Linear Metabolism - Repair System

NOT FOR QUOTATION WITHOUT THE PERMISSION OF THE AUTHOR

Linear Metabolism

-

Repair Systems

John Casti

September 1986 WP-86-51

Working Papers a r e interim r e p o r t s on work of the International Institute f o r Applied Systems Analysis and have received only limited review. Views o r opinions expressed herein do not necessarily r e p r e s e n t those of t h e Institute o r of its National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

The Theory of Manufacturing study had as one of i t s objectives t o e x p l o r e a l t e r n a - tive formal system s t r u c t u r e s f o r c h a r a c t e r i z i n g modern manufacturing processes.

One such s t r u c t u r e i s based upon t h e metaphor of a living cell. In t h i s Working Pa- p e r , t h e a b s t r a c t mathematical s t r u c t u r e of t h i s metaphor is developed as a basis upon which t o build a formal t h e o r y of cellular processes.

T.H. Lee D i r e c t o r

Abstract

Metabolism-repair systems r e p r e s e n t a formal mathematical framework f o r r e p r e s e n t a t i n g c h a r a c t e r i s t i c p r o p e r t i e s of living systems such as r e p a i r , repli- cation, adaptation and s o f o r t h . In this p a p e r , t h e c o n c r e t e realization of such s t r u c t u r e s is developed in t h e c a s e when t h e system "metabolism" is l i n e a r . Expli- c i t r e s u l t s are given t o show when system r e p a i r operations c a n c o u n t e r a c t en- vironmental and metabolic fluctuations. Additional r e s u l t s pertaining t o t h e repli- cation operation and t h e possibility f o r "Lamarckian" inheritance are also given, t o g e t h e r with a formal demonstration of t h e i n c r e a s e in complexity as we p r o c e e d fron; t h e p r o c e s s e s of metabolism t o r e p a i r t o replication. The p a p e r concludes with a discussion of s e v e r a l application a r e a s , t o g e t h e r with a consideration of s e v e r z l conceptual and mathematical questions requiring attention f o r f u r t h e r development of t h i s non-Newtonian systems paradigm.

Linear

Metabolism

-

^Repair S y s t e m s

John Casti

1. Extending the Newtonian Paradigm

A number of a u t h o r s [I-31 h a v e r e c e n t l y (and n o t s o r e c e n t l y ) pointed o u t a v a r i e t y of deficiencies i n h e r e n t in t h e classical Newtonian paradigm of mechanics r e g a r d i n g i t s utility in d e s c r i b i n g living systems, both biological as well as s o c i z l and behavioral. In p a r t i c u l a r , it h a s b e e n r a t h e r convincingly a r g u e d t h a t t h e Newtonian world view, as exemplified in classical p a r t i c l e mechanics, s a y , h a s no r o l e f o r any t y p e of a n t i c i p a t o r y b e h a v i o r on t h e p a r t of e i t h e r o b s e r v e r s o r de- cisionmakers [4]. F u r t h e r m o r e , t h e c r u c i a l biological a c t i v i t i e s of r e p a i r a n d re- plicztion d o not f i t in any n a t u r a l way i n t o Newton's "WeLtanschauung," leading to t h e conclusion t h a t a n extension of t h e Newtonian paradigm, comparable in s c o p e and impact t o t h e e x t e n s i o n s o f f e r e d by both quantum mechanics and r e l a t i v i t y t h e o r y in physics, is long o v e r d u e f o r mathematically c a p t u r i n g t h e e s s e n c e of bio- logical, social a n d b e h a v i o r a l phenomena.

About 30 y e a r s a g o in 2 series of p a p e r s devoted t o r e l a t i o n a l c e l l models 15- 71, Rosen introduced t h e notion of a metabolism-repair (M,R)-network in a n a t t e m p t t o show formally how t h e f e a t u r e s of r e p a i r and r e p l i c a t i o n could b e naturally in- duced solely from a c e l l ' s metabolic machinery. In subsequent work, i t was a l s o pointed out how a n t i c i p z t o r y b e h a v i o r a l modes a l s o followed in a s t r a i g h t f o r w a r d manner from t h e (h! ,R)-f ormalism. Ucf o r t u n a t e l y , t h e formalism s e t up by Rosen and. developed by o t h e r s

18-91

was p u r e l y r e l a t i o n a l ; i.e., i t d e a l t with t h e mnc-

tional c h a r a c t e r i s t i c s of t h e cell independent of i t s s t r u c t u r a l organization. In o r d e r t o make c o n t a c t with r e a l m a t e r i a l o b j e c t s , i t w a s n e c e s s a r y t o e x p l o r e means f o r realizing a b s t r a c t (M,R)-systems in h a r d w a r e , o r g a n i c o r otherwise.

The initiz! a t t e m p t s in t h i s d i r e c t i o n led t o r e a l i z a t i o n s of (M, R)-systems a s automa- ta, but with a highly non-canonical state-space [LO-111. P e r h a p s due t o t h e discouraging r e s u l t s which followed from t h e s e somewhat u n f o r t u n a t e realizations, t h e t o p i c seems t o h a v e d i s a p p e a r e d from t h e l i t e r z t u r e and died, what in o u r opin- ion, i s a v e r y p r e m a t u r e d e z t k .

In t h i s p a p e r , we a t t e m p t t o r e s u r r e c t t h e t h e o r y of (K.R)-systems making use of t h e vastly d e e p e r understanding of t h e n a t u r e of canoniczl rezlizztions a c - q u i r e d o v e r t h e p a s t d e c a d e o r s o , s t a r t i n g with t h e pioneering work of Kalman in t h e e a r l y '60s 112-14). By making use of c o n c e p t s a n d tools t h a t w e r e totzlly unk- nown at t h e time of Rosen's initizl work, we put t h e (M,R)-set-up on f i r m e r system- t h e o r e t i c grounds, while at t h e same time answering a number of questions t h a t w e r e only in t h e realm of speculation at t h e time of t h e e a r l y p a p e r s [5-111. While o u r r e s u l t s are p r e s e n t e d only f o r t h e simplest c a s e of Linear (M,R)-systems, t h e extension t o nonlinear situations follows along t h e szme lines as t h e extensiozs from l i n e a r t o nonlinear in system t h e o r y as given, f o r example, in [15,20]. I t i s o u r e x p e c t a t i o n t h a t t h e framework set f o r t h h e r e will s e r v e as a point of d e p a r - t u r e f o r t h e development of t h e kind of extension t o t h e Newtonian paradigm t h a t wil! s e r v e t h e same r o l e i n biology and t h e social s c i e n c e s t h a t t h e S c h r o d i n g e r equation znd t h e Lorentz t r a n s f o r m a t i o n s e r v e d f o r physics.

The p z p e r is organized z c c o r d i n g t o t h e following scheme. Section 2 p r e s e n t s t h e basic ideas surrounding (M,R)-systems as originally developed by Rosen. In Section 3, 2 brief review of t h e r e a l i z a t i o n problem a n d t h e c o n s t r u c t i o n of canon- ical s t a t e - s p a c e models i s giver; f o r l i n e a r dynamical systems. Tne principal new r e s c l t s of t h e p z p e r zre p r e s e n t e d in Sections 4 a n d 5, w h e r e we give e x p l i c i t

c h a r a c t e r i z a t i o n s of t h o s e l i n e a r systems which c a n b e e x t e n d e d t o l i n e a r (M,R)- systems, t o g e t h e r with a discussion of how system complexity i n c r e a s e s as we at- tempt t o superimpose additional biological s t r u c t u r e upon t h e basic metabolic machinery. Along t h e way, i t is shown t h a t Rosen's original scheme f o r t h e c e l l ' s r e p l i c a t i o n mechanism c a n only b e possible f o r 2 v e r y limited c l a s s of (M,R)- p r o c e s s e s . Finally, in S e c t i o n s 6 a n d 7 w e discuss t h e extensions of o u r r e s u l t s t o nonlinear (M,R)-processes, as well as issues pertaining t o a network of cells a n d t h e stability a n d c o n t r o l problems t h a t such s t r u c t u r e s g e n e r z t e . The p a p e r con- cludes with a n indication of s e v e r a l application areas where (h!,R)-systems should p r o v e valuable in formalizing a v a r i e t y of important p r a c t i c a l questions.

