Analysis of eigen's equations for selection of biological molecules with fluctuating mutation rates

BIYLLETIN OF MATHENIATICAL BIOLOGY V O L U ~ 39, 1977

ANALYSIS OF E I G E N ' S E Q U A T I O N S F O R SELECTION OF BIOLOGICAL MOLECULES W I T H F L U C T U A T I N G MUTATION RATES

[] B. L. J o ~ s

Department of Physics, Simon Fraser University, Burnaby, British Columbia, Ganada

I n t h i s p a p e r w e consider E i g e n ' s e q u a t i o n s for t h e s e l e c t i o n a n d e v o l u t i o n of s e l f - i n s t r u c t ing, m a c r o m o l e e u l a r s y s t e m s . W e c o n s t r u c t e x a c t , a s y m p t o t i c solutions for t h e e q u a t i o n s w h e n t h e r a t e coefficients a n d e r r o r d i s t r i b u t i o n s are c o n s i d e r e d as f u n c t i o n s of t i m e . I m p l i c a t i o n s for selection are discussed.

l. Introduction. In a previous paper (Jones, Enns and Rangnckar 1976; hereafter referred to as J E R ) we examined a set of nonlinear rate equations (devised b y M. Eigen (Eigen 1971)) which describe the process of selection and evolution of a set of self-reproducing, macromolecular, information carriers. Exact solutions to the equations for the case of constant rate parameters and constant error distributions were constructed. Criteria for selection in steady state was discussed and compared with Eigen's approximate results. In the case of constant coefficients the macromolecular system, starting with any initial distribution, approaches a well-defined steady state. The steady state is characterized b y the largest eigenvalue of a certain matrix which determines the reproduction of the macromolecules in the presence of constraints. In the limit of relatively small error distribution the species with the largest "selective value", i.e. the largest eigenvalue referred to above, dominates in steady state. The question now arises as to what extent the above ideas remain valid when the error distribution is considered as time-dependent. In reality, of course, mutation rates m a y vary considerably in time. They will depend on external 311

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parameters such as temperature, pressure and radiation. New mutations m a y thus be induced as a result of fluctuations in these or other parameters. Also there is an inherent randomness in the error process itself. In this paper we consider selection of self-instructing, macromolecular systems when the rate parameters and error distribution are functions of time. We construct exact asymptotic solutions to the rate equations and show t h a t under certain restrictions, which seem appropriate for selection, the system is found to fluctuate in time about a well-defined state. The state is characterized by a kind of time-averaged "selective value". Again, for small error coefficients, the macromolecular species with the largest mean selective value dominates the population.

II. Eigen's Equations for Selection. We consider a collection of macromolecular information carriers in which each molecular species is able to instruct its own synthesis. It is assumed t h a t the system is open and maintained in a nonequilibrium state by a continuous flow of energy and matter used in the synthesis of the macromolecules. Eigen's equations for the generation and turnover of the various molecular species are N

Xe(t) = We(t)Zk(t)+ ~ q~kz(t)Xt(t)-E(t)Xk(t).

(1)

Xe(t) represents the concentration of species k (k = 1, 2, 3 , . . . N for N species) at time t and X~ = dXk/dt. W~ = AkQ~--Dk is called the "selective value". A ~ is a rate parameter which describes the synthesis of all substances (including mutants) which are formed as a result of specific instructions by templates associated with species k. Qk is a " q u a l i t y factor" which tells which fraction of the processes leads to precise copies of species k. The fraction 1 - Q~ determines the formation of mutants which still resemble the master copy k. D~ describes the decomposition of species k due to degradation processes. The quantities ~0~t represent the rates of spontaneous production of species k arising from errors in the replication of species 1. I t is assumed t h a t all noninstructed formation of any molecular species is completely negligible. As a result we have the following conservation relation for the total error production

Z A~(1 -Q~)X~ k

=

Z Z q~kzX~. k

(2)

l

The term EX~ in (1) is a dilution term which is used to impose external constraints on the system. The dilution flow is taken to be proportional to the concentration and the proportionality is assumed to be the same for all species. In the following we shall, following Eigen, consider the selection constraint

A l q A L Y S I S O F E I G E l q ' S EQUATIOI~IS

Xk = n = constant.

