MCA Perspectiva Teorica

7/29/2019 MCA Perspectiva Teorica

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Metabolic Control Analysis from a Control Theoretic

Perspective

Brian P. Ingalls

Department of Applied Mathematics

University of Waterloo

Waterloo, Ontario

Canada N2L 3G1

[email protected]

Abstract A theory of control of biochemical systemsdeveloped by the theoretical biology community is in-terpreted from a control-theoretic point of view. Thistheory, known as Metabolic Control Analysis (MCA), isbased on an extension of parametric sensitivity analysis

to stoichiometric systems. The main results of MCAare shown to correspond to particular solutions of anappropriately stated tracking problem. This presentationleads to some natural generalizations of existing results.

I. INTRODUCTION

Issues of regulation and control are central to the

study of biochemical systems. The maintenance of

cellular behaviour and the appropriate response to

environmental signals can only be achieved by sys-

tems which have evolved to be robust against certain

perturbations and sensitive to others. These behaviours

demand the use of feedback. It is then natural to

expect that the tools of feedback control theory, whichwere developed to address the analysis and design of

self-regulating systems, would be useful in the study

of these biological networks.

Biochemical mechanisms for implementation of

feedback control were first discovered in the biosyn-

thetic pathways of metabolism, and it was within the

study of metabolism that a quantitative theory of con-

trol and regulation of biochemical networks was first

developed. The fundamental tool used in this study

is parametric sensitivity analysis, applied primarily at

steady state. One approach, dubbed Metabolic Control

Analysis (MCA), or sometimes Metabolic Control

Theory (MCT), makes use of a standard linearization

technique in addressing steady state behaviour [5], [6],

[11].

This paper will focus on the straightforward lin-

earization method used by the MCA community. This

framework provides a direct connection between these

biochemical studies and the general theory of paramet-

ric sensitivity analysis. Moreover, linearization leaves

intact the stoichiometric relationships which are ex-

ploited in studies of these chemical networks. Indeed,

as will be shown below, it is this stoichiometric nature

which distinguishes the mathematics of MCA from

standard sensitivity analysis. As first shown in [17],

application of some basic linear algebra provides an

extension of sensitivity analysis which captures thefeatures of stoichiometry. Beyond these mathematical

underpinnings, the field of MCA deals with myriad

intricacies of application to biochemical networks that

demand careful interpretation of experimental and

theoretical results. Surveys can be found in [3], [4],

[7].

The analysis in this paper is based on the stan-

dard ordinary differential equation-based description

of biochemical systems in which the states are the

concentrations of the chemical species involved in the

network and the inputs are parameters influencing the

reaction rates. In most cases this model is a non-

minimal representation of the system. In addressing

metabolic systems, it is standard to take enzyme

activity as the parameter input. This choice of input

typically results in an overactuated system with

more inputs than states. However, in addition to the

species concentrations, there are outputs of primary

importance, namely the rates of the reactions within

the network. As will be described below, it is a

consequence of the stoichiometric nature of the system

that the degree of overactuation coincides exactly

with the degree by which this reaction rate output

is underdetermined as a function of the state. The

result is a control system in which the output enjoyssome autonomy from the state. Consequently, issues

of controlling the state and output can be, to a degree,

addressed separately.

Many investigations of metabolic redesign have

appeared in the MCA literature (e.g. [12], [19], [21]).

These results can all be seen as contained within

Enzymes are proteins which act as catalysts. In most biochemi-cal networks, each reaction is catalysed by a single enzyme and thereaction rate is directly proportional to the level of enzyme activity.


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the metabolic design approach developed by Kholo-

denko et al. [13], [14]. The current paper describes

these results as solutions to a tracking problem, thus

drawing parallels between MCA and control theory

as well as providing control-theoretic generalizations

of the main results of MCA, the Summation andConnectivity Theorems.

I I . PRELIMINARIES

Consider n chemical species involved in m reac-

tions in a fixed volume. The concentrations of the

species make up the n-dimensional vector s. The rates

of the reactions are the elements of the m-vector v.

These rates depend on the species concentrations and

on a number of parameters which are collected in

a k-vector p. The function v = v(s, p) is assumedcontinuously differentiable. The network topology is

described by the nm stoichiometry matrix N, whose

i, j-th element indicates the net number of moleculesof species i produced in reaction j (negative values

indicate consumption).

The system dynamics are described by

d

dts(t) = Nv(s(t), p(t)) for all t 0. (1)

In addition to the state, s(), and the input p(), thevariables of primary interest in this system are the

reaction rates v(s, p). Thus, in interpreting (1) as acontrol system, we choose an output y comprised of

these two vectors:

y(s, p) = s

v(s, p)

. (2)

Linearly dependent rows within the stoichiometry

matrix correspond to integrals of motion of the system.

Each redundant row identifies a chemical species

whose dynamics is completely determined by the

behaviour of other species in the system through a

conservation relation.

The consequences of linear dependence among the

columns of N will be explored below. If the stoi-

chiometry matrix has full column rank then steady

state can only be attained when v(s, p) = 0. Mostbiochemical systems admit steady states in which

there is a flux through the network. These correspond

to non-zero reaction rate vectors v which lie in the

nullspace of N.