2. MetaboIism

-

R e p a i r N e t w o r k s

Consider a collection of N "cells", e a c h of which a c c e p t s a v a r i e t y of inputs a n d p r o d u c e s z s p e c t r u m of outputs. Assume t h a t at l e a s t o n e c e l l a c c e p t s inputs from t h e "environment" and a t least one cell p r o d u c e s o u t p u t s t h a t are s e n t t o t h e environment. F u r t h e r , suppose t h a t e v e r y c e l l a c c e p t s e i t h e r environmental in- p u t s or h a s as i t s inputs a n output from at l e a s t o n e o t h e r cell; similarly, assume t h a t e a c h c e l l p r o d u c e s e i t h e r a n environmental o u t p u t o r h a s its output utilize6 as a n o t h e r c e l l ' s input. Such a network might look like F i g u r e 1 (with N = 5 ) . H e r e we h a v e t h e c e l l s

MI -

^M5, t o g e t h e r with t h e two environmental inputs ol and 02: as well as t h e single environmental output yl. We call such a network a "metabolic"

network.

I t is r e a s o n a b l e t o suppose t h a t any cell in s u c h a network will h a v e a finite lifetime a f t e r which i t will b e removed f r o m t h e system. When t h i s happens, all c e l l s whose input depen&s upon t h e output from t h e "dead" cel! will a l s o b e a f f e c t - e d , ultimately failing ic t h e i r metabolic r o l e , as weli. In Figure 1, f o r i n s t a n c e , if

Figure 1. A Metabolism Network

t h e cell MI fails, then s o will MS,

MJ, M4

and M5 all of whose inputs ultimately depend.

upon M l l s output. Any such cell whose failure r e s u l t s in t h e failure of t h e e n t i r e network i s called a central component of t h e network.

Now l e t us suppose t h a t we associate with each metabolic component Mi, a com- ponent Ri whose function is t o repair Mi. In o t h e r words, when Mi fails t h e r e p a i r component Ri a c t s t o build a copy of

Mi

back into t h e network. The Ri a r e consti- tcted s o that each Ri must receive a t least one environmental output from t h e net- work and, in o r d e r t o function, Ri must receive al of its inputs. Thus, in Figure 1 each Ri must receive t h e sole environmental output yl. Note zlso by t h e second

condition t h a t any c e l l M i , whose r e p a i r component Ri r e c e i v e s M i ' s o.;ltput as p a r t of i t s input, cannot b e built back into t h e network. We w i l l c a l l s a c h a cell n o n - r e e s t a b l i s h a b l e . Thus, t h e ce!: M 2 i s non-reestablishable, while c e l l M 5 is reestab- l i s h a b l e .

Introduction of t h e r e p a i r components f R i ] g e n e r a t e s t h e following b a s i c question: who r e p a i r s t h e r e p a i r e r s ? I t would l e a d t o a u s e l e s s infinite r e g r e s s t o i n t r o d u c e a n o t h e r l e v e l of r e p a i r mechanisms, but what is t h e a l t e r n a t i v e ? N a t u r e ' s solution t o t h e problem i s t o make t h e r e p a i r components self-replicating.

Before Ri d i e s , t h e r e p l i c a t i o n mechanism built i n t o Ri a r r a n g e s t o p r o d u c e a c o p y of Rl, which t h e n t a k e s R i l s p l a c e in t h e network. Such networks are called (X,R)- systems.

The elementary c o n c e p t s introduced a b o v e a l r e a d y allow t h e following in- t e r e s t i n g r e s u l t s t o b e e s t a b l i s h e d [ 5 , 7 ] :

Theorem 2 (Rosen). E v e r y f i n i t e (M,R)

-

n e t w o r k c o n t a i n s a t l e a s t one n o n - r e e s t a b l i s h a b l e component.

C o r o l l a r y . If an (M,R)

-

n e t w o r k c o n t a i n s e x a c t l y one n o n - r e e s t a b l i s h a b l e component, t h e n t h a t component i s c e n t r a l .

Thus, we s e e t h a t e v e r y (M,R)-network must contain some c e l l s t h a t cannot b e built back into t h e system if t h e y fail. F u r t h e r , if t h e r e are a s m a l l number of such cells, t h e n t h e y a r e likely t o b e of prime importance t o t h e o v e r a l l functioning of t h e system. This last r e s u l t h a s c l e a r implications f o r policies devoted t o keeping e v e r y component of a system alive (politicians and o t h e r s o c i a l r e f o r m e r s : p l e a s e note!). I t may b e much b e t t e r to.allow some cells t o f a i l r a t h e r t h a n r u n t h e r i s k of i n c u r r i n g a global system f a i l u r e by t r y i n g t o prop-up weak, non-competitive com- ponents which, by T h e o r e n 1, c a c ' t all b e saved in any c a s e .

- 6 -

Let us now t u r n t o a n examination of t h e simplest possible (M,R)

-

system com- posed of a single component (N = 1). diagram ma tical!^, we h a v e

o r , more a b s t r a c t l y ,

w h e r e C1

=

icput set,

r ⁼

o u t p u t set, f : R ^--,

r

⁽⁼ metabolic map),

Pf : --, H ( R ,

I?)

(= r e p a i r map) with H ( R ,

I?) =

set of all physicalLy feasible meta- bolic maps. H e r e we s u b s c r i p t t h e r e p a i r map by f t o indicate t h a t t h e r o l e of Pf is t o p r o d u c e t h e metabolism f when t h e metabolic p a r t of t h e system r e c e i v e s i t s

" c o r r e c t " input o E R. W e s h a l l r e t u r n t o t h i s point in d e t a i l in S e c t i o n s 4 and 5.

The f i r s t point t h a t arises i s how t o a b s t r a c t l y c h a r a c t e r i z e t h e system repli- cation map. Arguing biologically, t h e r e p a i r component

Pf

r e p r e s e n t s t h e system's genetic component 2x6 t h e job of t h e r e p l i c e t i o x mz; i s t o n s e t h e s y s t e r r , ' ~ a e t z - bolic machinery ( R , r , H ( R ,

T))

a n 6 g r o c e s s i t i n t o z copy of Pf E E ( r , H(S,

9).

Puttixg t h e s e r e m a r k s t o g e t h e r , we s e e t h z t t h e r e p l i c e t i o n map, call i t

Bf:

mnst act zs

Thns; t h e z b s t r a c t d i z g r a ~ c h z r z c t e r i z i n g t h e entiye (K,R)-systez is

f P f

s:

n - r -

^{H ( R , Y)}

- ^{~ ( r , ~ ( n , r))}

)metzSo!ism { [ r e p a i r { [ r e p l i c z t i o n {

In what follows, we shzl! b e c o n c e r n e d with putting c o n c r e t e "meat" on t h e a b s t r a c t

"skeleton" of t h i s diagrzm.

