone. If the stress in each cytogel zone is modulated
by the local trigger chemical concentration (calcium ion), and the trigger
chemical kinetics are coupled to cortex deformation, then this can lead to a
weakening of the contracted zone, a strengthening of the stretched zone, and
thence an automatic role reversal of contracted and stretched cortex zones.
This cycle can repeat or propagate...
To find whether this intuitively appealing idea has a chance to work, we
consider the caricature of a Physarum vein segment depicted in Fig. 4. Instead
of many finite elements, we focus attention upon just two. One can contract only
by oozing the endoplasm it contains through a (narrow) tube into its 'opponent'.
We assume that, at each instant, the trigger chemical has the same concentration
everywhere in the left cortex. Similarly for the right cortex. We assume that the
c-reaction specified by (3.2) occurs throughout each cortex and also throughout
the well-stirred endoplasm in the lumen of each cortex. For the endoplasm c
kinetics, we use e = 0 in (3.2). This decision strongly influences the dynamical
behaviour of the model, and may be an inappropriate choice. The chemical c can
diffuse between each cortex and the endoplasm it bounds. When endoplasm is
squeezed from one cortical chamber to the other, it convects trigger chemical
with it through the tube.
We assume that the volume in each cortical shell remains constant, so that the
cortical shell gets thicker as its radii decrease. Thus, while the cortex layer at each
flow resistance
Fig. 4. The two-cortex vein segment caricature. For k = 1,2, pk and Rk are inner and
outer radii of cytogel cortex k. Ck and ek are trigger chemical concentrations in cortex
k and in the endoplasm inside it, respectively. The connecting tube offers viscous
resistance, but cortical contractions can pump endoplasm back and forth through it.
274
G. M. ODELL
radius between pk and Rk experiences the same c-concentration, it suffers a
different strain, and thus, from Fig. 1, generates a different stress. We account
for this variation of stress with cortex radius by assuming that the circumferential
stress varies linearly from its computed value at radius Qk to its value at Rk.
Integrating circumferential stress divided by radius from pk to Rk gives the
hydrostatic pressure in the endoplasm within cortex k needed to balance the
cortical contraction. In this simplified model, we ignore radial stresses, [k ranges
from 1 to 2 in this paragraph.]
The mathematical version of these assumptions can be massaged into a system
of five non-linear autonomous differential equations. The five dependent variables are {Ri ,ci ,C2 ,ei ,ti}. From these, and the assumptions set in the above
paragraphs, R2 ,Q\ ,Qi can be computed algebraically. The equations of this
model are recorded (for completeness) in the Appendix. The parameters involved in the model can be found listed there.
Even for this highly simplified caricature of a Physarum vein segment, the
differential equations present formidable algebraic complexity. The reader who
exhumes them from the Appendix will find them altogether opaque. Their
derivation (which space limitations preclude) is straightforward but tedious.
Numerically computed solutions of these equations are presented and explained
in sections 5 and 6. Viewing many numerically computed solutions will not reveal
their conceptual explanation, however, just as viewingfilmsof Physarum shuttle
streaming does not generate an explanation for its cause.
In the next section we explore an intuitively transparent problem which incorporates almost all the essential physics of the one set in the Appendix and which
therefore provides a solid intuitive basis for understanding the dynamical
behaviour of the above problem.
B. Two soap bubbles connected by a pipe yield a simple metaphor containing
most crucial concepts
Consider two soap bubbles, each filled with air, connected by a tiny hollow
pipe through which the air can flow. If one bubble contracts, the air it forces
through the pipe must expand the other bubble. The soap film is the analogue
of the contractile cortex above, while the air serves as endoplasm.
