thd/etm ,# M,lthem,¢ticu/ Bu,/ogv Vol. 40, p p 549 579 Pergamon Pte,b Ltd. 1978. Printed in Great Britain (l- S~,uictx lot Mallicinatical Biolog 5

0007-49857809t -( s49 S02.000

D Y N A M I C A L SIMILARITY AND THE T H E O R Y OF B I O L O G I C A L T R A N S F O R M A T I O N S



ROBERT ROSEN

Dept. of Physiology and Biophysics, Faculty of Medicine, Dalhousie University, Halifax, Nova Scotia, Canada

The D'Arcy Thompson concept of biological transformations is developed in a form analogous to such physical concepts as the Law of Corresponding States in thermodynamics, and the Principles of Similitude found in engineering. We find that such concepts depend on a distinction between fiindament al and derived quantities, in which the values assigned to the fundamental quantities set the natural scales for the derived ones. Among other things, we see that critical phenomena, such as phase transitions, arise as an immediate consequence of this distinction. In a biological context, we explore the implications of Thompson's hypothesis that closely related organisms are phenotypically similar, assuming that the organisms we see are the result of selection processes operating on phenotypes.

1. Introduction. In previous work (Rosen, 1962, 1967) we examined the relations between the D'Arcy Thompson Theory of Transformations (Thompson, 1917) and the idea of optimal design as formulated by Rashevsky (cf. Rashevsky, 1960). The object of this work was to attempt to provide some underlying mechanism which would convert D'Arcy Thompson's observation, that related organisms can be converted into one another through continuous transformations, into an actual theory of biological structure and function. In the present paper, we wish to provide a more extensive analysis of these ideas, and at the same time to place the Thompson work into a broader physical and mathematical context. We will begin with some simple and well-known nonbiological examples, which will then be extended to biological cases. II. Laws of Corresponding States. We shall begin our exploration of physical situations cognate to the Thompson Theory of Transformations with an example drawn from thermodynamics. The point of departure for the application of thermodynamic concepts to 549

55(}

ROBERT ROSEN

specific problems is the equation o f state of a thermodynamic system. The equation of state is a functional relation among the state variables which characterize a thermodynamic system at equilibrium. A typical equation of state is the van der Waals equation, describing a particular kind of nonideal gas: (p + a/v 2 ) (v - b) = r T

where p, T, v are the thermodynamic state variables (pressure, temperature and volume respectively), and the parameters a, b, r are determined by the particular "species" of gas under consideration. The thrust of this kind of description is the following: given a particular kind of gas satisfying this equation, then the specification of two of the state variables (say p, T) allows the corresponding equilibrium value (or values) of t~ to be determined. If we regard the parameters a, b, r as fixed, and solve the equation of state for the volume v, then the equation of state becomes a cubic equation in v, whose coefficients are determined by the values given to the other state variables p, T. For a given choice of values for p and T, this equation will in general have one real root or three real roots. For one choice of values Pc, T~, however, the equation will have as solution v = t:c only one value, which corresponds to three coincidental real roots. The resultant state (vc, Pc, T~) corresponds to ¢&e critical point of the gas, and is, of course, related intimately to the gas-liquid phase transition. The geometric interpretation of the critical point is as follows. Given any temperature T, the equation of state becomes a function ofp and v alone and can be plotted as a curve in the (p, v) plane. Such a curve is called an isotherm. In general, the isotherms are of two types: they either possess no extreme points (maxima or minima), or they possess one maximum and one minimum. Separating the two types of isotherms is a critical isotherm (corresponding to T = T~), which possesses a point of inflection. The co-ordinates (vc, Pc ) of this point of inflection then give the critical point. Thus, we can explicitly find the co-ordinates of the critical point from the equation of state. At the critical point, we have three conditions to be satisfied: (a) since the critical point is an equilibrium point, the equation of state must be satisfied; (b) since the critical point is an extreme point in an isotherm, we must have

c3p CU

T= Tc = 0 ;

(c) since the critical point is a point of inflection, we must have ~2p T= (~[12

=0. Tc

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551

These comprise three conditions for the three quantities t,~,p~, T~, and it is an easy exercise to evaluate. We find

& = a/27b 2 ;

t,c = 3b;

T,. = 8a/27rb.

It is important to recognize that at the critical point, the values of the state variables p, v, T are determined by the parameters a, b, r alone. In other words, at this special equilibrium point, knowing the three quantities a, b, r allows one to compute the values of each of the state variables; at all other equilibrium points, five quantities must be known in order to compute the sixth from the equation of state. As we shall see later, this represents a crucial property of critical points. If we now define new variables by writing

rc=p/pc;

co = r/v~;

r = T/T•,

the van der Waats equation becomes (rr + 3/o)2)(3c0 - 1) = 8r, in which the constants (a, b, r) which characterize the gas do not appear explicitly. This is the reduced form of the equation of state. These considerations lead to the Law of Corresponding States for van der Waals gases. Specifically, suppose we make a transition from a given gas, characterized by the parameter values (a, b, r), to a different gas, characterized by new parameter values (a', b', r'). Then the critical point for the new gas is given by

p ~ = a / , . i o -;

t,c = 3b'"

T~=8a/27rb.

N o w let us observe that passing to reduced co-ordinates for the second gas gives us exactly the same reduced equation as did the first gas. This means that, for any state (p, v, T) of the first gas, there is a corresponding state (p'. v', T' ) of the second gas, satisfying the relations p/pc = p'/p;;

,/ve =

T~

= T'/T;.

Stated another way, if we make the co-ordinate transformation

p'= (pJp,.)p;

v ' = (vJ,,e)v;

T ' = (T;/T~)T

which maps each state (p, r, T) on its corresponding state, the equation of state is left int;ariant.

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ROBERT ROSEN

The Law of Corresponding States means precisely that a transition (a, b, r) - , ( a ' , b ' , r ' ) from one species of gas to another can be annihilated by an appropriate co-ordinate transformation of the state space. Another way of saying this is: all van der Waals gases appear identical, when the state variables of the gas are measured in the set of units appropriate to that gas. The scales appropriate to a gas (a, b, r) are determined by the critical values of the gas; indeed, it is easy to see that these critical values determine the vertices of the unit cube in (p, v, T)-space in the natural units. Thus, the quantities (el, b, r) set the scales for the measurement of (p, l,, T).

III. The Principle of Similitude (Dynamical Similarity). Considerations of dynamical similarity, or similitude, go back at least to Galileo and have been extensively developed since that time. An extensive discussion of elementary biological applications of this principle is to be found in D'Arcy Thompson (loe. cit. ). However, the first general consideration of this principle in its modern form was probably given by Buckingham (1915) and was based on the principle of dimensional homogeneity, which Buckingham credits to Fourier. In the present section, we shall briefly review these ideas. Suppose that we are given some physical process which we wish to describe, and suppose further that we know the process depends entirely on a certain number of physical quantities, or observables, associated with that process. Then the law governing the process must be expressible as some functional relation between the observables which are involved in it ; i.e. it must be of the form qS(x, .... x , ) = 0 ,

(1)

where the xi are the observables in question. This relation will be called the equation of state of the process, for reasons which will soon become apparent. N o w in mechanical systems, all scales are fixed by the scales in which we measure the basic quantities of mass, length and time; in non-mechanical systems, we may require other basic quantities, such as temperature, charge, etc. For mechanical systems, in general, we can choose three of the observables of our process (say x,, x2, x3 ) asfimdamental quantities (in dimensional terms, it is only necessary that mass, length and time be expressible as definite functions of x,, x2, x3). Then these suffice to set the scales for all of the remaining xv Indeed, we may write --

Xi--X

-:~1,

Or2, ~3i

1 X2 X3

(2)

which is interpreted not as a numerical relation, but in dimensional terms; that is, the units in which xi is to be measured must be expressible as a monomial in the units in which the fundamental quantities Xl, x> x3 are measured.

