Biologicai Journal of the Linnean Society (1990), 39: 125-134. With 4 figures
The epigenetic landscape and evolution
P. T. SAUNDERS
Department of Mathematics, King’s College, Strand, London WC2R 2LS
Waddington’s epigenetic landscape illustrates such characteristic features of development as
homeorhesis and the existence of alternative developmental pathways. Simply recognizing that these
are typical allows us to make inferences about evolution, for example that macroevolution is often a
different process from microevolution. We can account for the origin of these properties by assuming
that many processes in development can be modelled by non-linear differential equations. The
assumption then leads to two further predictions: that multiple speciation may be relatively common
and that phenocopying is likely to occur in one direction only.
KEY WORDS:-Evolution
equations - non-linearity.
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speciation - development
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epigenesis - phenocopies - differential
CONTENTS
Introduction . . . . . . .
The epigenetic landscape and evolution
Speciation and multiple speciation . .
Phenocopies . . . . . . .
References
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INTRODUCTION
The epigenetic landscape was devised almost fifty years ago by the
embryologist C. H. Waddington (1940). He had noticed that certain features
were characteristic of many developmental, or as he put it, epigenetic systems,
and he illustrated these by means of a picture like Fig. 1. The development
system is portrayed as a mountainous terrain, with valleys standing for possible
developmental pathways. The shape of the landscape is determined by a
complex network of guy ropes underneath it, which represents the complicated
way in which the genes influence development. A ball rolls down the landscape,
and the path it follows stands for the actual development of the organism.
The simple model illustrates remarkably well some important properties of
developing systems. For example, embryos are capable of withstanding many
perturbations. They have to be, for otherwise they could not develop except in
ideal conditions. This stability is not quite the same as the ball in the bottom of a
cup that we are familiar with from elementary physics, because the embryo does
not return to the state it was in before the disturbance. Instead, it continues to
develop and eventually reaches more or less the state it would have achieved if it
had been left alone. Waddington gave this property the name homeorhesis.
0024-4066/90/020125+ 10 SOS.OO/O
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0 1990 The Linnean Society of London
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P. T. SAUNDERS
Figure 1. The epigenetic landscape. The butterfly symbolises the role of catastrophe theory. From
Saunders & Kubal (1989).
I t is easy to visualize homeorhesis on the epigenetic landscape. A ball rolling
down the bottom of a valley represents an organism developing normally, and a
perturbation is represented by the ball being pushed up the side of the valley. If
the disturbance is not too great, then the ball returns not to where it was but
further down the valley bottom.
What happens if the disturbance is large enough to take the ball out of the
valley? T h e most likely outcome is that it will reach a dead end, but it is also
possible that it will pass over a watershed and then continue down a different
valley just as stable as the original one. This illustrates a property which is
actually observed: sometimes when an embryo is perturbed it neither dies nor
returns to normal development but switches to an alternative pathway which
leads to the production of a viable, though significantly different organism.
The model also illustrates how a perturbation can have the same effect as a
mutation. A suitable genetic change, represented in the model by a change in
the position or tension of a guy rope, could alter the landscape just enough to
send the ball down the same neighbouring valley that could have been reached
by forcing it too far off the bottom of the original one. This represents the
phenomenon of phenocopying: if, for example, genetically normal Drosophila
embryos are immersed for a short time in ether vapour, some of them will
develop into adult flies that resemble the bithorax mutant.
THE EPIGENETIC LANDSCAPE AND EVOLUTION
From an evolutionary point of view, the most significant property of an
epigenetic system is almost certainly its stability against both environmental
THE EPIGENETIC LANDSCAPE AND EVOLUTION
127
perturbations and mutations. In the picture, not only are the valleys relatively
deep, but also the complexity of the network of guy ropes means that many
changes in individual ropes would not alter the landscape significantly. A slight
change in the topography of a valley would be unlikely to have any effect on the
final position of the ball, which would simply return to the unaltered valley floor
further down. The most that might happen would be that the end result would
be very slightly changed, either because the ball did not have time to return to
the precise trajectory or if the change in the landscape was very near the bottom,
where the valley floor has flattened out.