Let Ri(t) and R2(t) denote the radii of the two bubbles, and suppose that, initially, at t = 0, the two radii have identical values, Ro. Since no air gets lost, the
sum of the volume of the two bubbles must remain constant and we must have:
(4*/3)[Ri (t) 3 + R2 (t)3] = 2(4*/3)Ro3
(4.1)
Let Pi and P2 denote the pressure inside bubbles 1 and 2, respectively, and let
Zi and £2 denote the surface tension coefficients for bubbles 1 and 2. 2 is the
analogue of <E>. As told in any physics text, we have:
Pi (t) = 22i (t)/Ri (t) and P2 (t) = 22 2 (t)/R 2 (t)
(4.2)
Mechanochemical oscillations in mathematically modelled cytogel
275
(4.2) is a crucial relationship. When the surface tension is constant, the
pressure inside the bubble increases when its radius decreases.
Assuming the connecting pipe offers sufficiently great viscous resistance to air
flow the inertial effects can be ignored, then whenever Pi ^ P2 air will ooze
through the pipe at the volume flow rate Q(P\-P2), where Q is an inverse
measure of the viscous resistance the pipe imposes upon air flow through it. This
volume flow rate is simultaneously the rate of volume shrinkage of bubble number 1 and the rate of volume expansion of bubble number 2. Thus:
-Q(Pi-P2)
(4.3)
When we carry out the differentiation, use (4.2) to eliminate Pi and P2, and
express R2 in terms of Ri using (4.1), then (4.3) becomes:
d p . - - Q [Si_
Z2
1
dt
2;rR, 2 LR l (2Ro3 - R, 3 ) 1 / 3 J
(44)
^ ' '
It is trivial to show that, if the surface tension coefficients, 2i and 22, are
constant and identical, then an initial configuration with equal-sized bubbles is
unstable. If any perturbation reduces R i , then Pi will always exceed P2 and
Ri(t) will shrink to zero, pushing all the air into bubble 2. Thus the initial
configuration is unstable to any perturbation. If you doubt the mathematical
reasoning, get two tubes, blow soap bubbles on one end of each, then quickly
stick the open tube ends together; one bubble always wins.
Suppose, however, that some surfactant chemical, c, in the bubble film acts to
change the surface tension coefficients, so that Z = 2(c,R). Suppose that the
surfactant concentration changes not only because stretching the film dilutes c
but also because the surfactant participates in a chemical reaction. Then, in (4.4)
we would substitute Zi = Z(ci ,Ri) and22 = 2(c2 ,R2)- We would have to solve,
simultaneously with (4.4), two more differential equations of the form
[analogous to two copies of (3.2)]:
^
^
R
O
, (2R O 3 -Ri 3 ) 1 / 3 )
(4.5)
(4.6)
Equations (4.4), (4.5), and (4.6) constitute an autonomous system of three
ordinary differential equations. If we specify the initial conditions and the functions 2(c,R) and H(c,R) then we could solve this system numerically to discover
its behaviour.
Surfactant effects can stabilize the initial equilibrium with equal-sized bubbles
if a decrease of one bubble's diameter leads to a sufficiently great increase in the
surfactant concentration and thence a sufficiently great reduction of the surface
276
G. M. ODELL
tension coefficient. Indeed, properly tuned functions 2(c,R) and H(c,R) can
lead to oscillations in which each bubble shrinks then expands, squeezing the air
back and forth through the pipe. All that is required is that 2(c,R) and H(c,R)
be such that (i) small departures of the bubble radii from initial equality leave 2
roughly constant, and (ii) large changes of radii cause 2 eventually to attenuate
greatly for the smaller radius bubble, while 2 eventually increases greatly for the
larger bubble. Then condition (i) ensures instability of the initial equilibrium,
while condition (ii) ensures that a bubble which gets blown up will eventually
blow up its opponent, and so o n . . .