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It follows that the expressions

~Ii=xiXI=Iix2:Qix3 :~3i,

i=4,...,n,

(31

are dimensionless. Indeed, we can rewrite the equation of state governing our process in the dimensionless form ~(FI~ ..... F[,) = 0 .

(4)

From this form of the equation of state, it is immediate that any two situations in which each of the dynamical expressions FI~ are identical will behave identically. This fact leads to a precise formulation of the principle of similitude, or dynamical similarity: two systems governed by the same equation of state are dynamically similar if and only if the corresponding expressions Fli are numerically equal. This situation means the following. Let us envisage two systems (Xl ..... xn), (x'~ ..... x', I governed by tile same equation of state. Then these two systems will be dynamically similar if and only if we have X 1 ~ I i x 2 : ¢ 2 i x 3 ~3i XI=

. -~ x I

,-a,. . 'x 2 ~x 3

~i Xi"

i = 4 ..... n.

(5)

Another way of stating this is as follows: given a system governed by our equation of state, if I make a change x , ---, x',,

---' x l ,

--,x;

of the fundamental quantities, that transformation can be almihilated by a corresponding transformation of the remaining derived quantities x4,...,x, given by (5) above. It should be noted that all of these dimensional arguments are valid only up to numerical factors; thus the dimensionless, or reduced, form of the equation of state, given by (4) above, may contain numerical factors which we cannot write down explicitly from dimensional considerations alone. However, such numerical factors in the i-Ii will cancel from the transformation law (5) expressing tile principle of similitude; hence this transformation is entirely determined by the argument which we have sketched. It follows from (3) that, up to a numerical factor, a choice of fundamental quantities xl, x2, x3 determines a special value of each of the remaining derived quantities x,, ..... xn; namely, that value ofxi which renders the corresponding I1; equal to unity. These values of the derived quantities will be called the critical values, for reasons soon to become apparent. It is evident that the critical values

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ROBERT ROSEN

of the derived variables represent situations in which the value of a derived variable depends only on the values assigned to the fimdamental variables. As such, the critical values can be expected to be associated with some kind of degeneracy. Ordinarily, we need to know n - 1 values of the variables xi in order to determine the nth from the equation of state. At critical values, we need to know fewer than n - l values in order to determine all of the others. Thus, additional relations must be satisfied at the critical values, besides the equation of state; when we move away from the critical values, these relations are violated. Such considerations are quite general, and as will be seen, they are of great significance. Before leaving this subject, we may make two observations: (a) critical values will exist only f there exist at least two derived quantities. For if there exists only a single derived quantity, its value is by hypothesis uniquely specified by the equation of state and the values assigned to the fundamental quantities. Further: (b) if there are r derived quantities, then there are at most r non-zero critical values. This number r has an objective significance, since it specifies the greatest number of special relations which may be satisfied at a critical state, and hence the order 01 the degeneracy which we may expect to find manifested when we perturb such a state. These ideas, of course, are closely related to Thorn's considerations on unfoldings (Thorn, loc. cit.). However, as our examples will show, such arguments can only give us upper bounds.

IV. Example: The van der Waals Equation. It is most instructive, and illuminating, to consider the van der Waals equation from the standpoint developed in the preceding section. In the process of doing so, we shall see that the Law of Corresponding States arises as a corollary of the principle of similitude. We shall also obtain some insight into the significance of the critical point. The van der Waals equation is a relation between six quantities, and is of the general form q){p, v, T,a, b, r ) = 0 . Let us consider these quantities in dimensional terms. We note that, in addition to the basic mechanical units of mass m, length I and time t, there is also a unit d of temperature, in which T is measured. In dimensional terms, we have

p=ml-lt

2;

v=13;

T=d;

for the parameters a, b, r, we easily compute that we must have

a=lSmt-2;

b=13;

r=ml2d-lt

2.

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555

We shall now choose the parameters a, b, r as fundamental quantities (i.e. corresponding to xl, Xz, x3 in the discussion of the preceding section), so that the state variables p, v, T are the derived quantities. We must then be able to write, in dimensional terms, p = aal ba2ra3,

v = a ~lb~2rp3, T =aYlb'/zr 73.

Substituting for each of these quantities their corresponding dimensions and equating exponents allows the nine exponents ~1 ..... 73 to be uniquely determined. We find p = a/b 2 ;

v = b;

T = a/rb,

(6)

always up to numerical factors. According to our definition of the preceding section, the critical values of the derived quantities are those for which the relations (6) are numerically satisfied. For these values the dimensionless quantities H 1 =p/(a/b2),

lq[2

=v/b,

FI3 = T / ( a / r b )

take on the value unity. If we compare these values with those obtained directly from the van der Waals equation, we see that they are exactly the same (again, up to numerical factors). This justifies the employment of the term "critical" in the contexts of the previous section. Moreover, it is seen that the Law of Corresponding States is the precise statement of the principle of similitude in this context. Namely, a change (a, b, r ) - , (a', b', r') of fundamental quantities can be undone by a corresponding transformation (p, v, T)--*(p', v', T ' ) as before, in such a way as to leave the dimensionless quantities, and afortiori the equation of state, invariant. It is instructive in this context to see why the equation of state of an ideal gas, pt~ = A T,

exhibits no critical point. F r o m dimensional considerations, it is clear that the available parameters (there is here only one, namely A) do not form a set of fundamental quantities, which set the scale for the state variables p, v, T. Indeed, for such a set we would have to take three of the four available quantities; say (p, A, T). There is only one dimensionless quantity here, and in fact the value ofv is uniquely determined by the values assigned to the fundamental ones. Thus, the

556

ROBERTROSEN

entire argument becomes inapplicable. The absence of a critical point thus arises, from this viewpoint, from the fact that there are too few derived quantities, or alternatively, too many fundamental ones. Finally, it should be noted that the degeneracy which occurs at the critical point in the van der Waals equation is essentially that of the cusp catastrophe of Thom (Thom, 1972; Fowler, 1972).

Y. AnotherExample: The Linear Oscillator. The undamped harmonic oscillator provides a classic illustrative example of the power of the kinds of dimensional ideas described in the preceding sections. Suppose that we wish to obtain the relation between the frequency v of the oscillator and its basic parameters, the mass m and the stiffness k. If the frequency depends only on m and k, then a functional relation of the form

@(v, ,71,k ) = 0

(7)

must be satisfied, which plays the role of the equation of state. In this situation, it is sufficient to pick m, k as a set of fundamental quantities, leaving only v as derived quantity. Dimensionally, we have v=t-l:

nl=l~l;

k = m t -2.

In dimensional terms, then, it is easy to see that v = k~m ~ Hence the equation of state can be written in the form

v = T( x / k / m ). We obtain this result witl:out having to know or solve any differential equation, or indeed any laws of physics at all, beyond the principle of dimensional homogeneity. We note explicitly that, since there is only one dimensionless quantity, and hence only one derived quantity, there is no critical value of v. On tile other hand, the differential equation which, in mechanical terms, provides a complete picture of the undamped harmonic oscillator, is m2 + kx = O.

(8)

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557

The solution can be written as x (t) = x (0) cos x/(k/,n )t + 2(0) x/mlk sin v/( k/m)t

(9)

so that the explicit solution for the frequency v is v = (1/27r) x/k/m.

(10)

If we consider the case of the harmonic oscillator with damping/~, governed by the differential equation mY +/72 + kx = 0,

( 11 )

with solution x(t)=e

~i,,, a c O S / m

t+bsin

t

(12)

(where a, b, depend on x(O), 2(0)), then the frequency v in this case is given by

1

/k

v=2~x] m

,'7 4m

(13)

2 .

The same kind of dimensional arguments which yield the frequency in the undamped case seem useless in the damped case currently under consideration. To apply these arguments, we would write an equation of state in the form @(v,/~, m, k) = 0 .