Major changes might occur by long sequences of small ones, if the landscape
slowly changed so as to deflect a valley gradually, but the complicated
interconnections make this difficult to achieve. If pulling a little on one rope
shifts the landscape in an appropriate direction at first, after a while the tensions
from the other ropes to which it is linked will stop it from continuing the same
effect. O n the other hand, a relatively small change in the landscape, if it occurs
near a bifurcation point where one valley splits into two, could divert the ball
down a different valley and so to a very different end position than before.
In evolutionary terms, the first possibility corresponds to microevolution, and
the second to macroevolution. The model suggests that these are generally
different, that large evolutionary changes are not usually the result of long
sequences of small ones. This is consistent with the experience of plant and
animal breeders that even artificial selection cannot continue bringing about
change indefinitely, that eventually one seems to run up against a limit and that
this is usually well before anything one could call macroevolution occurs.
Large changes are more likely to occur fairly rapidly, when a system is
diverted into a new developmental pathway. This implies that we should expect
to observe in evolution long periods of time in which only minor changes take
place. Occasionally, however, major changes could occur, often not related in
any obvious way to the sequence of minor ones. Thus the model of evolution
suggested by the epigenetic landscape is precisely that of punctuated equilibria
(Ho & Saunders, 1979).
Some Neo-Darwinists do allow the possibility of large phenotypic changes,
which they ascribe to mutations in ‘regulatory genes’, that is, genes that affect
several chemical reactions in the developmental process. There is, however,
nothing in this account to explain how the net effect of a number of random
changes will be to produce a viable organism, let alone one that will be in any
sense fitter than the normal one. It is simply an attempt to explain away
awkward observations, and certainly not a prediction of the theory.
The Waddingtonian view is crucially different. I t involves not a large random
change but a diversion to one of a limited number of alternative pathways. The
mutation that causes the major change need not affect more than one reaction;
what matters is that it diverts the process down a different pathway. That large
changes can occur is thus a definite prediction, not a vague possibility.
Of course it is not enough to refer to a sketch; we have to demonstrate that
alternative pathways exist and, if possible, explain how they originate. The first
is straightforward, because there is ample empirical evidence. For instance, we
know that if an embryo is disturbed during its development it can develop into a
viable but significantly altered organism. Moreover, the abnormal form can be
the same as can be caused by a mutation; these are the phenocopies mentioned
P. T. SAUNDERS
128
above. This is easy to explain if it is a matter of either the mutation or the
perturbation pushing the devlopment into a n already existing alternative
pathway, but not if we believe the mutation is in a regulatory gene which affects
many subsequent reactions. In the latter case, it is necessary to suppose that the
perturbation-ether vapour, heat or cold shock, or whatever-somehow affects
the same reactions in the same way.
We can also see why we would expect alternative pathways to exist, even
though we do not yet have models for most of the events that occur during
development. We need only suppose, as seems reasonable, that many of them
will be based on differential equations. Some of these will be non-linear, and it
turns out that this means they can have a number of interesting properties which
are quite different from what we have come to expect from our experience of the
linear equations with which we are more familiar.
A linear differential equation, or a set of them, typically has a unique
equilibrium point. If this equilibrium is stable (as it must be if the system it
represents is to persist and so be recognized as a system) the system will move to
the corresponding steady state and stop. If it is perturbed away from this state it
will return to it.
In contrast, non-linear differential equations often have multiple solutions,
and these can include kinds not found as solutions of linear equations. This is not
really surprising, since the same is true of algebraic equations. The linear
equation ax+ b = 0 has a unique real solution, x = - b/a, whereas the simplest
non-linear equation, ax2 bx c = 0, has two, which can be either real or
complex. I n the case of non-linear differential equations, not only can there be
several equilibrium points, there can also be stable trajectories, that is, sequences
of states. A system represented by non-linear differential equations need not
come to a halt but may move along one of the stable trajectories. If it is
perturbed, it will tend to return to the trajectory, though not back to the point at
which it was disturbed. If the system is forced beyond the range of stability of the
original trajectory, it may be drawn to another stable one instead. As this is
precisely the behaviour that Waddington illustrated in the epigenetic landscape,
we can now understand why the developmental system should have these
properties: any system which can be modelled by non-linear differential
equations is likely to possess them.