5. HOW AND WHY THE TWO-CORTEX MODEL OSCILLATES
5A. Phase portrait display of numerical solutions exhibits the dynamical system's
behaviour
The fifth order differential equation [ODE] system recorded in the Appendix,
which makes the model described in section 4A mathematically precise, is so
algebraically complicated that only numerically computed solution techniques
can reveal its behaviour. For each set of initial conditions we can compute a
solution trajectory, namely five functions of time, t: (Ri(t),ci(t),C2(t),ei(t),
e2(t)}. We are interested in whether the ODE can exhibit stable oscillations that
is, whether one of those solutions is periodic and whether all solutions emanating
from initial points 'near' that periodic solution approach it as t increases. The
model system, indeed, has that property when its parameters are properly tuned
to lie within a certain (large) subset of parameter space. Fig. 6 shows three such
sets of stable periodic solutions, each set corresponding to a different value of the
cortex-to-endoplasm c diffusivity parameter, D m . I postpone until section 6 a
discussion of the bifurcation behaviour which occurs as Dm gradually varies.
In the special case when there is no diffusion of trigger chemical between cortex
and endoplasm (i.e. D m = 0) the fifth order system collapses to a third order
system [because ei and Q2 no longer matter] which is easy to analyse. Fig. 6A
shows one way of graphing the corresponding periodic solution. The best way to
view the different kinds of solutions the system has, however, is to construct a socalled phase portrait. Fig. 5 shows one, using stereoscopic views. This is built as
follows. Given a solution (Ri(t),ci(t),C2(t)}, we interpret the three functions as
specifying the cartesian coordinates of a point in three-dimensional space which
moves as t increases, thus tracing out a three-dimensional curve. The ensemble
of all such solution curves (trajectories) constitutes the phase portrait. Every
point in the three-dimensional phase space can serve as initial conditions for a
trajectory, so the phase space is filled densely with interwoven trajectories. We
draw enough of these curves to allow a viewer tofillin the rest by imagination. In
this sense, a phase portrait shows, in a single drawing, all the solutions of an ODE
system. For more thorough discussions of the concept and construction of phase
Mechanochemical oscillations in mathematically modelled cytogel
277
Fig. 5. A phase portrait for the Dm = 0 case, shown with stereo pair images. The
right frame is the right eye's view. There is a unique limit cycle. This limit cycle
appears again in Fig. 8, labelled A. The functions (Ri(t),ci(t),C2(t)} corresponding
to this limit cycle are shown in Fig. 6A. It is stable, but is not a global attractor.
Trajectories starting in the positive orthant flow either to the limit cycle or to the
'lower', small ci = C2, Ri = 1 stable critical point.
portraits designed for biologically oriented readers see Odell (1980) or Segel
(1984, Appendix 5).
A stable oscillatory solution corresponds to a closed orbit in the phase portrait
with the property that all nearby trajectories are 'attracted to it'. In Fig. 5 there
is a unique periodic orbit (the only closed curve in the picture); it is traversed in
the counter-clockwise direction. Fig. 6A shows (Ri(t),ci(t),C2(t)} corresponding
to this periodic solution. Physically, this means our model system can move so
each cortex alternately contracts and expands.
Not every solution eventually approaches the periodic orbit in Fig. 5. A substantial subset of the solution trajectories (i.e. a substantial set of initial points)
end up at a stable equilibrium point. The dashed straight line, defined by Ri = 1
and ci = C2, consists of four solution trajectories, and intersects three
equilibrium points marked with crosses. The 'lowest' of these is a stable
equilibrium point to which many solutions are attracted. The other two are
unstable. This means the two-cortex model has a stable equilibrium in which
both cortices have identical radii, and 'low' trigger chemical concentrations, and
that a fairly large perturbation is needed to 'kick' the system into oscillatory
behaviour once it approaches that stable equilibrium. This feature is biologically
interesting. The model Physarum is capable of remaining motionless and chemically quiescent until it suffers a fairly large-amplitude provocation, after which
278
G. M. ODELL
10-
o-o10-
6E
00Fig. 6. Three oscillatory solutions. Outer cortex radius, R, trigger chemical concentrations in cortex and endoplasm, C and E, and total force, F, per unit length of
cortex shell are graphed as functions of time. Different values of the cortex/
endoplasm diffusion parameter, Dm delineate the three cases. In case A, Dm = 0, in
case D, Dm = -00425, and Dm = -005 in case E. See text for full discussion of the
graphs. Using the notation in the Appendix, F is computed by a numerical
quadrature approximating:
it will commence oscillations (compare item [2.4]). On the other hand, certain
externally imposed perturbations can quench an oscillation in progress.