B e f o r e discassing some of t h e questioxs s d r r o u n d i n g t h e b e h e v i o r of such a system, two importznt points should b e noted: 1) if w e d e l e t e t h e r e p z i r 2nd r e p l i - cztior, corzponents of t h e dizgram, w e z r e !eft with t h e s t a n d a r d s t a r t i n g point of Iu'ewtonizn mechznics 2nd modern systerr. t h e o r y , nzmely, p u r e metabolism; t h u s . t h e single-component (M,R)-network r e p r e s e c t s a genuine extension of t h e classi- cz! 7zradigm; 2) t h e r e i s no s e t - t h e o r e t i c d i f f e r e n c e between metabolisn? a n d r e p z i r : t h e y both r e p r e s e n t mz?s between a b s t r a c t sets. Bioiogiczlly, t h i s sug- g e s t s t h a t t h e r e may b e n o i n t r i n s i c d i f f e r e n c e between a c e l l ' s metzbolic a n d i t s genetic a c t i v i t y . W e s h a l l e x p l o r e t h i s point ir, more d e t a i l l a t e r or,.

The i m p o r t a n t questions surrounding t h e r e p z i r a s p e c t s of t h e 2b0ve t y p e of (K,R)-system r e v o l v e a b o n t t h e d e g r e e t o which t h e r e p z i r znd r e p l i c a t i o n com- ponents of t h e system c z n p r e s e r v e t h e metabolic b e h z v i o r ir, t h e f a c e of fluctua- tions i:! t h e system's i c p n t c: o r d i s t u r b a n c e s t o its metzbolism f .

L e t ' s t a k e 2 look at 2 few a s p e c t s of t h i s question t h z t we s h z l l a d d r e s s in cozsid- e r a b l y more d e t a i l in Section 4.

S t a b l e Metabolic m e r a t i o n s i n C h a n g i n g E n v i r o n m e n t s

-

imagine t h e situa- tior, in which t h e cell's "usuz!" input o is d i s t u r b e d t o a new input

z.

The conditio:!

f o r s t z b l e operatior, of t h e ce?! is f o r t h e environmect o t o b e such t h a t

Pr(f(o)>

=

f , ('1

i.e. t h e metabolic s t r u c t u r e f i s s t a b l e in t h e envirorment o in t h e s e n s e t h a t t h e r e p z i r mechanism Pf alwzys r e g e n e r a t e s f when t h e environmeztz! input i s o. W e woulc! s z y t h z t z?! ^t: E 3 sztisfying (*) form 2 s t z b l e environment f o r t h e cell.

Now s u p p o s e t h a t t h e new environment E f o . Then (*) will hold only if e i t h e r f (0)

=

f ( E ) o r Pf(f(Ej))

=

f .

The f i r s t c a s e i s t r i v i a l in t h e seme t h z t t h e o b s e r v e d p r o d u c t s of t h e c e l l a r e in- v a r i a n t t o t h e c h a n g e of environmental inputs. If f ( o ) # f ( z ) t h e n t h e c e l l ' s out- p u t s are n o t s t z b l e with r e s p e c t t o t h e c h a n g e of environment a n d we must c o n s i d e r t h e r e p a i r mechanism t o see w h e t h e r o r n o t t h e environmental a l t e r a t i o n s c a n b e compensated f o r in t h e s e n s e t h a t

P f ( f ( Z ) ) =

-

f # f ,

with

T(Z) =

f ( o ) , i.e., w h e t h e r t h e g e n e t i c mechanism w i l l p r o d u c e a new metabol- ism

i

which duplicates t h e o u t p u t of f , b u t with t h e input 3 r a t h e r t h a n o. In t h i s c a s e , t h e e n t i r e metabolic a c t i v i t y of t h e cell would b e permanently a l t e r e d if we h a 6

On t h e o t h e r hand, if we h a 6 :(E)

=

f ( o ) o r , more g e n e r a l l y , pf(i(z))

=

f ,

t h e n t h e cel!'s metabolism would only u n d e r g o p e r i o d i c c h a n g e s in time.

Finzlly we could h a v e t h e situation in which

A

-

P,(T(E))

=

f

z

f , f

a n d , i t e r a t i n g t h i s p r o c e s s , w e may see t h a t a n environmental c h a n g e will c a u s e t h e c e l l t o wznder a b o u t in t h e set

3(n, r),

changing i t s input/output b e h a v i o r t h r o u g h a sequence of metabolic p r o c e s s e s f (I) ,f ^(') ,f (3),

... .

This "hunting" p r o - c e s s will t e r m i n a t e if e i t h e r

(i) t h e r e e x i s t s a n N s u c h t h a t

P f (f(N)(E))

=

^?(N)

o r

(ii) t h e r e e x i s t s an N such t h z t

P, (f ("(Z)) = f (N -k) , k

=

^1,2,..

.

, N -1

.

In case (i) t h e cell becomes s t a b l e in t h e new environment

G,

while in c a s e (ii) t h e cell undergoes periodic changes in i t s metabolic s t r u c t u r e . If no such N exists, t h e cell is unstable and aperiodic. (Note: This last possibility c a n o c c u r only if t h e s e t of possible metabolisms H(R , F) is infinite).

A collection of r e l a t e d questions also a r i s e in connection with t h e replication

I

^map

^Bf.

For instance, w e can ask whether o r not Lamarckian changes are possible,

I

i.e., can an environmental change o --r

Z

g e n e r a t e a permanent change in t h e genetic map P, via t h e replication map

Bf

discussed above? In one p a r t i c u l a r con- struction of

&

due t o Rosen [?], it can b e shown t h a t such changes a r e not possi- ble. W e shall show t h a t Rosen's case is v e r y special a n d t h a t t h e general situation is f a r more complicated, even f o r linear maps.

Finally, we have a c i r c l e of issues relating t o t h e complexity of (M,R)-systems.

We can a s k , f o r example, how complex Pf and

pf

must b e in o r d e r t o r e p a i r a given metabolic map f , and t h e d e g r e e t o which this requisite complexity czn b e generat- ed within t h e bounds of biological and/or social constraints. W e shall e x p l o r e such c o n s i d e n t i o n s within t h e detailed confines of t h e linear framework developed in Section 4 .

3. Input /Output Maps and Realizations

Beyond any doubt, it can safely b e a s s e r t e d t h a t t h e fundamentzl problem of mathematical system t h e o r y is t h e construction of models from datz: the Realiza- tion Problem. In general terms, we a r e given a system's e x t e r n a l behavioral

description f (input/output behavior), and t h e task is t o c o n s t r u c t an internal state-space and dynamics s o t h a t t h e behavior of t h e resulting system C a g r e e s with f , C being in some s e n s e t h e "simplest" such system. The d e g r e e t o which this construction can b e c a r r i e d out, e i t h e r analytically o r computationally, depends upon t h e c h a r a c t e r of f , a s well as upon o t h e r problem boundary conditions (meas- urement e r r o r , constraints, input classes, etc.). H e r e w e shall give a brief sum- mary of t h e simplest and most well-understood c a s e when f is linear. For a fuller account of t h e s e r e s u l t s , as w e l l as t h e i r extensions t o nonlinear f , we r e f e r t o t h e works [16-181.

Let

n

b e a s e t of admissable system inputs, with

r

being t h e corresponding set of outputs. W e shall assume t h a t t h e elements of fl a r e sequences of v e c t o r s in Rm, while

l?

consists of sequences of v e c t o r s in RP, m,p ² 1. The behavior map i s speci- fied by a time-invariant, linear map f : fl ^-+

r.