313

(3)

/c

Then the dilution term takes the form 2V

E,(t) = y, E ~ X k / n , E~ = A ~ - D ~ .

(4)

k=l

The quantity ~(t) is called the mean productivity b y Eigen. In the following analyses we shall assume that the rate parameters do not depend further on concentration b u t that they m a y be functions of time.

I I I . Asymptotic Solutions of the Rate Equations. the solution to (1) can be expressed in the form Xk(t) = n Y ~ ( t ) / ~ Y~(t),

We have shown in JEI~ that

Xk(O) = Yk(O),

(5)

where Yk(t) satisfies the linear equation

Y~(t) = ~ Akz(t)Yz(t).

(6)

1

In (6)

A~(t) = W~(t)Sk~+gk~(t),

~p~ = 0

(7)

and 5 ~ is the usual Kroneeker delta, viz. 5~ = {10 k = l k#l. If the coefficients A k, are constants, (6) can be easily solved b y diagonalization of A ~z and the complete solutions are given in J E R . In the following we shall consider the time-dependent case. A general solution of (6) ibr arbitrary A(t) can not be obtained; however, provided that A(t) satisfies certain asymptotic properties (i.e. as t --~ co) exact asymptotic relations can be deduced (See Bellman 1953). We state the necessary restrictions below. We require that A(t) be represented as a sum of two matrices, A (t) = A o(t) + Al(t) where A 0(t) and A l(t) are chosen such that

(a) S°° [[AI(t)I [ dt < m, i.e. bounded. (b)

AO(t) - + c o n s t a n t m a t r i x as t --~ co a n d 5~ I]d/dt A0(t)l[ dt < oo.

(e) The time-dependent eigenvalues 2~(t) of AO(t) are assumed nondegenerate and satisfy 5t Re ~ ( t ) dt -~ either + co as t --~ m. In (a) and (b) the double bars denote the norm of the matrix i.e.[IAl[{ =

E,,j IAbl. Consider now the applications of the above to the selection problem. One can well imagine selection processes in which as a result of an environmental change B

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the matrix W(t) + ~o(t)might initially change with time and then fluctuate about some constant value, i.e. certain error coefficients ~pkz(t) might initially increase from zero thereby introducing new m u t a n t species into the system. Thus the properties (a) to (c) seem appropriate for selection problems and in the following we shall assume t h a t the matrix defined by (7) can be put into this form. We write

AO(t)--- WO(t)+~po(t),

Al(t) = Wl(t)+tpl(t)

(8)

Let qk~(t) denote the kth component of the eigenvector belonging to the eigenvalue 2~(t) of the matrix AO(t). (Note that q(t)-~constant matrix at large t.) Then by applying standard matrix transformation theory it is easy to show that the solution to (6) can be expressed (see Bellman, 1953) as y~(t) = ~ qej(t)Zj~(t)C~ exp f~ 2z(tl) dtl ju

j,l

(9)

I n (9) Ct are constants determined from the initial conditions by inverting at t--0.

Cz = E (q(O)Z(O))~ X~(O).

(10)

k

The quantities Z~(t) are the characteristic solutions of a set of properly chosen integral equations which we now define. Let ~ ( t ) = ~i(t)- 2~(t), c ~ = Lim Re t~Qo

~e(tl) dtl J

and define the matrix R~j(t) as components of

R(t) = q-l(t)Al(t)- q- l(t)q(t).