Not surprisingly, the converse of this statement is false: the epigenetic
landscape does not illustrate all the properties of non-linear differential
equations. For instance, Waddington drew the landscape as a two dimensional
surface because that is as much as can be shown in three dimensions. He
acknowledged that this was a simplification, but it actually makes much more
difference than he thought, because two dimensions is a very special case in
dynamics. This is because we are studying trajectories, which are onedimensional, and there are many things that are true only when the objects of
interest are of dimension only one less than the space they are in. For example, in
the landscape each valley has the only two neighbours, but in higher dimensions
a path can have many neighbours: imagine a multicore electric cable. Obviously
this greatly increases the number of things that can happen.
Our aim is to extend Waddington’s idea by investigating mathematical
systems that are non-linear or have more than two independent variables, or
both, seeing what typical properties they have beyond those that are obvious
+ +
THE EPIGENETIC LANDSCAPE AND EVOLUTION
129
from Waddington’s ingenious but necessarily over-simplified epigenetic
landscape, and then asking what the consequences are for development and
evolution.
SPECIATION AND MULTIPLE SPECIATION
Mathematicians have become quite interested in non-linear differential
equations recently, so much more is known about them than before. But we are
only just beginning the project of trying to work out the epigenetic and
evolutionary consequences, and besides, much of what we know about the
mathematics is not necessarily relevant to evolution. Consequently, we can give
only an indication of what we expect can be done. All the same, even at this
early stage it is possible to outline some concrete examples and we discuss two
here.
As mentioned above, non-linear equations typically have multiple solutions,
and this includes multiple trajectories. Now where there are multiple trajectories
we generally find also splitting of trajectories, which mathematicians call
bifurcations. These do not generally occur like a branch coming off a main
railway line (Fig. 2A) but rather as shown in Fig. 2B. The original trajectory is
still there, but it has become unstable and so no longer represents anything that
will be observed. Two new trajectories have appeared, one to either side of the
first.
Part of this is not hard to understand, because with trajectories, just as with
equilibrium points, stable and unstable generally alternate: between any two
valleys there is usually a watershed. What is less obvious is that i t is the old
trajectory that becomes unstable, when we might have expected a new stable/
unstable pair. O n the face of it, this suggests that speciation should occur not by
a branching but by the disappearance of the old species and its replacement by
two new- ones.
Figure 2. Bifurcations. The x-coordinate is some parameter which is increasing and the y-coordinate
measures the response of the system. Typical hehaviour is indicated not by A, hut by B.
P. T. SAUNDERS
I30
Now we have to be a bit careful in interpreting this biologically, because even
when we think in terms of equations rather than a picture we are simplifying the
situation considerably. We have not even identified the coordinates, so the state
corresponding to one of the new trajectories might actually be indistinguishable
from the original one. All the same, it is significant that splitting does typically
occur in this way, and at least it suggests that extinction and replacement by one
or more successors (it would be only one if one of the successors was unable to
survive, possibly in competition with its sibling) rather than the simple
appearance of a new species, is not especially surprising.
It is also possible to see how a number of new species might appear at about
the same time. For this, it is perhaps easiest to think of a simple physical
analogue. If you gradually increase the loading on a pillar, nothing much
happens for a long time. After a critical load is reached, however, the pillar
buckles, and the direction in which this happens depends upon some
imperfection in the pillar. If there were no defect at all, which is of course
impossible, it would remain in the now unstable vertical position until it was
disturbed. Sometimes the imperfection is obvious, as when a lumberjack cuts a
notch to ensure that a tree falls in the right direction, but often it is not, and then
it can be quite impossible to predict in which direction the pillar will buckle.