Thus, a typical element o E fl h a s t h e form

while a n element y E looks like

Y

=

( Y ~ ' Y2, Y ~ ' . . . ) ' ^# Y1 E R P *

Iu'otice t h a t w e assume that time is discrete with t h e input w s t a r t i n g at time t

=

0, while t h e f i r s t output a p p e a r s one unit l a t e r at time t

=

1. In view of t h e linearity assumption on f , we can a s s e r t t h e existence of a sequence of matrices

El

=

A A A . , A, E R P , ~

such t h a t t h e action o --+ f(w)

=

y can b e r e p r e s e n t e d a s

We call t h e sequence B , t h e b e h a v i o r sequence. F o r technical reasons, i t t u r n s out t o b e convenient l a t e r t o e x p r e s s t h e above input/output relation in component

t ^-l

T A ( ~ > ¹ A (2) ⁱ . . . _i

~ p j ]

s ( u i ) ,

~t

= C _

^{t-:I t-i}

I =o

where A$

=

jth colurzn of and S(ui) = "stack" of t h e v e c t o r u!, i.e. t h e vec- t o r forme2 by stacking t h e columns o i u, t o form z column of scalars. In t h i s situa- tior,, where u! is a l r e a d y e v e c t o r , S(u,)

=

zi and t h e o p e r z t i o n ^{' I S '} h a s no e f f e c t . i z t e r i t wil! b e important wher. i t i s o p e r z t i n g on m a t r i c e s .

The s t r u c t u r e of t h e a b o v e i n - , u t / o u t g ~ t r e l e t i o n c z n z!so b e writter, using a block Toep!itz a z t r i x F as y

=

F w , o r ,

In whzt follows, i t will also b e useful t o r e - z z z n g e t h e b e h a v i o r s e q u e n c e B in t h e block Hznke! form

We czr. now formulate t h e Realizztion Problem 2s:

Given the b e h a v i o r s e q u e n c e B, find ar, i n t e g e r ^r,, a vecto- s p a c e X of dimen- sion n? znf! m z t r i c e s F E R"", G E RnXm, H E Rpxr, s u c h t h z t

(1) _A,

=

Fr ^: 1 = IA,..,,... 7 ,

(2) The p a i r ((F,C;) i s complete!y r e a c h a b l e , i.e.

r z n k

:

^{G i FG i y 2 ~}

i

^{- .}

-

_{: ?"'G ]}

- -

^III

(3) t h e p z i r (P,G) i s completely o b s e r v a b l e , i.e.,

rznk [ H'

i

F'z' i F ' ~ ~ '

i .

.

.

F*~-:H/ 7 = n .

Dynzmicz!!~, we car, e x p r e s s t h e system

C

= (F, G , I?) 2s

The condition (2) s i n p l y meacs t h a t t h e behavior of C a g r e e s with t h a t of B while conditions (2)

-

(3) i n s c r e t h z t

C

i s t h e simplest poss:Sle l i n e z r system s s t i s - fying c o a l i t i o n (I), in t h e s e n s e t h a t t h e r e is no system whose s t a t e - s p a c e X h a s smzller dimension a n d whose b e h a v i o r z g r e e s with B The problem i s how t o con- s t r u c t t h e s?ace X and t h e system C

=

(F, G ,

H)

from B. The znswer hinges c r i t i - czl!y cpon w h e t h e r we know i n a d v a n c e w h e t h e r o r n o t t h e r e e x i s t s a n y n

<

with t h e r e q u i s i t e p r o p e r t i e s . If y e s , t h e n w e czn invoke a curzber of algorithms f o r determining

C;

i f not, we are in t h e reaLrn of t h e so-callec? " p a r t i d r e d i z a t i o r , "

problem, some of t h e d e e p e s t waters in n o d e r n system t h e o r y . We shzl! r e f e r t o t h e r e f e r e n c e s f o r 2 discussion of t h i s case and c o n s i d e r h e r e only t h e situation where n i s a s s u m e d finite an?, known.

Assuming t h e dimension c is known f o r a system

C

sztisfying conditions (1)

-

(3), t h e f i r s t , and sti!! o n e of t h e simplest! ? r o c e d u r e s f o r a c t u a l l y c o n s t r u c t i n g

(F, G , H) i s t h e KO Xealization Algorithm [14,16,19], developed by

B.L.

KO in 1968.

Let n

<

m b e givez. I t c z z b e shown t h z t t h e infinite Elankel 2 , r z y H i s such t h z t r z n k H = ^c. Thus, t h e r e e x i s t m z t r i c e s P and Q such t h z t

where I n

=

r,xn identity matrix. Let o (H) denote t h e infixite a r r a y obtained f r o x

H c y ieft-shifting e a c h row, i.e.

F u r t h e r , let Rt and CS b e "editing" matrices having t h e following actions:

Rt (A) = "retain f i r s t L rows of A,"

CS (A)

=

"retain f i r s t s columns of A,"

Then Ho's Algorithm shows t h a t a canonical (minimal) realization of B i s given by setting X

=

R n and taking

C

= (F, G , H) t o b e

Thus, zside from t h e t r i v i z l editing operatiocs R and C, t h e only r e a l computatior, involved in Ho's p r o c e d u r e i s t h e calculation o i t h e matrices P and Q reducing H t o Hermite form. A l l this is under t h e assumption, of c o u r s e , t h a t t h e all-important dimension X

=

n is known via o t h e r cocsiderations (e.g., a l l A,

=

0 f o r i

>

N). In what follows, we shall often invoke t h e existence of this algorithm ( o r i t s many equivalects) as a means f o r constructively realizing different behavior sequences t h a t we encounter

4. Linear (M.R)-Systems: Repair

Now w e r e t u r n t o t h e consideration of t h e metabolism-repair systems outlined in Section 2, with t h e additional assumption t h a t t h e metabolism, r e p a i r and repli- cation maps zre linear. For t h e moment, l e t us focus attention only upon t h e meta- bolic and r e p a i r s t r u c t u r e s .

The metabolic map f : fl-

r

is e x a c t l y t h e s t r u c t u r e discussed in t h e preceding section, with fl and

r

v e c t o r s p a c e s of input and output sequences, respectively. The r e p a i r n a p Pf :

r -

^H(R,

^{r )}

must a b s t r a c t l y produce f , given t h e outpnt 7 r

r

produced by f from t h e icput o E fl. Since we have s e e n t h a t t h e metzbolic map f is equivalent t o t h e behavior sequence B, i.e.,

B

=

[ A , , A , , A

,,...

j Z f , we conclude t h a t t h e s p a c e

I! ( 3 ,

r )

= tall possible behaviors B j

.

This i s a v e c t o r s p a c e under t h e obvious r u l e s f o r addition znd scalar multiplica- tior..

Since we have assumed t h e map Pf t o be l i n e a r , w e can r e p r e s e n t i t s action zs

7 -i

W,

= C

R,+ vi , T

=

^12,

..., (**I

i =o

where (wi, v i ) z r e t h e output and in3ut t o t h e r e p a i r systerr,, respectively, with t h e elements R; being l i n e a r maps determined b y 7 and f . However, since t h e r e p a i r systerr,, when i t o p e r a t e s p r o p e r l y , must a c c e p t t h e input 7 and p r o d u c e t h e outpnt f , we must have w T

=

A, and v T

=

S ( Y ~ + ~ ) where S = "stack" o p e r a t o r defined in t h e previous section. Note h e r e t h a t w e h a v e used a d i f f e r e n t time p a r a m e t e r T f o r t h e r e p a i r system, as it will usually b e t h e c a s e t h a t t h e time-sczle of operatior, of t h e r e p e i r s y s t e n is considerably slower t h a n t h e metabo!ic o;leratior.. We r e t u r c t o t h i s poizt again ir, connectior: w i t h replicatior, ir, t h e next sectior..