5B. The saddle-shaped stress response surface of Fig. 1 is responsible for oscillatory behaviour
Which qualitative features of the model system give rise to the oscillations?
Mechanochemical oscillations in mathematically modelled cytogel
279
The following assertions are drawn partially from numerical experimentation
and partially from intuition generated by the soap bubble metaphor of section
4B.
Fig. 7 shows constant <3> contours for the saddle-shaped function O(e,c)
graphed in Fig. 1, on top of which projections of the solution curves shown in Fig.
6 have been superimposed. Also shown are three equilibrium points for the
dynamical system in the Dm = 0 case. These correspond to the three equilibrium
values of c, when £ = 0, shown in Fig. 3B. Imagine all the parameters, which
control the shape of O(e,c) and the right-hand side of equation (3.2), to vary
C=l
Fig. 7. The curves without arrows are constant-O contours for the surface in Fig. 1.
The curves with arrows are projections of the periodic solution trajectories into the
(C,e) plane of these contours; the arrows show the direction in which the limit cycles
are traversed. Limit cycle cases A, D, and E, corresponding to those in Figs 6 and
8 are shown. The e value used to depict these periodic solutions is that corresponding
to the midline of the thick cortical shell.
280
G. M. ODELL
under genetic control. Then the oscillatory behaviour of the model can arise
whenever 3>(£,c) becomes saddle shaped and, simultaneously, one of the chemically stable equilibria at £ = 0 is 'close enough' to the saddle point in O(£,c).
When these coincidences occur, the equilibrium near the saddle point is unstable
because of the mechanical reasons discussed in the last paragraph of section 4B.
(The finite thicknesses of the cortices in our model complicate, but do not negate
this.) When, through mechanical instability, one cortex contracts, forcing its
opponent to expand, then the right-hand side of (3.2) for the contracted cortex
can become as graphed in Fig. 3A while the right-hand side of (3.2) for the
stretched cortex becomes as graphed in Fig. 3C. Whether these latter changes
actually occur hirtges upon the parameter values, so this prose has no value other
than to describe what actually happened when solutions were computed
mathematically.
When the c-kinetics in the stretched cortex are governed by the graph in Fig.
3C, c rises in that cortex toward the now unique high-c equilibrium. Due to the
saddle shape of O(£,c) and the assumed proximity of the equilibrium point to the
saddle point, this increase ofc in the stretched cortex can act to weaken the stress
it generates. This may be a part of the explanation for the experimental observation, cited in item [2.8], that maximum Ca2+ occurs simultaneously with
minimum total force.
The c-kinetics in the contracted cortex, governed by the graph in Fig. 3A,
cause a diminution of c in the contracted cortex, and thereby a dramatic reduction of tension in that contracted cortex (study Fig. 7 to see why). In the solutions
graphed in Fig. 6, the net tension force in the contracted cortex not only
diminishes. It even goes into compression and therefore sucks endoplasm back
from the interior of its inflated opponent. This is a much more important effect
than the stress changes in the stretched cortex in driving the oscillation. The net
result of the strain-induced modulation of the c-kinetics modelled in equation
(3.2) is to cause the contracted and stretched cortices to exchange roles forever.
5C. If c stimulates contractions, how can maximal c correlate with minimal tension?
Unfortunately, for the parameters chosen in this study, that biologically
measured counter-intuitive relationship does not emerge at first glance. But, we
take a closer look.