(14)

If we were to try to take m, k,/~ as fundamental quantities, and v as the derived quantity, we would very quickly find that there are not enough conditions to determine the exponents in the expression of dimensional homogeneity for (14). A moment's consideration will reveal, however, that/7 itself is a derived quantity, just as is v, from the fundamental quantities m, k. In fact, we can write the dimensional relation

It follows that the equation of state (14) can be expressed, in terms of two dimensionless quantities El i

m~ k_ ½

1

1

in the form O(FI1, FIz) = O.

(15)

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ROBERT ROSEN

Note that this equation can be written as [I 1 = ~(FI2),

or, in explicit dimensional terms, solving for v,

v=k~m -~ud(m-}k

~fi).

Now the exact expression (13) can be written in the form

Ilk V=2~ ~/m

f12

l/k/

fie

4m 2 --2re k/~/m" 1 . . .4kin ...

l jk / l ( ~ ) 2

=2d m l-4

So we see that dimensional arguments are in fact of considerable power, even in the damped case. Moreover, since there are now two derived quantities, all of the machinery of the principle of similitude can be applied. In particular, let us suppose that the numbers (v, fl, k, m) represent a definite oscillator. Let us change the fundamental quantities from (m, k) to (m', k') respectively. Then the principle of similitude says that the transformation of the derived quantities from (v, fl) to (v', fl') given by m~k ~

v'

m'~k'-Cv

fl'-

m- ~k- ~ m,_ =,k, ~_fi

(16)

keeps the equation of state (14) invariant (indeed, it is a useful exercise to check this fact on the more exact equation (l 3)). This expresses the analog of the Law of Corresponding States for damped harmonic oscillators. In addition, since there are now two derived quantities, there is also a unique set of critical values vc, fl,.;it will be recalled that these are the unique values which (up to a numerical constant) render the dimensionless quantities H1, [12 equal to unity. As was remarked previously, the critical wdues must generally involve some degeneracy. In the present situation, the critical value fi,. =2m~k ~, substituted into (13), is that value whicti renders the quantity under the radical, and hence the corresponding frequency v(fi,.), equal to zero. In other words, it is that calue of the damping fi which just annihilates the oscillation; it marks the transition from a damped oscillation to the aperiodic overdamped ("deadbeat")

T H E O R Y O F BIOLOGICAL T R A N S F O R M A T I O N S

559

situation. The degeneracy at/~c is related to the fold catastrophe of Thom (Ioc.

cir.). It should be noticed that this situation is completely analogous to that which occurs in the van der Waals equation. In that case, the critical values mark a state at which a pair of complex roots of a cubic equation laecome a pair of coincident real roots; in the damped oscillator, the critical value/~c likewise marks the state at which a pair of complex conjugate roots of a quadratic equation become a pair of coincident real roots. Thus, we get a further indication that the kind of critical behavior which occurs in the van der Waals equation is in fact of a very general character, and that it ultimately devolves not from any special physical situation, but entirely from considerations of similitude and from the fact that there is more than one derived quantity. Likewise, the critical frequency vc = ( l/2~z)(k ~m ~ ) is that value which renders p(vc)=0; it is that value which marks the transition from a damped to an undamped oscillator, or from a positively damped to a negatively damped one. It should be noted that here, unlike the van der Waals equation, the two critical values (/~c, v~)do not co-exist; i.e. do not satisfy the equation of state. Thus for this example we find two critical states (/~c,0) and (0, v~). However, it may be noted that here,just as in the van der Waals equation, we may solve for the values of the fundamental quantities in terms of the critical values of the derived ones; we find, up to a numerical factor, that

This, however, need not be true in general; it is necessary that there be no fewer derived quantities than fundamental ones. Before returning to more general matters, let us draw some consequences from the transformation ( l 6). First, we see that if/~ = 0, then/3' = 0: i.e. no damped oscillator can be corresponding to an undamped one. Likewise, if/~ ~ 0, then /~'-0; thus, no positively damped oscillator can be corresponding to a negatively damped one. Second, we note that the distinction we have drawn between fundamental and derived quantities cuts across the usual classifications we make of physical observables. For instance, ordinarily we would regard m, k, /~simply as independent variables, and v as the dependent variable, whose values are fixed by the equation of state. In our context, although fl is still indeed an independent variable, it is considered derived, since the scale in which /~ is measured is determined by the values assigned to the fundamental quantities m, k, as is also the scale in which the dependent variable v is measured. Further, we would ordinarily regard the parameter values m, k,/~ as "intrinsic" to the system. while v is a dynamical variable playing the role of a state variable. In our context, however, both/3 and v play the role of state variables,just as p, v, T represent the variables of state in the van der Waals equation.

560

ROBERT ROSEN

VI. Some Considerations of Similitude in Engineering. In the preceding sections, we have presented some simple physical examples of the principle of similitude based on dimensional arguments. We have seen that considerations based only on dimensional homogeneity allow us to make non-trivial assertions about the form of equations of state and also to formulate laws of corresponding states when possible. As a corollary, we find that a law of corresponding states carries with it the concept of a critical point, at which there will generally be some degeneracy; the physical manifestation of this degeneracy can in general be regarded as a "phase transition" which occurs where the derived variables are varied through the critical values. In engineering, there have been two main applications of these ideas. The first of these is the exploitation of the principle of similitude in the construction of "scale models" to determine the behavior of a prototype system, whose properties are too complex to be determined through direct argument. In general, the argument from model to prototype involves a general sort of analog computation, in which we learn about a system of interest by studying a different but analogous one. Basically, in terms of the development we have presented so far, the rationale for such "'scale model" arguments is the following. We suppose (a) that the model system and the prototype system satisfy the same equation of state; (b) that the model system and the prototype system differ from one another initially in the numerical values assigned to the fundamental quantities, which fix the scales for all the remaining derived quantities; (c) that these derived quantities in the model and the prototype are related to one another through the appropriate corresponding state transformation, which leaves the equation of state invariant. If these conditions are satisfied, then every behavior of the model system implies a corresponding behavior in the prototype, which may be computed when the model behavior, and the transformation law between corresponding quantities, is known. This transformation law is in turn determined completely by the manner in which the fundamental quantities fix the scales for the derived ones. Closely related to considerations of scale modelling is the specification of "dimensionless numbers". Such "dimensionless numbers" represent monomials (in the quantities involved in an equation of state) which are themselves free of dimension; i.e. are pure numbers. The quantities FIi which we defined previously are examples of such dimensionless quantities, and of course play an integral role in the formulation of the principle of similitude; indeed, we could express this principle by saying that two systems are similar if and only if each of the dimensionless quantities H i,Hi associated with their equation of state are respectively equal. Dimensionless numbers are also important in the description of the degenerate behavior, akin to phase transitions, which we saw could be associated with systems to which the principle of similitude applies. For instance,

THEORY OF" BIOLOGICAL T R A N S F O R M A T I O N S

561

we may consider the wellknown Reynolds Number of hydrodynamics; this number is of the form 1 R = pavq where v is the velocity of flow of fluid in a vessel; a is the radius of the vessel;/) is the density of the fluid, and I/is the viscosity. It is well known that when the Reynolds number exceeds some critical value depending on the fluid (i.e. when the velocity of flow in a particular vessel exceeds some threshold velocity) there is a transition from laminar flow to turbulent flow. Now a dimensionless quantity like the Reynolds number has all the properties associated with a quantity Fli appearing in some reduced form of an equation ofstate governing fluid flow; here we may suppose that a, p, and q are fundamental quantities, and v is a derived quantity. The fact that a critical value of v exists means precisely that there must be other (unknown} quantities involved in the equation of state; i.e. the identity of the Reynolds number in two situations is not sufficient to render the systems corresponding in all of their hydrodynamic properties. Conversely, as we shall see later, the kind of degeneracy associated with the transition from laminar to turbulent flow can be expected to throw some light on (a) the number of other derived quantities, and hence of dimensionless numbers, involved in the full equation of state; and (b) on the actual form of the equation of state. Considerations of this type, however, are far beyond the scope of the present paper; we merely wish to point out here that the kind of degenerate behavior associated with the principle of similitude, and exemplified by phase transitions and similar phenomena in physics, appear quite generally as a consequence of the same type of underlying phenomenon ; their study in a unified context may prove quite illuminating. Occasionally, it is found that a number of dimensionless quantities associated with a particular phenomenon are related. For instance, Stahl (1962) cites an example of three dimensionless quantities associated with the loss of heat from vertical metal plates; the Nusselt number N, the Grashof number G, and the Stanton number S. It is found empirically that N=(O.59)GS °'25. Such a relation between dimensionless quantities in fact constitutes an equation of state for the process under discussion. VII. Similarity and Dynamics. The examples provided in the previous few sections have been entirely devoid of dynamical considerations. There has been no mention of time, nor of time dependence of either the fundamental or derived quantities with which we have been concerned. In the present section, we wish to extend our considerations of similifude to a dynamical context.