I t i s a n e z s y e x e r c i s e t o s e e t h a t t h e elements R j must h a v e t h e form

S o , in component form we c a n write (**) as

w h e r e we h a v e written R/') = Bjs.

J u s t zs t h e metabolism f was r e p r e s e c t e d by t h e sequence IA1.A2,

. . 1,

we c a n now see t h a t t h e r e p a i r system Pf c a n b e r e p r e s e n t e d as

Pf = [R1,R2,R3,

. . 1.

Similzrly, we c a n a l s o identify Pf with t h e Toeplitz matrix

Remarks:

(1) If w e w r i t e e a c h Ai as

t h e J'complexityJ' of e a c k component of t h e metabolic map f is O(pm); t h e complexity of e a c k element R, of t h e r e p a i r map Pf i s 0(p2 m). Thus, a l r e a d y t h e o f t e n noted c o m ~ l e x i t y i n c r e a s e z s s o c i a t e d with living systems begins t o emerge through

n a t c r a l mzthemztical reqnirernents.

(2) A s t r z i g h t f o r w a r d calculation shows t h a t t h e assumption dim C

=

n

<

=

implies t h a t t h e s e t [ A l , A2,

...

^{, A Z n j} i s l i n e a r l y dependent (this follows from ele- mentzry p r o p e r t i e s of t h e Hanke! a r r a y H). I t is now easy t o see t h a t t h e condition

dim

C

= n

<

^a+, a l s o implies t h a t t h e canonical realizztion of t h e r e p a i r sequence IR2, R,,... ^{ h a s dimension np 5 n. Thus, we can again employ Ho's Algorithm t o pro- duce a system Cp = (Fp, Gp, Hp) realizing t h e r e p a i r sequence.

Example.

.

To fix t h e foregoing ideas, consider t h e situation ir. which t h e system's environmental input o is

with t h e metabolic output 7 = f ( 0 ) being given by

7

=

[1,2,3,4 ,... j

=

n a t u r a l numbers.

Since o and 7 a r e s c a l z r sequences, w e have m

=

^p

=

1 . W e easily obtain t h e behavior sequence

B = 11,1,2,2,3,3,4,4

,...

j

= I A ~ ,

^{A,, A,}

,...

j .

I t can b e shown t h a t t h i s behavior sequence h a s a canonical realization

C =

(F, G ,

H)

of dimension n

=

3, s o a n application of Ho's Algorithm yields t h e canonical system matrices

The dynamics f o r t h e metabolic subsystem are

Turning now t o t h e r e p a i r component, we must have Pf(7)

=

f which leads t o

Ri = { ^{I l l ,}

^{i odd}

[-I], i even Thus, t h e Toeplitz o p e r a t o r f o r Pf _is

with t h e a s s o c i a t e d Hankel a r r a y

Since w e know t h a t t h e r e p a i r sequence h a s a finite-dimensional r e a l i z a t i o n of dimension np ^{I ri} = 3, experimentir.g a b i t with Ho's Algorithm ( o r computing r a n k Hp) gives n p

=

1, with t h e r e s u l t a n t canonical r e p a i r r e a l i z a t i o n Cp = (Fp, Gp, Hp), where

The r e p a i r d y n z x i c s a r e t h e n

2 ,

=

[ - l ] z , + l ] v , , z o = o , 7 = 0 , 1 , 2

,...

W,

=

l l ] ^2,

.

From o u r e a r l i e r r e m a r k s , we c o n n e c t t h i s system with t h e metabolic map f via inputs and o u t p u t s as w,

=

A , , v,

=

YT+I

Remarks

(1) At f i r s t glance, t h e r e z 7 p e a r s t o b e a contradiction h e r e t o o a r e a r l i e r clzim t h a t t h e r e p a i r system is more "corr.plexJ' t h a n t h e metabolism. In t h i s exam- p l e , w e see t h a t dim

C, =

1

<

dim

C =

3 , s o if one m e a s u r e s complexity b y state- s p a c e dimension, t h e n Cp i s actuzlly n e v e r more complex thzri

C.

In fact, as w e h a v e a l r e a d y noted, t h i s will always b e t h e c a s e . However, o u r earlier r e m a r k used a d i f f e r e n t notion of complexity, one involving t h e o b j e c t s of t h e behavior21 d e s c r i p t i o n s , t h e elements Ai and Ri. Unless p

=

1, t h e o b j e c t s IRi ] always contain more elements t h a n t h e

I A : ~ .

Thus, by t h i s m e a s u r e of complexity, t h e r e p a i r sys-

tem is always at least as complex as t h e metabolism. Roughly speaking, it is more difficult to describe t h e behavior of t h e r e p a i r process than t h e metabolism, but is simpler t o r e a l i z e i t s dynamics. In engineering t e r m s , t h e r e a r e fewer "integra- tors", but of a more complicated type.

Now l e t us r e t u r n t o a consideration of t h e main function of t h e r e p a i r mechanism: t o r e s t o r e t h e c o r r e c t input/output behavior ( o , y ) in t h e f a c e of changes in e i t h e r t h e environmental input o o r t h e metabolic machinery f . There a r e s e v e r a l cases and subcases t o examine:

C a s e I. FYxed environment o* a n d a fixed genetic m a c h i n e r y

Pi

with v a r i a b l e metabolism f.

In t h i s c a s e , w e a r e concerned with changes in t h e metabolic machinery from some nominal, o r basal, metabolism f*. In o t h e r words, we consider those metabol- isms f such t h a t Pf. (f ( o m ) )

=

f o r f*. In t h e first c a s e , t h e r e p a i r machinery

P;

stabilizes t h e system a t t h e new metabolism f; in t h e second c z s e , 2; a c t s to r e s t o r e t h e nominal metabolism f*.

To study this situation, i t i s useful to consider t h e map

e,,,

: H ( n , r 1

-

H ( n , r )

f

+

P,(f (o*))

-

The c a s e in which t h e r e p a i r system stabilizes t h e system a t t h e new metabolism f c o r r e s p o n d s t o finding t h e fixed points of t h e map , i.e., those metabolisms f such t h a t

The situztior: is which t h e r e p z i r systerr. r e s t o r e s t h e design metabolism f* hy

"repziring" t h e p e r t c r b z t i o ? f f - + f , co,responds t o finding those perturbations f suck: t h z t

Note t h a t by c o n s t r u c t i o n we must h a v e

i.e., f* i s a t r i v i a l fixed point of +,,,p as i s t h e null metabolism f=O, by v i r t u e of t h e f a c t t h a t \ko.,T is l i n e a r , being induced from t h e l i n e a r map Pr.

Since e a c h f E H ( R , r ) h a s t h e form f

=

iA1,A2,A 3,... { , w e c a n r e p r e s e n t \k,,,p by t h e infinite matrix

i , j = 1 , 2 ,

...

S i n c e +fl,p i s induced from t h e r e p a i r map Pp, t h e

I+; +z2

^6;3

¹

elements +;j will b e determined by t h e elements ~R;.R;.R;

,...

J a n d o*

=

fu,,ul.

. . ...

{

+u=:?-=

- -

determining Pp

.

I t should b e noted t h a t in genera!, as with t h e c h o i c e of t h e

*i2 . ^{. . .} I

^I

I . .

m a t r i c e s !A: { defining f x , t h e r e is some level of a r b i t r a r i n e s s in t h e elements

IR;

j.

where

c;:

E R ~ ~ ~ ,

Unless t h e input CY.* h a s s p e c i a l s t r u c t u r e , t h e r e will b e p(m-1) degrees-of-freedom in t h e c h o i c e of e a c h A:; similarly, e a c h R; will h a v e pm(p-1) degrees-of-freedom in its elements, non-uniqueness t h a t is i n h e r i t e d by t h e elements 4'; comprising

+,=,?=

W e c a n now make t h e following o b s e r v a t i o n s a b o u t t h e p r o p e r t i e s of +,,,p and t h e b e h a v i o r of t h e r e p a i r system Pp in t h e form of

Theorem 2. (r) h e r e q u i r e m e n t t h a t f*

=

~A;,A; ,A:,...