The total force generated in the cortex is graphed in Fig. 6. As each cortex
traverses minimal diameter (hence maximal thickness) this force exhibits a
strong maximum tension spike followed by a strong compression spike. The
tension spike occurs while the c concentration is falling, and is half way between
its minimum and maximum values. The compression spike occurs at minimal
c-concentration.
The oversimplified two-cortical-shell model treated here is not a valid
approximation of the experimental situation, reported by Kamiya (1981), which
Mechanochemical oscillations in mathematically modelled cytogel
281
2+
gave rise to the conclusion, cited in item [2.8] above, that the Ca and tension
oscillations are 180° out of phase. Nor would I describe the c(t) and F(t) functions graphed in Fig. 6 simply as being 180° out of phase. Nevertheless, notice
that in Fig. 6D and 6E the maxima in c-concentration coincide with local minima
in total contractile force. In Fig. 6D, at least, the maximum tension occurs very
close to the time of minimum c-concentration.
A. K. Harris (1971) discovered that fibroblasts exposed in culture to 10~5 Mvinblastine, lose their adhesivity to the substrate and exhibit dramatic shuttle
streaming. Vinblastine disrupts the microtubule struts in the fibroblast cytoskeleton, but leaves the actomyosin machinery intact. When treated, the
fibroblasts adopt the shape of a short string of sausages about three or four links
long, and commence periodic contractions in at least two links. This results in
one link squeezing most of its contents into an adjacent link which then contracts
reversing the flow. The period of this oscillation is about 16 seconds and many
cycles occur. The two cortex model explored in this paper can represent such
fibroblast shuttle streaming more faithfully than it represents Physarum streaming because the geometric simplicity of the model is quite close to what is observed in Harris's experiment.
6.
GRADUAL
PARAMETER VARIATIONS CAUSE ABRUPT
CHANGES OF OSCILLATORY BEHAVIOUR
QUALITATIVE
Variation of any of the parameters listed in the Appendix cause at least quantitative changes in the model's behaviour. That is, for example, the period of the
oscillation, the amplitudes of the R(t), g(t), ci(t), C2(t) functions will change. In
this paper, I vary only D m , which measures the rate of c diffusion between cortex
and endoplasm, while all the other parameters remain fixed. The D m = 0
behaviour is shown in Fig. 6A. When D m is increased to -00425, the oscillation
period doubles, and other quantitative changes occur as shown in Fig. 6D.
Changes in other parameters {which control the right-hand side of (3.2) and the
shape of O(e,c)} can change the amplitude of the oscillations. Such quantitative
tuning of the model's behaviour is useful for fitting experimental data (something
I have not here attempted), but is of little conceptual interest.
In contrast, certain infinitesimal alterations of parameter values induce striking qualitative changes in the model's behaviour. One such progression of
parameter tuning (not illustrated here) which gradually eliminates the saddle
shape of the O(e,c) surface can abruptly snuff out the existence of oscillatory
behaviour. So can diminution of a and/or y in (3.2).
A more complex change in the model's behaviour, due to gradual increase of
D m is shown in Figs 6,7, and 8. Namely, as D m increases past a threshold value
(between -00425 and -005), a single stable limit cycle (case D in Figs 6, 7, 8)
bifurcates into two distinct stable limit cycles (case E), and the period of each is
halved. In one of these, R(t) oscillates with small amplitude around a value near
282
G. M. ODELL
1, while c(t) never falls below 1, and the net force in the cortex remains low.
Refer to Fig. 6E. In the other limit cycle, R(t) oscillates with large amplitude
around 0-6, while c(t) never exceeds 1. That is, in correlation with item [2.8], the
cortex with the constantly high value of c generates constantly low tension. The
dynamic behaviour of the model in this case involves one cortex running around
each of the two possible limit cycles.
Fig. 8. Typicalfilmframe showingfivelimit cycles. See text for explanation.
Mechanochemical oscillations in mathematically modelled cytogel
283
Case E (in Figs 6, 7, and 8) corresponds to non-symmetric shuttle streaming
which has biological counterparts.