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ROBERT R O S E N

Although dynamics has not been mentioned explicitly so far, it has permeated our examples in several kinds of ways. For instance, the van der Waals equation in fact represents the set of limiting states (steady states) of a gas governed by a particular kind of potential function; these states thus represent the asymptotic behavior of trajectories under a dynamics specified by the gradient of this potential function. As such, it is natural to inquire whether the Law of Corresponding States can be "lifted" from equilibrium behavior to the dynamics which generates it (e.g. in terms of a correspondence between equipotentials). Again, our discussion of the linear oscillator concerned only quantities which are time-invariant and in fact independent of the state variables in terms of which the oscillator is generally described (the displacement x, and the velocity or m o m e n t u m p), although our exact discussion of that system proceeded in terms of integrating the equations of motion governing the oscillator. Thus here too there is an underlying dynamics (albeit not in this case generally specified by a potential), and we may ask again whether our discussion of time-invariant quantities associated with the system can be "lifted" to a discussion of timedependent quantities. We may further ask : if we are given an equation of state which admits a principle of similitude, can we in general associate the state equation with some kind of dynamics in such a way that the similitude can be extended to the time-varying quantities ? The present section will be concerned with these kinds of questions. It is convenient to begin with a reconsideration of the linear oscillator. Let us consider the undamped case first; we recall that the full mechanical description of such a system is given by (8) and (9) above. If we wish to pursue an argument analogous to the kinds we have used so far, we would have to write down an equation of state of the form • (x, t, m, k, x(0), 2 ( 0 ) ) = 0 .

(17)

Intuitively, this would correspond to the relation (9), which in effect describes a traj ectory of the system, specified by the appropriate initial conditions x (0), 2 (0). Arguing dimensionally, we find that m, k no longer constitute an adequate set of fundamental quantities; we must augment this set by the addition of a quantity containing the length I. Assuming x(0)4:0, we can use this quantity for the purpose; ifx (0) = 0, we can take 2 (0) as fundamental; if both x (0) and 2 (0) = 0, it is clear that the system degenerates completely (i.e. x = 0 for any choice oft, m, k). If we choose m, k, x(0) as fundamental quantities, then these set the scales for the derived quantities x, t, 2(0); in dimensional terms, we have x=x(0),

t=m½k -~,

x(O)=x(O)k-m +

This gives us the dimensionless quantities Fl l = x ( 0 ) - ax;

1-12=k~m-½t;

[I3=x(O)-lm~k-~2(O),

THEORY OF BIOLOGICAL TRANSFORMATIONS

563

so that the equation of state (17) can be written in the form

(18) From this, we can immediately apply the principle of similitude; namely, a change in the fundamental quantities of the form m ~ m', k---,k', x ( 0 ) ~ x (0)' can be annihilated by the transformation of the derived quantities given by

X --~ X ' --

t~t'

x(0)' X, x(0) ~/ k m t,

2(0)~2(0),_x(0)'

/mk' "0

x(O) X / ~ x( )"

It is an easy exercise to show that the solutions, or trajectories, of the system, given exactly by (9), are indeed invariant under this transformation. Such a transformation as the above is of fundamental significance in dynamical terms, for it is intimately related to the structural stability of the dynamical system being described. Roughly speaking, a dynamical system is called structurally stable ira "small change" in the equations of motion leave the trajectories qualitatively the same; this in turn means that there is a continuous deformation, or transformation, of the state space which carries the unperturbed trajectories onto the perturbed ones. Or, stated another way, structural stability means that a modification in the equations of motion can be annihilated by a corresponding coordinate change in the state space, so that the perturbed system in the transformed coordinates is identical with the unperturbed system in the original coordinates. We can already see (and Will develop this point in detail subsequently) that the principle of similitude is merely a statement of structural stability, and that the dimensional arguments which we used to formulate the principle must be of a far more general significance than has appeared so far. For the moment, however, we will only point out that the above example allows us to draw a further conclusion beyond the mere invariance of trajectories: namely, that there is a transformation of the time variable, such that two systems (m, k, x(0)), (m', k', x(0)') are in corresponding states at corresponding instants. " In the present case, of course, we are considering only a conditional structural stability, in which we perturb the equations of motion only by the parameter modifications m ~ m', k - . k', and the initial condition modification x (0) ~ x (0)'.

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The principle of similitude means here that the undamped oscillator is structurally stable to these perturbations, or alternatively, that through appropriate changes of scale of the derived quantities, all such oscillators can be made to appear identical; i.e. satisfy the same reduced equation (18). A word should be said about the significance of the concept of critical values for the equation of state (18). We pointed out above that the existence of more than one derived quantity, and hence more than one dimensionless quantity, implies the existence of critical values for the derived quantities, at which the values of the derived quantities are so chosen as to make the corresponding dimensionless quantities equal to unity. At these critical values, the derived quantities are numerically determined by the fundamental ones alone, and hence further relations are satisfied at these critical values beyond the one given by the equation of state; this in general leads to some degeneracy, as these relations are violated when we move away from the critical values. In the present case, which is of a very special character, no such degeneracy arises. This is because the critical time tc is simply the period of the oscillation. At the critical time, the situation is exactly what it was at the initial time t = 0, and it is easy to see from the exact solution (9) that at this time the equation of state becomes independent of 2 (0). Thus, any special condition satisfied by .~(0) at this critical time is irrelevant. In a sense, the example we are considering is thus already degenerate, and this inherent degeneracy "cancels" that arising at the critical values; the intrinsic degeneracy arises ultimately from the fact that the fundamental quantities which set the time scale are not involved in setting the scales for the other derived quantities. Thus, this example shows that the presence of more than one derived quantity is necessary, but not always sufficient, for degenerate behavior at the critical values. We leave it as an exercise to the reader to consider the case of the damped harmonic oscillator in dimensional terms, analogous to that given above for the undamped oscillator; particular attention should be given to the relation of the principle of similitude, as a mapping between trajectories, to the structural stability of the oscillator (noting that the quantity fl is a derit~ed quantity, not a fundamental one) and to the meaning of the critical values. In any case, we can see from the simple examples we have considered that the principle of similitude can be applied equally to dynamical problems, and that in this context it is closely related to considerations of structural stability. Moreover, it is easy to see that, if two dynamical processes are similar, so that they are in corresponding states at corresponding instants, then their limiting behat, ior as t--+ v¢ are also similar; hence any equation of state governing the steady states of the system, and a corresponding principle of similitude arising from this equation of state, can be "lifted" back to the dynamics which generated the steady states. This embodies the general proposition that if the limiting sets

(i.e. the asymptotic behavior) of a dynamical system is stable to perturbations of