1

be a m e d p o i n t of

*,,,

m e a n s t h a t t h e v e c t o r [A;,A;.A;.

...I'

i s a c h a r a c t e r i s t i c v e c t o r of

*u*,P w i t h a s s o c i a t e d c h a r a c t e r i s t i c v a l u e 1.

h e elements

+rj

a r e r e s t r i c t e d o n l y i n t h a t t h e y m u s t be selected to s a t i s f i t h i s c o n d i t i o n ;

(2) the p e r t u r b a t i o n metabolism f

=

fA1,A2,A 3,...

1

w i l l be a fizea' p o i n t of

4',,,r i f a n d o n l y

zf

t h e v e c t o r [A1,A2,A 3,...]' i s a c h a r a c t e r i s t i c v e c t o r of 4',=,r w i t h a s s o c i a t e d c h a r a c t e r i s t i c v a l u e I ;

(3) t h e p e r t u r b a t i o n f w i l l be " r e p a i r e d " , i - e . , 4',x,r(f)

=

f * i f a n d o n l y i f f has t h e f o r m f = f* + k e r \k,,,r. In o t h e r w o r d s , f o r repair w e m u s t h a v e t h e v e c t o r [Al-A,: .A,--;,A,--A;

,... 1'

^E k e r qu=,r

.

The l a s t two points have deep implications f o r t h e ability of t h e r e p a i r system t o function effectively in t h a t they a r e diametrically opposed: if we want t o b e able t o r e p a i r many different types of perturbation f , then by ( 3 ) we need t o have ker\k,,,f, "large"; if ker\kfl,f, i s l a r g e , then t h e r e a r e relatively "few" charac- t e r i s t i c v e c t o r s with associated c h a r a c t e r i s t i c values 1 implying t h a t t h e r e a r e only a "small" number of perturbations f t h a t will b e stabilized by t h e r e p a i r sys- tem. The sum total is t h a t we can e i t h e r z r r a n g e t o have k e r \k,,,T 'largeu and r e p a i r many disturbznces, o r we can have k e r \k,,,f, "small" and b e a b l e t o s t a b i l - i z e many metzbolic disturbances, but not both! The amount of flexibility we have in choosing t h e k e r \kUxlr is dictated by t h e degrees-of-freedom we have in determin- ing Pf which, as noted above, is proportional t o t h e quantity pm(p-l), where p and n a r e t h e number of metabolic outputs and inputs, respectively. (It should b e noted t h z t this is t h e number of degrees-of-freedon: a f t e r satisfying t h e condition in p a r t (1) of Theorem 2).

I t is impossible t o speak any more precisely about t h e r e p z i r mechanism in t h e absence of more specific detzils about t h e s t r u c t u r e of 4',,,,. So, l e t us examine

t h e p r o c e s s deternining in g r e z t e r detail.

From t h e component representation of (**), we can s e e t h a t

This is c l e a r l y a t r i a n g u l a r (in fact, Toeplitz) r e p r e s e n t a t i o n as A, depends only upon t h e elements A1,A2,

. . ^.

^, Ar in a l i n e a r , Toeplitz manner. As long as all p c o m - ponents of Air, U; are not z e r o , we can always find a solution to t h i s equation in t h e components of t h e m a t r i c e s I R ; - ~ ] a n d t h e elements juy {, i=0.1,2,.

. .

, T -1; j =1,2,.

. .

,m. In fact, generically t h e r e is a pm(p-1)-parameter Jam- ily of s u c h solutions, after we h a v e s e l e c t e d some of t h e e n t r i e s of t h e R's in o r d e r t o satisfy t h e r e q u i r e m e n t t h a t

w h e r e t h e elements R* d e n o t e t h e p a r a m e t r i z e d family of solutions satisfying t h i s r e l a t i o n .

On t h e o t h e r hand, t h e induced r e l a t i o n !Pu,,r s a y s t h a t we must h a v e

f o r some triangular c h o i c e of

*;, ^.

In p a r t i c u l a r , t h i s means t h a t

*;=o,

j

>

i a n d we h a v e

But, we a l s o h a v e t h e e x p r e s s i o n f o r A: from a b o v e involving t h e elements {R:-~{.

S e t t i n g t h e s e t w o e x p r e s s i o n s equal, we obtain

The r e l a t i o n ( t ) t h e n e n a b l e s u s t o pin down some of t h e elements f q T k { , k=1,2,

....

The a r b i t r a r y elements in j\k:kj will usually t h e n b e d i c t a t e d by t h e a r b i t r a r y ele- ments in t h e

IR:-~

^j in o r d e r t o make t h e k e r \kflvP ',large" o r "small", as t h e case may b e .

One case in which w e c a n b e v e r y specific about t h e s t r u c t u r e of \kelp i s when m=p=l. In this case w e can easily solve t h e relation ^("8') f o r t h e elements

\k;y obtaining t h e t r i a n g u l a r Toeplitz a r r a y

H e r e t h e r e a r e no degrees-of-freedom in t h e

IR;],

s o t h e s p e c t r a l s t r u c t u r e of qfllP is fixed.

EzampLe (continued]

We can make use of t h e above s c a l a r input/output c a s e t o examine t h e r e p a i r mechanism f o r o u r e a r l i e r sample problem. Before w e had

Let us suppose t h a t t h e metabolism f* is p e r t u r b e d t o t h e new metabolism

i.e., t h e r e i s a change only in t h e 2nd element. The system output under f i s now 7

=

f ( o X )

=

11.3,4.4.5.6.?

,... 1.

Thus, t h e metabolic change r e s u l t s in a change of output from 7=

=

n a t u r a l numbers t o t h e closely r e l a t e d sequence 7 , which d i f f e r s from 7' only in t h e 2nd and 3 r d e n t r i e s . The question is what e f f e c t t h i s seemingly minor change h a s upon t h e r e p a i r mechanism.

To a d d r e s s t h i s issue, we compute t h e matrix \k,,,p which, using t h e require- ment t h a t f * must b e a fixed point, gives

identity.

Consequently, appealing t o Theorem 2 w e find t h a t t h e metabolism f is also a fixed point of 4',,,f= with c h a r a c t e r i s t i c value 1; hence, t h e r e p a i r mechanism will pro- cess 7 into f and thus stabilize t h e system at t h e new metabolism f . In f a c t , t h i s will be t r u e f o r a n y metabolic perturbation f of this system: t h e r e p a i r p r o c e s s will immediately ''lock-on" t o t h e new metabolism f and stabilize t h e system t h e r e . Thus, f o r this system t h e r e i s no " r e p a i r " but only a n immediate stabilization at t h e new metabolism.

Another important point t o note about t h i s s c a l a r case i s t h a t w e must have t h e product

~f

u:

=

1 or 0 f o r t h e possibility of e i t h e r r e p a i r o r 'locking-on" t o a new metabolism. Otherwise, w e cannot e v e r exactly r e s t o r e f* or exactly lock-on t o a new metabolism, but only obtain a scalar multiple of fL or f. Technically, of course, t h i s i s not an important distinction; in p r a c t i c e , i t may or may not b e signi- ficant.

Case 11. k f l u c t u a t i n g environment u w i t h f i z e d nominal metabolism f*

a n d fixed genetic m a c h i n e r y P, .