2
K
I
0.1
S
Figure 3. The bifurcation set for equation (2). Equilibrium is possible only at a low value of g in
region I and at a high value in region 111; in region I1 there are two stable equilibria. If K = 1, a
sudden jump to a high value of g occurs as S increases through 0.041 but there is no return to the
low value if S is decreased through that value.
THE EPIGENETIC LANDSCAPE AND EVOLUTION
131
Now imagine a large number of very similar pillars, each made without
obvious imperfections, and suppose each is given the same, gradually increasing
load. For a while, they would still all look much the same. Eventually, however,
and more or less simultaneously, they would all buckle. But they would not all
buckle in the same direction, because the imperfections that determine the
direction of buckling would be different. When the stability of a system breaks
down, previously unimportant differences can become crucial.
In the same way, the stability of the epigenetic system, that is, the canalization
which is necessary for normal development, means that individual organisms will
develop in the same way even though there are considerable genetic differences
among them. What is more, so long as evolution involves only minor changes,
canalization would mean that most individuals would change in much the same
way. For speciation to occur, the stability has to break down, and when that
happens these genetic differences may come into play and bring about the
almost simultaneous appearance of a number of new forms. This will not always
happen but we should not be surprised when it does.
The idea that genes that previously had no effect on the phenotype can later
come into play is not new, but here we are not envisaging silent genes being
turned on. This may indeed sometimes happen, but it is hard to see why it
should lead to multiple branching unless a number of such genes are turned on
at the same time. What we have in mind is that the genes were being expressed,
but that the canalization of the epigenetic system prevented them from affecting
the phenotype. When this stability breaks down, these previously unimportant
(but not unexpressed) genes have the opportunity to influence development,
especially since the relatively unstable system is susceptible even to small genetic
differences. Thus genes can become significant simultaneously without having
either to appear or first be expressed simultaneously.
PHENOCOPIES
The other result to be discussed here is also quite general, but to make the
argument easier to follow we shall work with one particular model. We omit
some of the mathematical details; for a full account see Saunders & Ho (1985).
The equation, which is taken from Lewis, Slack & Wolpert (1977), is supposed
to represent the production of a gene product g which is activated by a ‘signal
substance’, S, possibly calcium ions, or cANP. The rate of change of g is given by
-dg_ - S+-- g2
dt
K+g2
0.4g.
We suppose that everything happens slowly enough that we may take g to be at
very nearly its steady state value for the current value of S. This is a common
assumption in the study of reaction kinetics. Then for any given S we can find
the corresponding value of g by solving the equation dg/dt = 0, i.e.
(2)
Because this is a cubic equation, it has three roots. For some values of S and g all
three are real, but for others there is only one real root and a complex conjugate
pair. Figure 3 shows this: in the region marked I the only stable equilibrium is at
2g3-5g2(1 +S)+2Kg--5KS = 0.
132
P. T. SAUNDERS
low g, in the region marked I11 the only stable equilibrium is at high g, but in
region I1 both are possible. T h e boundary of region I1 is called the bifurcation
set because it is the set of all points at which a unique possibility splits into two.
Suppose we take K = 1 and set S initially zero. The system is then in region 11,
so there are two possible stable steady states, g=O and g = 2 , and an unstable
one, g=0.5, between them. If we set up the system with g=O, the system will be
a t equilibrium and nothing will happen. If we now increase S slowly, we alter
the steady states slowly. Consequently g will increase slowly, remaining all the
time close to equilibrium. When, however, S becomes greater than about 0.041,
the system crosses the bifurcation set and the the stable steady state with the
lower value of g disappears (by combining with the unstable equilibrium). T h e
system then moves rapidly to the high valued equilibrium, that is, there is an
abrupt increase in g.