THEORY O F BIOLOGICAL T R A N S F O R M A T I O N S

565

the equations of motion, then the dynamics which generate these limiting sets is structurally stable ~o those perturbations. In particular, then, if we can establish a similarity between sets of steady states of a dynamical process (as for instance in the case of the van der Waals equation), then this implies a corresponding principle of similitude for the dynamics which generates these steady states. As we shall see, this is a very powerful result. VIII. Similarity and Optimality. Let us suppose that we have a process governed by an equation of state of the form O(xl ..... x , ) = 0. Suppose further that there is a partition into fundamental and derived quantities, so that the equation of state can be written in the reduced or dimensionless form • (H1 .... , l-I,)=0. Then as we have seen, the specific properties of any particular specific representative of that process, obtained by assigning specific values to the xi, is completely determined by the numerical values of the quantities II i. If this is the case, then any criterion by means of which we select from among the representatives of the process, in terms of their particular behaviors, must be a function only of the values assumed by the Hi. More specifically, let us suppose that such a selection is made on the basis of the minimization of some index of performance of the system. Then such an index must consist of one or more (realvalued) functions defined on the space of r-tuples ([I 1.... , Fir) which can arise from our process. However, it is clear that such a minimization or optimization process cannot distinguish between situations which give the same values of the Hi; or in other words, that any situation which corresponds (in the sense of dynamical similarity, or similitude) to an optimal situation is itself optimal. Moreover, if the optimality criterion we are using has a unique minimum, the converse is also true: an 3, two optimal situations must be dynamically similar. In general, letX be the manifold in Euclidean n-space consisting of all n-tuples (xl,..., x,) which satisfy the equation of state in question. Then this manifold is partitioned into equivalence classes through the relation of dynamical similarity. The quotient setX/R can be identified with the manifold in Euclidean r-space, consisting of all r-tuples (H 1,..., I-I,) which satisfy the reduced equation of state. What we are asserting is that any minimization process, selection procedure or optimality problem which is based on performance criteria is defined on X/R, and not directly on X. These simple results allow us to explore

566

ROBERT ROSEN

the nature of the equivalence relation R on X by means of any optimality criterion based on performance. It should be noted that the same argument holds when the optimality criterion involves only a subset of the 1-Ii. For each quantity FIi imposes a corresponding equivalence relation on X; any criterion involving the value of that [Ii will be defined on the corresponding quotient set, and exactly the same argument goes through. It should also be noted that if we modify the performance criterion (i.e. change the basis for the selection procedure) then in general the systems which are optimal for the new criterion cannot be dynamically similar to those which were optimal for the original one. This remark needs to be qualified to some extent, but the qualifications are obvious and are left to the reader.

IX. General Aspects of Dynamical Similarity. Both historically, and in terms of our presentation above, dynamical similarity has been tied to considerations of dimension. Indeed, whenever dimensional considerations can be applied, they seem to provide effective procedures for the establishment of principles of similitude relating different systems satisfying the same equation of state. They also give us effective methods for predicting and studying critical behavior, such as phase transitions and related phenomena. On the other hand, there are many ostensibly similar situations in which dimensional arguments are inapplicable and yet similitude principles should be present. For instance, we can treat physical oscillators such as the harmonic oscillator by means of dimensional considerations with relative ease, while homologous biochemical oscillators and other types of biological rhythms seem unreachable by these techniques. These considerations suggest (a) that there is something very special about the applicability of dimensional concepts and the manner in which they guarantee the establishment of similarity criteria, and (b) that there is nevertheless some underlying mathematical structure shared by both dimensional and dimensionfree situations, which can be extracted and applied even when dimensional arguments are inapplicable. In the present section, we shall consider some ramifications of these questions. The essence of the application of dimensional ideas to a specific problem lies in the distinction between what we have called fundamental and derived quantities. As we have seen, a choice of fundamental quantities sets the scale for all the remaining quantities;i.e, provides a natural system of units in terms of which the derived quantities can be expressed. These natural scales provide the basis for the dimensionless quantities which are the essential ingredients required to establish the principle of similitude relating different systems satisfying the same equation of state. At the same time, the consolidation of the original set of variables into a smaller number of dimensionless quantities represents a dimensional reduction which is of great practical and conceptual significance.

THEORY OF BIOLOGICAL TRANSFORMATIONS

567

We have already seen that principles of similitude are expressions of the structural stability of certain mappings. Since concepts of structural stability are generally applicable even when dimensional considerations are not, it is suggested that such concepts will provide the generalizations we seek. In the development of these considerations, we will make contact with the work of Thorn (loc. cit.); however, our emphases will be quite different. Thorn is basically interested in failure of structural stability, while we are concerned with its preservation. As a point of departure, let us suppose that we are given an equation of state of the form (1)(X 1 . . . . , Xk, bt 1 . . . . .

l~r ) = O

(19)

to which dimensional considerations apply and such that the u's represent the fundamental quantities, the x's the derived ones. Then by definition, we have the following situation: given a particular choice of values of the fundamental quantities ul ..... ur, we have a manifold of k-tuples (xl ..... Xk ) which satisfy the equation of state. If we now change the values of the fnndamental quantities from u; to u~, then this modification can be annihilated by means of a co-ordinate transformation in Ek, given by the requirements of similitude. This transformation maps the uz-manifold isomorphically onto the u~-manifold, in such a way as to leave the equation of state invariant. The equation of state (19) can be written as an r-parameter family of functions •£ , . 2 . . . . Ix1 . . . . . x k - , ) = x k

from ( k - 1 )-dimensional Euclidean space to real numbers. What is guaranteed to us by dimensional considerations is that all of the fimctions in this family are identical, up to a co-ordinate transformation in their domain and range. That is, if we perturb any function in the family by modifying the values assigned to the parameters ui, we obtain a function which is formally identical to the given one. Moreover, the principle of similitude gives us precisely the transformations which establish the identity of the functions in the family. It must be noted that we are not saying anything about the stability of the functions in the family to arbitrary perturbations; as functions of the xi, they may be arbitrarily pathological. All we are saying is that ifa function arises from an equation of state to which the dimensional considerations apply, then in every neighborhood of£ there is at least an r-dimensional manifold full of functions identical toJ~ we are restricted to this r-manifold by virtue of the fact that we are dealing with systems satisfying the given equation of state. Thus, the mappings J" must appear stable under these circumstances, regardless of how unstable they may be to arbitrary perturbations. Considerations of this type contrast sharply

568

ROBERT ROSEN

with the emphasis on transversality which dominates Thom's discussion. Indeed, considerations on the behavior of t h e f ' s at the critical values, with the attendant degeneracy which we have noted, guarantees that in general the mappings f cannot generally be stable to arbitrary perturbations. We may remark parenthetically that such conditions of relative stability, in which situations appear stable because we are not allowed access to perturbations which would provoke instability, are closely akin to such things as countable models of set theory, in which countable objects appear uncountable because we are not allowed access to mappings which would put them in 1-1 correspondence with integers. Thus, stability is not an intrinsic, but a contingent property, depending entirely on the context in which it appears. This is a crucial epistemological point involved in the interpretation of stability arguments in the physical world, but we shall not pursue it here. Let us see how these ideas generalize those arising from considerations of dimension. Let us suppose that we are given any function • (xl ..... x , ) = 0 ofn variables, which we will think of as representing the equation of state of some process. It is clear that we can think of any of the variables xi, say xl, as being a derived variable and all the others as fundamental. F r o m our standpoint, this is a trivial situation, for we can write the equation of state as an 07 - 1 ) parameter family of mappings

J~ex3...x,(xl ) = x l ; these are all constant maps, whose domains and ranges consist of a single point. That is, assigning arbitrary values to the fundamental variables x2 ..... x, determines the value of the derived variable completely through the equation of state. Now suppose that we can treat two of the variables, say x~ and x2, as derived quantities, and the remainder of the xi as fundamental. Then the equation of state can be regarded as an ( n - 2 ) - p a r a m e t e r family of mappings of the form f~,~,+..... (x2) = x 1, and moreover, all of the m a p p i n g s f in the family must be identical. This means that, given any set of values of the fundamental variables x3 ..... x,, if we modify ' ., x,' respectively, there is a transformation Xz~X2, x l ~ x l these values to x3,.. such that !

fx;

...X

:(xl)=xl

t

THEORY OF BIOLOGICAL TRANSFORMATIONS

569

this transformation holding for any pair of values (X 1 . . . . . Xn) which satisfy the equation of state. The pairs ix1, x2 ), (x'l, x~ )are thus corresponding states, or are dynamically similar, in the circumstances specified by the respective values (x3 ..... Xn), (X; ..... X',) chosen for the fundamental quantities. Moreover, the transformation equations for the derived variables can be written in the form t

t

!