Numerical experimentation shows that gradual further increases in Dm eventually cause both of these limit cycles to be snuffed out abruptly (via a dual Hopf
bifurcation).
Fig. 8 shows, in a coordinate box with R i , ci, and C2 axes, a three-dimensional
section of the five-dimensional phase space of the differential equation model
specified in the Appendix. Within this box, three-dimensional projections of the
limit cycles corresponding to five different values of Dm , are shown. They are
labelled A, B, C, D, and E in increasing order of the value of D m . Fig. 8 is a single
frame from a computer-generated film which depicts the dumbell-shaped twocortex model on the floor of the box, oscillating back and forth as the phase point
[with coordinates (Ri ,ci ,C2 ,ei ,e2)] runs around one of the limit cycles. While all
this occurs, the movie camera circumnavigates the phase portrait box to exhibit
the three-dimensional topology of the limit cycles displayed within it.
The kind of abrupt bifurcations discussed above represent much more than
mathematically interesting curiosities. They are specific illustrations of what is
probably the main mechanism by which small quantitative changes in reaction
rate constants, force generation intensities, etc., wrought by subtle genotype
variation, are amplified into very great distinctions in phenotypic behaviour.
This is one facet of the complexity amplification spiral discussed in section 3A.
Much of the work reported herein was done in collaboration with George Oster. It was
supported equally by National Science Foundation Grant number MCS 83-01460, and a
Guggenheim Fellowship, both of which the author gratefully acknowledges. Karen Odell
kindly read and re-read the manuscript, finding many errors and suggesting many improvements;
REFERENCES
DAWKINS, R. (1976). The Selfish Gene, Oxford University Press.
HARRIS, A. K. (1971). The role of adhesion and the cytoskeleton
in fibroblast locomotion.
PhD diss - full ref to come in proof.
KAMIYA, N. (1981). Physical and chemical basis of cytoplasmic streaming, Annual Rev. Plant
Physiol. 32, 205-236.
KAMIYA, N., YOSHIMOTO, Y. & MATSUMURA, F. (1981). Physiological aspects of actomyosinbased cell motility. In International Cell Biology volumes 80-81, (ed. H. G. Schweiger), pp.
346-358. Berlin: Springer-Verlag.
ODELL, G. M. (1980). Qualitative theory of systems of ordinary differential equations, including phase plane analysis and the use of the Hopf bifurcation theorem, In Mathematical
Models in Molecular and Cellular Biology, (ed L.A. Segel), pp.649-727. Cambridge: Cambridge University Press.
ODELL, G. M., OSTER, G. F., BURNSIDE, B. & ALBERCH, P. (1981). The mechanical basis of
morphogenesis, I: Epithelial folding and invagination. Devi Biol. 85, 446-462.
OSTER, G. F., ODELL, G. M. & ALBERCH, P. (1980). Mechanics, morphogenesis and
evolution. In Lectures on Mathematics in the Life Sciences, Vol. 9, Some Mathematical
Questions in Biology, (ed. G. Oster), pp. 165-255. Providence, Rhode Island: American
Mathematical Society.
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G. & ODELL, G. (1983). The mechanochemistry of cytogels. In Fronts, Interfaces and
Patterns, (ed. A. Bishop). Amsterdam: North Holland, Elsevier Science Division.
OSTER, G. F. & ODELL, G. M. (1984a). A mechanochemical model for plasmodial oscillations
in Physarum. In Proc. Conf. on Pattern Formation (ed. W. Jager). Heidelberg: SpringerVerlag.
OSTER, G. F. & ODELL, G. M. (19846). The mechanics of cytogels I: Plasmodial oscillations
in Physarum, (submitted to Cell Motility).
OSTER, G. F., MURRAY, J. D. & ODELL, G. M. (1984). Patterns of microvilli and stereocilia
on hair cells of the inner ear, (submitted to Biophys. J.).