In this situation, w e have a chznge of environment o m

-

w, and wznt t o find zl! those environments o such t h a t

P,. ( f m ( o m ) )

=

P,. (f'(w)) (= f*) implies

lr, o t h e r words, we want t o know when P is 1-1.

f*

But t h e matrix representation of Pf. is

i i

j ~ ; R; o . . .

I

P,.

=

i R ; I R; R; . . .

i l

^{R,= E ,}

1

implying t h a r P,. is 1-1 i f a n d only if k e r R; = 10

1.

This will b e t h e case i f and only if m = 1 a n d r a n k R;

=

p.

H e r e w e only c o n s i d e r t h e situation when PI. (I '(o*))

=

PI. ( ~ ' ( o ) ) , s i n c e i i t h i s i s not t h e czse, t h e n we are back in Case I , i.e., t h z t of a metabolic chznge. We c a n now conclude

Theorem 3. If m = 1 a n d r a n k R;

=

p, all e n v i r o n m e n t s o s u c h t h a t f ' (0)

=

f ' (a*) a r e g i v e n b y o = ow + k e r f';

Dn the other h a n d , i f m

>

1 a n d / o r r a n k R;

=

r

<

^p, t h e n a n y environ- mental change of the form o

=

x + ^00, where x i s a n y s o l u t i o n of the e q u a t i o n f ' (x)

= 7

^,

7

^E k e r R;, w i l l be r e p a i r e d b y P,.

.

Proof. Let m

=

1 a n d r a n k R;

=

p. Then t h e o p e r a t o r Pf. i s 1

-

1 a n d a l l t h e environments o such t h a t PI, (f '(0))

=

PI, (f '(ow)) implies f w ( o )

=

f ' (a') consist of t h o s e o satisfying o

=

o w + k e r f'.

Now l e t m

>

1 a n d / o r r a n k R;

=

r

<

p , i.e. k e r R; i s non-empty. Let

5

^E k e r R: and l e t x b e a solution of t h e equation f' (x)

= 7.

Then any environrnen- t a l change of t h e form ol"--, o

=

x + o* will b e r e p a i r e d by t h e g e n e t i c mechanisn P f b s i n c e

P,. (fW(o)) = P,. (f '(XI + f *(a'))'

=

P,. ($1

+-

Pf. (f '(o*))

=

0 + f ' = f W

Theorems 2 znd 3 c h a r a c t e r i z e a l l t h o s e metabolic and environmental c h a n g e s t h a t c a n b e " r e p a i r e d " by a fixed genetic machinery

P,..

L e t u s now consider t h e wzys in which t h i s g e n e t i c z p p z r a t u s itself czn c h a n g e by mear,s of r e p l i c a t i o n .

5. Linear

(P.R)-

Systems : Repiication

The system replication map

8: ^: H(R,I')--+H

(I',

H ( n , I ' ) )

can be formally cocsidered in much t h e same fashion as just discussed f o r t h e r e p a i r mechanism P,. However, since t h e functional r o l e of

Pf

is quite different from t h a t of PI, z number of interesting questions a r i s e t h a t are a b s e n t in t h e case of r e p z i r , questions involving mutation, adaptation, Lamarckian inheritance and so f o r t h . We shall consider t h e s e m a t t e r s in more detail ir, z moment, but f i r s t l e t us look a t t h e formal realization of

Bf.

Since

8,

is z l i n e a r map accepting inputs of t h e form f

=

!Al, A2, A 3 , . . . J 2nd producing o c t p c t s

Pf

= {Rl, R,,...

!,

we must hzve ^s: rezresentztior: of t h e action of

Bf

2s

f o r zn a p p r o p r i a t e s e t of mztrices ! U j j , where t h e iz;)zt e, = S(AI) and t h e outpct ci = Ri. Arguing jest z s f o r t h e r e p z i r map , we conclude t h z t U! must have t h e form

where e a c h Cj, E

R?

^m?. In what follows, we shall write

u$') =

Cjr So. just zs with f and. ?:, we hzve t h e r e p r e s e n t z t i o n of

6:

as

t o g e t h e r wiih t h e associzted Toeplitz identificztior,

and t h e associated Hankel a r m y

Note t h a t in t h e above set-up, since t h e inputs f o r t h e replication system must correspond t o t h e metabolism f , while t h e outputs must b e t h e associated r e p a i r map Pf , we have t h e relations

e ,

=

, c,

=

R, ,

with S being t h e "stacking" o p e r a t o r . These relations are e x p r e s s e d in t h e time- s c a l e o of t h e r e p l i c a t o r system. Here we have introduced still a t h i r d time-frame

o t o distinguish between t h e scale t f o r metabolism and T for r e p a i r . Usually, we will have At S AT ^S Ao.

Using t h e same arguments as f o r PI, i t can b e established t h a t if f has a finite-dimensional realization, s o does

Bf

and t h e dim

Bf

^S dim f . So, in connection with t h e example given in t h e last section, w e find t h a t if

then, a f t e r a bit of a l g e b r a ,

Bf =

{I,-2,1,0,0

,...

^j

.

Thus, only t h e terms U1, U2 and U3 a r e non-zero. Note t h e a p p a r e n t d e c r e a s e in complexity of t h e sequences f , Pf and

of

^{as we} pass frorn netabo!ism t o r e p a i r t o replication. We will r e t u r n t o this point below

Applying Ho's Algorithm t o

Bf

yields t h e realization of t h e replication map

Bf

io o o! i l i

c

⁰ qo

+

^{! O} I ^e,, c ; o = O * C c E R 3

90+1=1'

l o ]

10 0 0,

The machinery outlined above provides a systematic p r o c e d u r e f o r generation of z canonical replication system via Ho's Algorithm (and a r e p a i r mechanism, too) f o r any metabolism, provided only t h a t t h e metabolism possesses some finite- dimensional realization; t h i s is t h e only condition needed f o r t h e existence of a finite-dimensional r e p a i r and replication p r o c e s s constructible d i r e c t l y from t h e metabolic components

n,

l? and H(R, r ) via "natural" mathematical operations. In t h e p a p e r [73, Rosen suggests a n o t h e r construction f o r t h e replication system, o n e which imposes n o assumptions on t h e metabolism but which entzils some s e v e r e con- ditions of a n o t h e r n a t u r e o r d e r t o make t h e scheme work. S i n c e Rosen's construc- tion brings f o r t h many of t h e a s p e c t s of replication we want t o examine, a n d is of some i n t e r e s t in i t s own r i g h t , we briefly summarize h i s argument.

Recall t h a t f o r replication w e need a map

Bf

: H(Q, r ) --, H ( r , H(R, l? )) pos- sessing t h e p r o p e r t y t h a t

Bf

(f)

=

Pi. L e t X and Y b e a r b i t r a r y sets. Then t h e r e is a naturally defined map

x^

: H(X, Y) ⁴ Y , given by

x^

(f)

=

f (x) ,

f o r all x E X. This i s t h e so-called "evaluation map" on I!

(X,

Y). Assume t h a t

x^

is 1-1. Then t h e r e e x i s t s a mzp

2-I

such t h a t

x^ ^-1 _: Y --, H (X, Y)

.

Now w e need only set X =

r,

Y = H

( a ,

r ) t o obtain t h e d e s i r e d replication mzp, call i t ?-I:

9-I

^{: r i} ( R , r ) --, H ( r , H(R,

r)) .