X 1 =

q51 ( X 3 . . . . .

Xn, X 3 . . . . .

Xn, X 1 ) ,

X 2' =

qb 2 ( X 3 , • • ",

' X 2 }" Xn, X 3' , • " ", Xn,

Let us suppose that these transformation equations can be expanded as a power series, so that for instance

X'~ =Bo + B1x2-F B2X2-F...

(201

where the coefficients B i are functions of the original and modified values of the fundamental quantities: Bi=Bi[x

.3 . . . . . x , , x -'3 , . .., x'o).

These coefficients must have the property that B i ( x 3 . . . . . x n, x ; . . . . . x ' n ) = O,

Bl (X3,...,Xn,

X '3 . . . . . x ' , ) = l ,

i=/= 1,

(21) (22)

because our transformation must become the identity when the chosen values of the fundamental quantities are unperturbed. The general way for the second condition to be satisfied is for B 1 to be of the form t7(x; .... , x;,)

B1

h(x3,...,x,)

(23)

To see how the other conditions must be satisfied, let us retreat again to the dimensional case. If x2 were a dimensional quantity, then x2 and x2 must be of the same dimension. This fact would place dimensional restrictions on the coefficients Bi appearing in the expansion (20); indeed, it is easy to see that (a) Bo must be of the same dimension as x2, and (b) Bz must be of the dimension of (x2)1 i for i > 0 . Since the number B1 must be dimensionless, we could write, from (23), that x2 is of the dimension ofh(x3 ..... x,. )- 1. This in turn would mean that the coefficients Bi would have to be of the form

B i = N i h ( x 3..... Xn)l i,

(24)

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ROBERT ROSEN

for all i@ 1, where Ni is a pure number; i.e. dimensionless. Thus the B~ must be independent of the primed values x;,..., x',, and yet must satisfy the condition (21). This can only be the case if all the N~ = 0 for i 5a l ; this is the ultimate reason why all the transformations occurring in dimensional arguments turn out to be simple changes of scale. Now the argument leading to (23) is perfectly general; i.e. independent of dimensional considerations. If we define the quantities N~ by the relation (24), then we must still generally take all the N~ as vanishing, since otherwise the infinite number of relations (21)on the finite set of quantities (x3 .... , x,,x; .... , x',) would generally be inconsistent. Thus, by making certain further weak assumptions, the arguments applicable in the dimensional situation can be extended far more widely. Now the quantity h(x3 ..... Xn) appearing in (23) is the analog of the dimensionless expressions of the form (2), and the quantity x z h ( x 3 . . . . ,Xn)

1

is the analog of the dimensionless quantities I-Ii which we have considered above. Thus we can take over our dimensional discussion in toto to the more general situation ; in particular we can see how the fundamental quantities set the scales for the derived ones through the functions we have called h. Now we can keep iterating this argument; if the quantities xl,x2 can be considered as derived, then we can try regarding the quantities xl,x2,x3 as derived, and the remaining xis as fundamental. It is clear that the treatment of a quantity like x3 as fundamental, when in fact it can be regarded as derived from the remaining xis, will not affect the transformations obtained for Xl, x2; the diagnostic of this situation is in fact that the quantity x3 will not appear explicitly (i.e. will cancel) from the quantities h which fix the transformations. Thus we can proceed stepwise, until we reach a quantity x~ for which the (n - r)-parameter family of functions f~+, ..... (Xa,...,x~) obtained from the equation of state are not all identical (i.e. such that there is no transformation of (xl ..... xr) which can annihilate an arbitrary modification of the values of the quantities (xr + a,..., x, )). We may note that it is precisely at this point that Thom's analysis begins and that his classification theorem comes into play (Thorn, loc. cit.). One final remark may be made to conclude the present section. We have been considering how dimensional arguments involving the principle of similitude can be extended to more general (i.e. dimensionless) situations. This involves in effect the consideration of different "species" of systems satisfying the same

THEORY OF BIOLOGICAL TRANSFORMATIONS

571

equation of state; i.e. differing from one another in the values assigned to the fundamental quantities. We can regard the above arguments as widening these considerations, in the following way: suppose we are given two classes of systems (e.g., biochemical oscillators and mechanical ones) which can be regarded as satisfying the same equation of state (in the sense that we can establish a mapping between the respective observables involved in the two situations which maps one equation of motion onto the other). Then if one of them can be treated by dimensional considerations and a class of fundamental quantities extracted thereby, these same considerations can be carried back to the non-dimensional systems. In particular, if we can build a mechanical or electrical analog of a biological behavior and we can analyze the behavior of this analog in dimensional terms, then the dimensional conclusions obtained thereby also pertain to the biological behavior with which we started. This is an important generalization of the concept of "scale model", or similitude, which transcends the apparent restriction of such considerations to systems belonging to the same dimensional class.

X. The Theory of Biological Tran~s(ormations. The material in the preceding sections provides the basis for a more comprehensive discussion D"Arcy Thompson's concept of biological transformations (Thompson, loc. cir.). The basic idea is to regard such transformations as similarity transformations of precisely the same character as those occurring in the Law of Corresponding States in thermodynamics. The essence of Thompson's idea was to regard a biological organism as if it were a geometric object;i.e, as a subset A of a topological space S. A continuous mapping (i.e. homeomorphism) of this space onto itself continuously deforms each subset A into another subset A'. Thompson's discussion of biological transformations was couched in terms of such deformations A ~ A ' . In more quantitative terms, such subsets are characterized in terms of a set of numerical measurements which can be performed on them; i.e. in terms of a family of observables Pi. It should be noted that these observables are not defined on the underlying space S, but rather on some family of subsets orS. Each subset A in this family thus defines an n-tuple of numbers (pl(A) ..... p,(A)); the identification of A with its corresponding n-tuple maps the family of subsets into Euclidean n-dimensional space. A continuous mapping of S onto itself manifests itself as a continuous mapping orE, into itself. The details of this correspondence are far from trivial, but for our present purposes, it is sufficient to point out that the geometric transformations considered by T h o m p s o n can be faithfully mirrored by associated transformations in an appropriate n-dimensional Euclidean space. Our next observation is that, in biological terms, the T h o m p s o n transformations establish relations between phenotypes. Thus, the observables pi

572

ROBE RT ROSEN

introduced above would be interpreted biologically as pertaining to phenotypic qualities of the geometric objects (i.e. organisms) on which they are defined. Therefore, we can think of the set of n-tuples I(Pl (A) ..... p,(A )~ as defining a phenotype space p, each point of which specifies the object, or organism, A, on which the observables Pi are evaluated. The transformations introduced by Thompson had the following property: if a transformation T: S ~ S (i.e. a co-ordinate change in the underlying space S) is given, which carried a given subset A ~ S onto another subset A' = T(A ), then in terms of the observables p;, the transformation T induces a mapping