Note that if S then decreases so that the system recrosses the bifurcation set,
the high equilibrium does not disappear. There is no reason for g to move back
to the lower equilibrium and so it remains a t the higher one; even if S returns to
zero, we still find g = 2. This makes the process robust because there is no need to
maintain the value of S once the high value of g has been attained.
To see how a system of this kind might evolve, consider the effect of a change
in the saturation constant K. Suppose that initially K is small. Then if S
increases to about 0.03, say, g will reach a high equilibrium value which, as
explained above, is stable even if S subsequently returns to zero. Now suppose
that in evolution, K increases, to about 1.4, say. Then for the same S, g will
stabilize at a low value. But if S is perturbed to a higher value, g will move to a
high value which it will remain at even if S subsequently falls back to 0.03, or
even zero. Finally, if K increases even further, to much over 2 say, then the
system is insensitive to a perturbation in S and will return to the low equilibrium
value of g (see Fig. 4).
What happens in the other direction? Suppose the original form is K > 2 ,
S = 0.03. Then if we reduce K to 1.4 there is no change; we still get the low value.
It can be perturbed to the high value but that is as before; we can still only
phenocopy in one direction. So the model predicts that if it is possible to produce
a phenocopy of a mutant organism from an individual which is genetically
normal, it will not be possible to produce a phenocopy of a normal organism
from a mutant.
If K changes gradually in evolution, so that it passes through a number of
intermediate values, then there will be a period during which it is possible to
phenocopy the old form (or the one that is about to appear). After a while this
ability will disappear or (in the latter case) the new form will appear as a mutant
and it will not be possible to phenocopy the old one. Note also that in the
intermediate range of K there are two separate mutations that will produce the
same phenotypic change: a decrease in K or an increase in the maximum value
attained by S. This increases the probability that this particular phenotypic
change will occur. By the same token, the instability of the normal form in this
range means that phenocopies should be relatively common.
I t must be stressed that this result applies not just to one equation but to a
very large class, viz. those representing the simplest mechanisms which can lead
to a sudden change from one equilibrium state to another, even in a very
complex system. This is why we can put it forward as a n explanation of
T H E EPIGENETIC LANDSCAPE AND EVOLUTION
133
S
Figure 4.The bifurcation set for equation (2). The line S = 0 . 3 is shown with three different values of
K indicated. The short paths with U-turns indicate how the highest and lowest values of K give
robust equilibria but the middle one does not. It is therefore in the intermediate range of values that
we would expect phenocopying to occur.
observations on real organisms such as Drosophila while not claiming that
equation (1) applies to any process in their development. We can obtain similar
results for the next simplest case as well; this has three stable equilibria and can
be used to model situations such as that in which a single undetermined state can
go to one of two alternative determined states.
It seems to be a general rule that where there are alternate pathways,
perturbing the system is easier in one direction than in the other. I n terms of the
epigenetic landscape, it is generally easier, or perhaps only possible, to cross a
watershed in one direction. That we can obtain a bithorax phenocopy from a
genetically normal Drosophila should not lead us to expect that we can obtain a
normal phenocopy from a genetically bithorax embryo.
REFERENCES
HO, M. W. & SAUNDERS, P. T., 1979. Beyond Neo-Darwinism: An epigenetic approarh to evolution.
Journal of Theoretical Biology, 90: 515-530.
LEWIS, J., SLACK, J. M. W. & WOLPERT, L., 1977. Thresholds in biology. Journal of Theoretical Biology,65:
579-590.
134
P. T. SAUNDERS
SAUNDERS, P. T. & HO, M. W., 1985. Primary and secondary waves in prepattern formation. Journal of
Theoretical Biolqw, 114: 491-504.
SAUNDERS, P. T. & KUBAL, C., 1989. Bifurcations and the epigenetic landscape. In B. C. Goodwin & P. T.
Saunders (Eds), Theoretical Biology: Epzgenetic and Euoltdi~naryOrder f r o m Complex Systems; 16-30. Edinburgh:
Edinburgh University Press.
WADDINGTON, C. H., 1940. Organirers and Genes. Cambridge: Cambridge University Press.