This is Roser.'s constructior., which m i r r o r s t h e nsnz! p r o c e d u r e f o r construc- tiol: of t h e dun1 s p a c e of r . Note, however, t h z t t h e success of t h i s p r o c e d u r e f o r producing z replication map hinges entirely upon t h e map 2 being 1-1. Rosen

a r g u e s t h a t t h i s is a mathematical expression of t h e c e l e b r a t e d "one-gene, one- enzyme" h y p o t h e s i s from molecular g e n e t i c s , a n d u s e s t h i s i n t e r p r e t a t i o n as sup- p o r t i n g e v i d e n c e f o r h i s c o n s t r u c t i o n . Let u s examine t h i s argument in light of t h e l i n e a r s t r u c t u r e s i n t r o d u c e d above.

In o u r terminology, Rosen's c o n s t r u c t i o n involves t h e injectivity of t h e map

If

7

is 1-1, t h e n we h a v e a map

?-I

^{: H(R,}

r ) -

H ( r , H(R,

r))

f

+

^P,

IA18 A 2 t - - . j

+

!RIP R2t-.-j

-

But, t h i s means t h a t

5-I

i s equivalent t o t h e matrix

Thus, s u c h a map e x i s t s if a n d only if t h e matrix

4,

is invertible. But, s i n c e e a c h U, ^E R P X P ~ : c a n b e i n v e r t i b l e if a n d only if: 1) p

=

pZm2, i.e. p

=

m

=

1

and 2) U l

+

0. Consequently, we see t h a t Rosen's scheme c a n work only in t h e case of a single-input/single-output metabolism, a n d e v e n t h e n only if U1

+

^{0 ;} t h i s is a v e r y s e v e r e r e s t r i c t i o n .

In summary, t h e c o n s t r u c t i o n we h a v e given f o r t h e r e p l i c a t i o n o p e r a t i o n works f o r all finitely r e a l i z a b l e metabolisms. The c o n s t r u c t i o n d u e t o Rosen will work f o r any metabolism, provided t h a t t h e r e is only a single-input and a single- o u t p u t (assuming U 1

+

^{0 ) . We shall} see t h e implications of t h e s e d i f f e r e n t situa- tions momentarily.

Within t h e c o n t e x t of r e p l i c a t i o n , t h e r e a r e two b a s i c questions of i n t e r e s t :

1) When c a n environmental changes o --, o' r e s u l t in changes in t h e replica- tion map

Pf

?

2) If e x t e r n a l disturbances modify

&,

what kinds of changes in f can r e s u l t ?

The f i r s t of t h e s e is t h e question of Lamarckian inheritance, while t h e second a d d r e s s e s problems of mutation. We consider only t h e Lamarckian question h e r e , d e f e r r i n g a treatment of t h e second, vastly more complicated question t o a f u t u r e p a p e r .

From t h e diagram

i t i s evident t h a t

Pf ( f ( o > >

=

f =

[Bf

( f > l ( f ( o > >

Suppose w e have a change of environment o --, a'. This r e s u l t s in a change y

=

f ( o ) --, f (0')

=

7'. Assume t h a t

p, (7)

=

p, (7')

=

f ,

i.e. t h e r e p a i r mechanism is czpable of c o r r e c t i n g f o r t h e environmental change.

The2 w e have

(Bf ⁰ P f ) (7)

= (&

⁰ P f > (7')

=

Pf ,

implying t h a t t h e replication operation i s unaffected by t h e environmental change.

That is, Lamarckian-type changes in

Ff

cannot o c c u r under any t y p e of environ- mental change t h a t can b e c o r r e c t e d by t h e r e p a i r operation P f . Theorem 3 c h a r a c t e r i z e s just what s o r t s of changes fall into t h i s category.

Under Rosen's scheme, it is shown in [7] t h a t no environmental change of any s o r t can lead t o Lamarckian changes in

Bf,

a vastly s t r o n g e r r e s u l t but, as noted, under extremely r e s t r i c t i v e hypotheses.

6 . Linear (M,R)-Systems : a Summary

Our development of t h e realization theory f o r linear (h',,R)-systems has been somewhat lengthy, in o r d e r t o allow considerable commentary on t h e basic set-up and properties of t h e s e o b j e c t s . Here w e summarize t h e entire development in t h e following diagram.

Pi

(Repzir) :

-

³

^{(R, r )}

p:

(Replication): H(R, r ) ^--3 H(ru H(91

n)

lU1,

u2 *...

j

f = )A1, A,, ^{a a} . j I--+ Pf

=

) R I B R2,...

1

V ! E R P T ~

0 -1

I u ( ~ ? - u

( A ) , 4

=

^1,2,

...

R,

= C

^{rUo-l 0 -1}

¹

1 =o

Assuming t h a t t h e metabolic component has a finite-dimensioned realization, s o do t h e r e p a i r and replication components, znd t h e s e canonical realizations c a n all b e computed by means of Ho's Algorithm. Furthermore, t h e dimensions of t h e realiza- tions f o r t h e r e p a i r and replication systems will be no l a r g e r t h a n t h a t of t h e metabolic subsystem. Thus, any finitely realizable metabolism can b e a metabolism-repair system using t h e constructions detailed h e r e .

7 . Discussion

The formalism given h e r e f o r l i n e a r (M,R)-systems g e n e r a t e s a long list of questions, problems and extensions of t h e classical "metabolism-only" Newtoniar.

framework meriting f u r t h e r study. High on this list a r e problems concerned with networks, mutation and nonlinearity. Here we only touch upon a few of t h e major issues.

A. Networks

-

w e began in Section 2 with a discussion of (M,R)- n e t w o r k s , emphasizing t h e r o l e of t h e r e p a i r mechanism z s ar; object whose inputs generally

come from o t h e r cells in t h e network. In p a r t i c u l a r , we noted t h a t a r e p a i r com- ponent needed t o r e c e i v e all of i t s inputs in o r d e r t o function, s o t h a t if one of t h e inputs was from i t s own associated metabolism, then t h e removal of t h a t metabolism would also incapacitate t h e r e p a i r subsystem. W e t h e n immediately shifted atten- tion away from networks and considered only a single (M,R)-unit. This clearly involves a different interpretation of how t h e r e p a i r and replication components i n t e r f a c e with t h e metabolism. A s w e have noted above, instead of imagining t h e metabolism t o be removed, we consider what happens when t h e r e is a n environmen- t a l change o r when t h e metabolic machinery a c t s , but imperfectly. These con- siderations bring us up against t h e question of just how t o i n t e r p r e t t h e action of t h e serially-connected metabolism-repair-replication subsystems.

Naively, w e could imagine t h a t t h e time-scales of operation of t h e subsystems are s o disparate that t h e systems o p e r a t e non-concurrently. In o t h e r words, t h e metabolic subsystem f i r s t processes o into 7 . When t h i s operation is complete, t h e output 7 is processed by t h e r e p a i r system and, finally, when t h e r e p a i r operation terminates. t h e replication p r o c e s s begins. Of c o u r s e , real cells never o p e r a t e in this fashion and this simple scheme can only be thought of as a convenient approxi- mation when t h e time-scales are such t h a t At

<<

A t

<<

Ao.

More realistically, t h e t h r e e subsystems o p e r a t e concurrently with t h e differ- e n c e s in time-scales introducing time-lags into t h e r e p a i r and replication opera- tions, relative t o metabolic time. In this case, we must d r o p t h e mathematical fic- tion of infinitely long input and/or output sequences and assume t h a t o i s of finite duration, with t h e metabolic output 7

=

f ( o ) also of finite length. In t h e s e situa- tions, t h e mathematical formalism r e q u i r e s t h e full machinery of t h e so-called Par- tial Realization Problem and i t s attendant version of Ho's Algorithm [14,16]. Since this is a matter of some delicacy, w e d e f e r i t s treatment t o a later paper.