(Pl (A )..... p,(A ) ) ~ (Pl (TA )..... p,(TA )) in the phenotype space P. The effect of this transformation of P can be annihilated by making a coordinate change in P; that is, we can introduce new observables Pl =P~(P~, T) such that for each i, we have

p~(TA)=p~(A). In other words, the new phenotype TA, measured with the new observables p~, appears identical with the original phenotype A, measured with the original observables Pi. Thus, we can already begin to discern a close relationship between the Thompson transformations and the general principles of similitude and corresponding states which we have developed above. Stated roughly, we can say that there is a sense in which the organisms related through Thompson transformations should be dynamically similar. If we pursue this line of reasoning, we see quickly that if this is to be so, then the phenotypic observables Pi must play the role of derived quantities (in the dimensional sense). Thus, there must be underlyingfimdamental quantities, and an equation of state relating them; the transformations T are thus not arbitrary, but arise from variations in the numerical values assigned to the fundamental quantities. Thus, in order to proceed further, we must identify more exactly the biological nature of the fundamental quantities, and of the equation of state which relates them to the derived quantities pi. We can resolve this question by asking: on what does the manifestation of a particular phenotype depend biologically? It certainly depends upon the properties of the genome of the organism, and upon the characteristics of the environment in which this genome is expressed. Moreover, if we accept the hypothesis that a particular genome G expressed in a particular environment E always gives rise to a uniquely determined phenotype, then the quantitative expression of this fact is simply the assertion that there exists an equation of state which relates the phenotype of an organism to its genome, and to the environment in which this genome is expressed.

THEORY

OF BIOLOGICAL

TRANSFORMATIONS

573

Now an environment can be characterized, as was a phenotype, in terms of an appropriate family of observables el ..... % Thus, each environment E is associated with an r-tuple of real numbers (el (E) ..... er(E)); i.e. with a point in Euclidean r-space Er. We shall call the set of all r-tuples which arise in this fashion the environment space d. Likewise, a genome can be represented quantitatively: ultimately, the effect of a genome resides in its capacity to determine the rates at which biological processes occur. Thus, the effect ofa genome can be described in terms of a family of system parameters (rate constants), say gl ..... gm"The space of all of these is a subset of Euclidean m-dimensional space E,, ; we will refer to it as the genotype space G. Under these circumstances, the relation between phenotype, genome and environment assumes the form of an equation of state q ) ( P l . . . . . Pn, g l . . . . . gin, e l . . . . . e r

) = 0.

(25)

Moreover, since we are assuming that each of the phenotypic quantities Pi is i,dit, idually uniquely determined by the genome and environment, this equation of state must also be expressible as n separate relations of the form

q~i(Pi, gl ..... gin, el ..... er) = 0.

(26)

It is significant that already from the relations (26), which appear to embody no special hypotheses, we can begin to draw non-trivial conclusions. For instance: we have seen above that the phenotypic quantities p~ must be regarded as derived quantities in order for a principle of similitude to hold between the phenotypes of related organisms. The relations (26) express the fact that each phenotypic quantity p~ must be regarded as satisfying its own equation of state, relating it with the genome and the environment. But we saw in Section III above that a principle of similitude can hohl only if there exists more than one derit~ed quantity in an equation of state. Thus, in each of the relations (26), there m u s t b e at least one other derived quantity besides p~. Now we have seen that the role of the fundamental quantities is to set the scales for the derived ones. Thus, we can already conclude: either not all genes g~ are fundamental, so that their scales are set by other genes and/or the environment; or not all environmental quantities e~ are fundamental; their scales are set by genes and/or other environmental quantities; or both. Further, we may note that the above formulation allows us to treat the Thompson theory of transformations in truly comparative terms. Thompson's idea has often been paraphrased as follows: "the phenotypes of closely related organisms can be deformed continuously into one another". The crux to making this kind of formulation operational is to assign a definite meaning to the phrase "closely related". In the formulation given above, we can suppose that two organi,~ms are closely related !f their genomes are close: more specifically, given

574

ROBERT ROSEN

two organisms characterized by genomes ( g l . . . . , gin), (g'l ..... g;,) respectively, then they are to be considered "closely related" if and only if we have Igi - g~] < e, for some pre-assigned measure of closeness e. As we shall see, this can have most significant ramifications when (a) not all genomic characters gi are fundamental, and (b) we assume that there is always a Thompson transformation between phenotypes which are closely related in the above sense. It should be noted that this partitition of the variables occurring in an equation of state into genomic, environmental and phenotypic is not without meaning even for non-biological examples. For instance, in the case of the van der Waals equation, we might interpret the parameters a, b, r as comprising the genome of the gas (in the sense that they determine the species of gas under consideration), the state variables p and T as constituting environmentally determined quantities, and the state variable v as representing a phenotype determined by the genome (a, b, r) in the environment (p, T). In this case, it happens that all the quantities comprising the genome may be regarded as fundamental; these set the scales for both the environmental and the phenotypic quantities, which are all derived. Two gases (a, b, r), (a', b', r') are closely related if and only if a, a'; b, b'; r, r' are respectively close; a"mutation", or modification of genomic quantities characterizing the "species" of the gas can always be annihilated by an appropriate change of environment (p, T). In the case of the damped harmonic oscillator, a phenotypic quantity like the frequency can be r e g a r d e d as determined by the "genome" (k,m) and the environmental quantity/~. However, a time-dependent phenotypic quantity, such as x(t), or even a time-independent quantity such as the steady state value x ( w ), requires the introduction of further environmentally determined quantities (in the form of initial conditions) as fundamental, along with the "genome". Moreover, it is useful to regard t itself as being a "genetic" quantity, so that the phenotypic quantity x (t) can be regarded as representing the "expression" of the instant t; and yet we always treat time as a derived quantity. Thus from these examples we can see, as before, that a classification of quantities as fundamental or derived can cut across their classification as "genome" or "environment". Before examining in more detail the application of these concepts to the biological situation, let us briefly review the manner in which natural selection is represented in our situation. Intuitively, given an environment (el ..... e,), selection can be regarded as specifying an optimal phenotype Pi, which minimizes some function F(pi, el ..... er) defined on environments and phenotypes. We can regard the cost function F as a potential function on the possible phenotypes in a fixed environment, which imposes a dynamics on these phenotypes according to the relation dpi dt

OF @i'

i = 1..... n.

(27)

THEORY OF BIOLOGICAL TRANSFORMATIONS

575

On the other hand, the phenotypes are tied to the genome through the equation of state (26). In a fixed environment, the only way in which the phenotype p~ can change according to (26), is for the genotype to change. Thus, selection is actually imposing a dynamics on genotypes; this dynamics can, from (27), be written as

k ~ 1

cpi dgk ?gk dt

CF

i=l,

.,n,

(28)

?Pi

a relation to which we shall return shortly. For our purposes, the important point is that selection allows environments to operate on genotypes. Now we know that the equation of state (26) can be written in the form ~ ( l q 1 . . . . . l-Is) = 0

(29)

where the FI~ are the analogs of dimensionless quantities. We know further, from the preceding section, that each FI~ can be regarded as a function of the fundamental quantities operating on exactly one of the derived quantities. We also know, from Section VII above, that the cost function F, which is based on system behavior, must also be of the form F = F ( H 1 ..... Hs).