SATO, M., WONG, T. & ALLEN, R. D. (1983). Rheological properties of living cytoplasm:
endoplasm of Physarum plasmodium. /. Cell Biol 97, 1098-1097.
SEGEL, L. A. (1984). Modeling Dynamic Phenomena in Molecular and Cellular Biology.
Cambridge: Cambridge University Press.
TAYLOR, D. L. & FECHHEIMER, M. (1982). Cytoplasmic structure and contractility: the
Solation-contraction Hypothesis. Phil. Trans. R. Soc. Lond. B, 299, pp. 185-197.
OSTER,
TAYLOR, D. L., HELLEWELL, S. B., VIRGIN, H. W. & HEIPLE, J. (1979). The solation-
contraction coupling hypothesis of cell movements. In Cell Motility: Molecules and Organization, (ed. S. Hatano, H. Ishikawa & H. Sato). Tokyo: University of Tokyo Press.
TYSON, J. (1982). Periodic phenomena in Physarum. In Cell Biology of Physarum and
Didymiun, Vol. 1, (ed. H. Aldrich & J. Daniel), pp. 61-109. New York: Academic Press.
WOHLFARTH-BOTTERMAN, ?. (1977). Oscillating contractions in protoplasmic strands of
Physarum: simultaneous tensiometry of longitudinal and radial rhythms, periodicity
analysis and temperature dependence. J. exp. Biol. 67, 49-59.
YOSHIMOTO, Y. & KAMIYA, N. (1978). Studies of contraction rhythm of the plasmodial strand
III. Role of endoplasmic streaming in synchronization of local rhythms. Protoplasma 95,
111-121.
APPENDIX
The differential equations for the model involving two opposing thick cortical
shells
We first list the variables and parameters involved in the model in dimensional
form, using asterisk superscripts to denote dimensioned quantities; in the main
text these dimensioned quantities appeared without asterisks. Refer to Figure 4
for further clarification.
t*
Rk* (t*)
Ck* (t*)
ek* (t*)
Ro*
do*
No*
/i*
v*
D*
r*
=
=
=
=
time.
outer radius of cortical shell number k, k = 1,2.
concentration of trigger chemical in cortex k, k = 1,2.
concentration of trigger chemical in endoplasm within cortex k,
k = l,2.
= initial (relaxed) outer radius of each cortex.
= initial thickness of each cortical shell.
= cortex fibre density (contractile fibres/unit cross-sectional area),
= viscosity coefficient for cortical cytogel.
= kinematic viscosity of endoplasm fluid.
= diffusion coefficient of trigger chemical between cortex and
endoplasm.
= resequestration rate of trigger chemical.
Mechanochemical
oscillations in mathematically modelled cytogel
285
y*
= release rate of trigger chemical per unit strain.
A*
= leakage rate of trigger chemical from vesicles.
a*,(3* = parameters controlling autocatalytic trigger chemical release.
w* ,r*
= length and radius of the flow resistance tube between cortices.
<1>* (£,c*) = static stress per contractile fibre ("empirical" input from Figure
We now define dimensionless combinations of the various parameters and
dimensionless versions of the variables. From now on, quantities without asterisk superscripts are dimensionless.
As suggested by Fig..4, we want to account for non-spherical geometry of the
cortices. We therefore introduce a parameter, m, which allows the geometry of
the shell to vary from cylindrical, when m =2, to spherical when m = 3. Any
value of m between 2 and 3 is allowed.
=
t
=r*t*
Rk (t) = Rk* (t*)/Ro* =
=
Ck(t) = Ck*/?*1/2
=
ek(t) = ek*/?*1/2
dimensionless
dimensionless
dimensionless
dimensionless
time.
outer radii, k = 1,2.
cortical trig. chem. cone, k = 1,2.
endoplasm trig. chem. cone, k = 1,2.