(11)

I f aik --~ ~ then Z,~(t) satisfies

Z~(t) -- [ 5 ~ - ~ j

exp

f:

]

2i~(s) ds Rij(tl)Zj~(tl) dtl

• 1

(12)

and if ~ } --~ - co,

Z~(t) = ~ , f~at [ e x p ~tt 2~(s ) as 3 R~,(t~)Z,~(t~)dt~

(13)

where a is a properly chosen constant. Equations (12) and (13) can be solved by successive interactions. Bellman shows that as a result of the conditions (a) to (c) that such a series converges and in particular as t --~ or,

Z~Tc(t) --~ 5~ + A~(t),

(14)

where A~(t) is a matrix of order unity. Using (9) in (5) we m a y now write the

A N A L Y S I S OF E I G E N ' S E Q U A T I O N S

315

complete time-dependent solutions of (1) for the concentrations as

~t~(t)C~ exp ~ t ~(tl) dtl X~(t) = n ~

3o

(15)

k,l

where we have put f~(t) =

q(t)Z(t).

IV. Selection with Fluctuating Error Coefficients. We now consider (15) in the limit of large t. Since q(t) approaches a constant at large t, ~(t) remains finite. Thus the time-dependence of Xk(t) at large t is determined by the time integral of the eigenvMues ,~(t). It is convenient to define the time averaged eigenvMues Iz = T

;t~(tl)dtl,

t -+large.

(16)

We let Im denote the largest mean eigenvalue. (Note since A ° is a positive matrix, the eigenvalue with the largest real part will be real (see Bellman 1970).) I t follows from (15) that when ,~mt >> 1,

xk(t) ~

n~(t)

(17)

'E a ~ ( t ) ' k

where n(t) -+ q(m ) (1 + A(t)) as t -~ m. Thus the distribution fluctuates about a state characterized by the largest mean eigenvMue ~ . The parameter ~ is tho appropriate generalized "selective value" for this time-dependent problem. In the limiting ease where mutation effects are small, i.e. if ~okz(t) ~ W~(t), standard perturbation methods give

ot

o t

,~k(t) ~ W°(t)+ Y~ ~0z ( ) ~ ( ) 0 z W g t ) - W~(t) o (p~t(t) qkl(t)'z 5 ~ + 'W~( o~----o ) - Wk(t) •

(Is) (19)

Iteration of (12) and (13) gives

A~l(t) ~ f t f e x p f: ~(s)ds]R~(tl)dtl,

(20)

where

R~,(t) ~ 5~W~(t)+ q~(t)-q~z(t). 1 Thus Ftkm(t) will be nearly diagonal and (17) shows that one species k = will dominate the population.

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N o w consider t h e stability of this state. We n o t e t h a t t h e coefficients of exp St A~(8)ds in (15) are nonvanishing for all l, i.e. all species, which were e i t h e r present initially or were i n t r o d u c e d prior to t i m e t via a m u t a t i o n . Consider now a s i t u a t i o n where a m u t a n t ~h + 1, with A2+i > ~ , is i n t r o d u c e d as a result of a r e p r o d u c t i o n e r r o r a t a t i m e to >> 0. T h e n for t < to t h e system reaches the stable state defined b y (17) a n d is d o m i n a t e d b y species ~h. H o w e v e r , for t > to a new e x p o n e n t i a l f a c t o r - - e x p ~ + i t appears in (15). T h u s an instability appears a n d grows u n t i l e v e n t u a l l y the m u t a n t ~ + 1 dominates t h e population. T h e system n e v e r r e t u r n s to t h e original states so long as 5~+i > ~&. T h e above illustrates d i r e c t l y how evolution can proceed as a series of instabilities due t o fluctuations which produce m u t a n t s w i t h larger a n d larger "selective values". Some of these results have been discussed q u a l i t a t i v e l y b y Eigen. The explicit t i m e - d e p e n d e n t solutions, however, give a clearer p i c t u r e of the e v o l u t i o n a r y b e h a v i o u r . LITERATURE Bellman, R. 1953. "Stability Theory of Differential Equations". New York: McGraw-Hill. • 1970. "Introduction to Matrix Analysis". New York: McGraw-Hill. Eigen, M. 1971. "Self-organization of Matter and the Evolution of Biological Macromolecules". Naturwiss., 58, 465-523. Jones, B. L., R. H. Enns and S. S. Rangnekar. 1976. "On the Theory of Selection of Coupled Maeromolecular Systems". Bull. lYlath. Biol., 38, 15-28.

RECEIV]~D 2-27-76