(30)

Thus, natural selection can be looked upon in the following manner: in a fixed environment el ..... er, selection imposes a dynamics on genotypes, given by (27) or (28), which generates an optimal phenotype for that environment. When the environment changes (i.e. when we are given a transition el ~e'l ..... er~e¢), selection re-optimizes, generating thereby a new phenotype x~, which is optimal in the new circumstances, and a new genotype, related to the original one by a similarity transformation. Now the D'Arcy Thompson concept of biological transformations asserts that these respective optima are dynamically similar. That is, the environmental transition e i ~ e ~results in a genomic transition gk~ g;, from (27) or (28) such that the His appearing in (30) are left invariant. In other words, the transformation arising from selection must be of the form of a similarity transformation. This indeed was the upshot of our previous work (Rosen, 1962), arguing that selection mechanisms can automatically give rise to the similarity transformations postulated by D'Arcy Thompson. But now let us see how it can happen that an environmental transition can be coupled with a corresponding genomic transition in such a way as to leave the Hi invariant. A moment's consideration will reveal that this is generally possible only if the environmental quantities ei are all fundamental. For it is precisely the definition of the fundamental quantities that arbitrary changes in them can be

576

ROBERT ROSEN

annihilated by an appropriate change, represented by a similarity transformation, in the derived quantities. Thus, if an arbitrary change in environment leads (through selection) always to a phenotype which is similar to the initial one, this means precisely that the environmental quantities are fundamental. This implies further that, with respect to the choice of the ei as fundamental quantities, not all of the genomic quantities can befundamental. For we saw earlier that we require at least two derived quantities in order to have a meaningful concept of similarity at all. Furthermore, any genomic quantities gi which are Jimdamental cannot be inroh, ed in the selection mechanism leading to the generation of an optimal phenotype xi in a given environment. This is because such quantities, being genomic, are not altered by the environmental shift, and, being fundamental, can remain constant while the appropriate similarity transformation is carried out on the remaining derived quantities. This curious consequence means that genes ./imdamental Jor a phenotypic quantity are unaffected by selection mechanisms operating on that quantity; but if such a gene g should change (e.g. through mutation), all the derived quantities whose scale is set by g must change by a similarity transformation (i.e., selection will result in a modification of every such derit:ed gene in the genome). Now let us look again at the selection dynamics (27), (28) in the light of(30) and the discussion given above. In terms of the Hi, we can write this dynamics as dFI i dt

~F 0H i "

(31)

Now by the remarks of the previous paragraph, it is no loss of generality to treat all the genomic quantities as if they were derived (since the fundamental ones simply remain fixed throughout the discussion). Thus, by previous arguments, each H will contain exactly one derived quantity, operated on by a function of the fundamental quantities: thus we can write l-Io = ~bo(eI ..... er)Pi, l-I l = (/)1 (el .....

er)gi,

(32)

FIr =~r(e~ ..... e~jg~. Now since dHi dt

_ ~I]i . dpi t Pi

dt

-q- ~ PHi~ dgk dt k = 1 Cgk

i= O, . .., r,

(33)

THEORY OF BIOLOGICAL T R A N S F O R M A T I O N S

577

we see from (32) that ?F [?FI i

dpi dgi - 4 o ( e l ..... e~) dt +~i(el,...,e~) dt "

(34)

Now the functions qSk(e1..... er) are simply the scale factors, whereby the fundamental quantities set the scales for the derived ones; they are known if the similarity transformation relating phenotypes is known. Moreover, we have such a set of relations of the form (33) for every phenotypic quantity p~; there are by hypothesis n such quantities. At optimality (i.e. when ~T/'?[Ii=0) the relations (33) for all the phenotypic quantities become a system of equations linear in the derivatives dp/dt, dg;/dt, which we would like to possess a unique solution in which all these derivatives vanish. But it will be seen that this places strong restrictions on the m,nber of phenotypic quantities which can in general be controlled by a genome of definite size, in order for this to be true. An exact argument along these lines is complicated, because the number of derived quantities appearing in each of the equations of state (26) can vary, but the main line of this kind of argument should be clear in any case. And it should be also observed that, if the rates of change of phenotypic quantities and genetic quantities are known from empirical observation, then it is in principle possible to solve the equations (33) for F, to explicitly determine the criterion of selection actually embodied in the situation being considered. This too is' a more general form of an argument given previously (Rosen, 1967). Let us look at another ramification of the above argument. We noted earlier that one formulation of the Thompson hypothesis, in evolutionary terms, is that "closely related organisms should be similar"; and we agreed that "closely related" should be defined in terms of the genomes of the organisms. On the other hand, we have argued that a change in environment governed by a selection mechanism imposes a dynamics on the genome, such that the optimal organism in the new environment shall be similar to the original organism in the original environment. What guarantee do we have, in general, that the original and transformed genomes will be close? For fundamental genes, "this is no problem, since as we have seen, they are not affected by the selection dynamics. On the other hand, for derived genes, we have only the similarity relation, of the form g~_ qS(el..... e~) gi. qb(el ..... er) /

/

Thus, how close the transformed genome is to the original one is a flmction entirely of how the scales are set by the fundamental quantities, and of how close the transformed environment is to the original one. On the other hand, we call see

578

ROBERTROSEN

that since at least some genes must be regarded as derived, even a small change in these derived genes can lead to phenotypes which are not similar in any em',ironment. We shall conclude this section by briefly considering the concept of critical values in an evolutionary context. As we saw, the critical values represent situations in which the values of derived quantities are determined entirely by the values of the fundamental quantities; they thus represent states in which additional conditions are satisfied besides the equation of state. As such, they are typically associated with degeneracy, so that small perturbations of the derived quantities away from the critical values can lead to situations which cannot be related by similarity transformations. Such situations are associated with phase transitions, and similar qualitative changes of behavior. In a fixed environment, in which all the environmental quantities are taken as fundamental we have seen that arbitrary variations in the fundamental genes cannot, by definition, lead to such behavior. On the other hand, arbitrary changes in the derived genes can, in general, manifest such qualitative changes of behavior. In a situation governed by selection, the phenotypes corresponding to such qualitatively different organisms must necessarily be non-optimal, and hence will be eradicated by the selection. It is only if the selection criterion changes that such qualitatively different behavior can be maintained. We can turn this kind of argument around: if two organisms are dissimilar, then (a) the dissimilarity must arise from genomic modifications involving derived genes, and (b) the selection mechanisms imposed upon them must be different. In the preceding sections, we have barely scratched the surface of the implications of the general concept of dynamical similarity and its implications for biological processes. For instance, we have not explicitly considered what happens when the equation of state governing a biological process involves temporally varying quantities, as in development (beyond a few hints in Section VII and the remark that time could be regarded as a genetic quantity). We intend to do this in subsequent papers, particularly in constructing developmental analogs of the selection arguments given above, by considering similarity transformations between subsequent developmental stages. However, enough should have been said to indicate just how powerful is the confluence of D'Arcy Thompson's concept of biological transformation, the theory of similarity, and a dynamics imposed by selection.

LITERATURE Buckingham, E. 1915. "On Physically Similar Systems." Phys. Rev., 4, 345 370. Fowler, D. H. 1972. "'The Riemann Hugoniot Catastrophe and the van der Waals Equation." In Towards a Theoretical Biology, Ed. C. H. Waddington, Vol. 4, pp. 1-7. Chicago: Aldine Atherton Inc. Rashevsky, N. 1960. Mathematical Biophysics. 3rd Edition. New York: Dover Publications.

THEORY OF BIOLOGICAL TRANSFORMATIONS

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Rosen, R. 1962. "'The Derivation of D'Arcy Thompson's Theory of Transformations from the Principle of Optimal Design." Bull. Math. Biophys., 24, 279- 290. - . 1967. Optimality Principles in Biology. London: Buttcrworths. Stahl, W. R. 1962. "'Dimensional Analysis in Mathematical Biology." Bull. Math. Biophys., 24, 8 l 108. Thorn, R. 1975. Structural Stability and Morphogenesis. (Trans. D. H. Fowlerl. Reading: W. A. Benjamin, inc. Thompson, D.Arcy W. 1917. On Growth and Form. Cambridge: The University Press. RECEIVED 3-17-77 REVISED 4 - 2 1 - 7 7

Dynamical similarity and the theory of biological transformations.

thd/etm ,# M,lthem,¢ticu/ Bu,/ogv Vol. 40, p p 549 579 Pergamon Pte,b Ltd. 1978. Printed in Great Britain (l- S~,uictx lot Mallicinatical Biolog 5 00...
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