—y f— = dimensionless inverse now resistance
Qm = - —
8 U ( m + l)J Lw*R*oVJ through pipe
D m = —:
_
= dimensionless cortex to endoplasm diffusivity
L m + 1 JRo*r*
a = a*/[i*p*i],
y = y* j8* */r*,
A = A* j8**/r*
Y(g,c) = O*(e,c*)/(r*ju*) = dimensionless version of function graphed in
Figure 1.
When we exploit algebraic constraints (constant volume of each cortex, and
constant total endoplasm volume), we find that the model described in section
4. A involves five dependent variables. I record the. equations of motion of this
fifth order autonomous system of ordinary differential equations in the following
form (in which underlining denotes vector quantities with five components):
—Y = F(Y), where Y is a vector of the five dependent variables.
That is Y =
Y2
Y3
Y4
Ri
C2
To be able to solve initial value problems numerically (or otherwise) we need
to be able to compute each of F's five components given any (valid) Y. For lack
of space, I will not derive the equations of motion for the model, but merely write
EMB 83S
286
G. M. ODELL
them down. The author will supply a written derivation on request [please send
a self-addressed stamped shipping crate].
The most difficult equation to derive is the one for Fi, which is analogous to
obtaining (4.4) from (4.3). We compute the pressure in the endoplasm contained
in cortex k as follows:
where £k(£) is the strain at dimensionless radius £in cortex k for k = 1,2. In this
formula, we approximate Y by assuming that it varies linearly between its values
at the integration limits. This simplifies the derivation tremendously, at the small
cost of introducing the natural logarithm function appearing in the formulae
below.
Let (5=1 - (1 - d ) m . Below U is just a (dummy) dimensionless variable, which
will usually stand for Ri or R.2.
We define the following dimensionless functions:
g(\J)
= [Um - d]1/"1 = inner radius of cortex with outer radius U
B(U) = l b(U) = - 1
0(e,c) = ac 2 /(l + c2) - c + ye + A
(I) computation of Fi =dRi/dt:
SetRi = Yi, R2 = [2-Ri m ] 1 / m .
Compute:
Q\ = g(Ri), and gi = g(R2), which are inner radii of the two cortices.
= Y(Ri - l,ci), and FR2 = Y(R2 - 1,C2), which are stresses on outer surfaces of the two cortices.
Fpi = Y(pi/[1 - d] - l,ci), and F p2 = Y ( ^ / [ l - d] - 1,C2), which are stresses
on the inner surfaces of the two cortices.
Finally, we have:
_ B(R 2 )FR2 + b(R 2 )r p2 - B ( R I ) F R I - b(Ri)F e i
F
1
R r " VQm + £(Ri) + (Ri/R2) m " * « R )
(II) computation of F2 = dci/dt and F3 = dc2/dt:
Compute:
F 2 = 0(Ri - 1, ci) + D m pi m " X (ei - ci)/6
F 3 = 0(R 2 - 1, c2) + DmQz"1-1 (e2 - c 2 )/6
Mechanochemical
oscillations in mathematically modelled cytogel
287
(HI) computation of F4 = dej/dt and F5 =dei/dt:
Endoplasm flow from one cortical chamber to the other convects trigger
chemical with it. This convective flux changes the c concentration only in the
chamber into which the endoplasm flows. We cope with this issue together with
the discontinuous jump between trigger chemical concentrations within the two
cortical chambers as follows:
If dRi/dt >0, then set J12 = 0, and J21 = mRi m - 1F1 (e2 —
otherwise,
If dRi/dt <0, then set J2i = 0, and J12 = m R r ^ F i (e2 Compute:
F 4 = 0(O,ei) + J21 + Dm(ci -
/
F 5 = 0(0,e2) + J12 + Dm(c2 The system of five differential equations we have just written down is so
algebraically complicated that only numerical solution methods deliver
solutions. The system is stiff. The solutions shown in sections 5 and 6 were
computed using a variable order (up to sixth) Adams-Moulton type predictorcorrector integration scheme which controls the integration step size automatically in order to keep local truncation errors smaller than specified (small)
bounds.