I I I . CONSEQUENCES OF RAN K DEFICIENCIES

Networks which describe metabolic systems often

have highly redundant stoichiometries. As an example,

consider a metabolic map from Escherichia coli pub-

lished in [18] which has a 770 931 stoichiometrymatrix of rank 733. Clearly, in attempting an analysis

of such a system it would be worthwhile to begin with

a reduction afforded by linear dependence.

A. Deficiencies in row rank

As mentioned, structural conservations in the reac-

tion network reveal themselves as linear dependenciesamong the rows of the stoichiometry matrix N. Let

r denote the rank of N. Following the standard

treatment (presented in [17]), relabel the species so

that the first r rows of N are independent. The

species concentration vector can then be partitioned

as s = (sTi , sTd )

T, where si Rr is the vector of

independent species and sd Rnr contains the

dependent species. Next N is partitioned into two

submatrices. Calling the first r rows NR, we can write

N = LNR where the matrix L is referred to as thelink matrix. Following [17] we arrive at a reduced

version of (1)

d

dtsi(t) = NRv(Lsi(t) + T, p(t)) (3)

where T depends on the initial condition, and s(t) =Lsi(t) + T. It follows that the n-dimensional stateenjoys only r degrees of freedom. Thus the original

description in terms of n state variables is nonminimal

(provided r < n), regardless of the form of the

reaction rates.

B. Deficiencies in column rank

Recalling that r denotes the rank of the stoichiom-

etry matrix N, relabel the reactions so that the first

m r columns of N are linearly dependent on theremaining r. Partition the vector of reaction rates v

correspondingly into m r independent (vi) and rdependent (vd) rates as v = (v

Ti , v

Td )

T. In analogy

to the procedure outlined above, one might hope to

reach a reduced description of the system dynamics

in which some of these reaction rates are eliminated,

but this is an impossible task. (Such an elimination

could decouple an input channel from the dynamics.)

However, an analogous reduction can be made after a

change of variables in the reaction rate space Rm.

To generate such a transformation, begin by choos-

ing a matrix K of the form

K =

Imr

K0

such that the columns ofK form a basis for the kernel

of N. Next, introduce the invertible m m matrix

P =

Imr 0(mr)r

K0 Ir

,

and define the transformed reaction rates (s, p) by

(s, p) = P1v(s, p).


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It will prove useful to partition these transformed rates

into independent (i Rmr) and dependent (d

Rr) components: = (Ti ,

Td )

T.

Letting NC denote the submatrix of N consisting

of the last r columns, the dynamics (1) can be written

in terms of the transformed rates asd

dts(t) = NP(s(t), p(t))

= [0n(mr) NC](s(t), p(t))

= NCd(s(t), p(t)). (4)

We conclude that the values of the independent trans-

formed rates have no bearing on the system dynamics.

To relate this fact to the true reaction rates v, note

that i = vi. Thus this transformation characterizesthe r combinations of reaction rates which impact

the dynamics of the state (i.e. d = vd K0vi)while identifying m r rates whose values can be

set independently of the dynamics (i.e. i = vi).This explicit transformation in reaction rates is a

novel construction. However, the decomposition of the

reaction rate vector into independent and dependent

parts is standard, as outlined in, e.g., [7]. In most work

on MCA, attention is restricted to the values of the

reaction rates at steady state, which are referred to as

the system fluxes J. By partitioning N as described

above, the vector J is separated into dependent and in-

dependent components J = (JTi , JTd )

T corresponding

to the partitioning of the reaction rates. These steady

state rates satisfy

J = KJi =

ImrK0

Ji.

This is an immediate consequence of (4).

C. Input-Output Interpretation

System (1) enjoys only m degrees of freedom in the

m+n-dimensional output (2). This could be describeddirectly by observing that the m reaction rates are

free to vary independently, while any such choice

prescribes the evolution of the n species concentra-

tions. The proceeding analysis has described a more

useful characterization of freedom in r independent

concentrations and m r independent reaction rates.

We now address how the motion in these m degrees

of freedom might be influenced by the parameter

input p(). It will be illustrative to consider an overlyambitious control objective. Suppose given an m-

dimensional signal r(), partitioned into r1() Rmr

and r2() Rr, defined for t 0. Consider theproblem of forcing the system to track r perfectly,

in the sense that vi(t) r1(t) and si(t) r2(t).Clearly this will only be possible if r() satisfiescertain conditions.

Complete authority over the evolution of the inde-

pendent species concentrations lies with the dependent

transformed rates d = vd K0vi. To force theoutput to track the target according to (vi(t), si(t)) (r1(t), r2(t)) the input p() must satisfy

vi(Lr2(t) +T, p(t)) = r1(t), (5)

vd(Lr2(t) +T, p(t))= K0r1(t) + NCR

1 d

dtr2(t) (6)

for all t 0.

Equations (5) and (6) represent an extreme control

objective, but serve to illustrate the possibility of

addressing the behaviour of the independent concen-

trations and reaction states rates separately. A solution

could be reached by solving these m equations in

p at each time t. In the absence of redundancy

this is infeasible if the vector p has fewer than m

components.

In the metabolic setting, the standard choice ofenzyme activities as parameter inputs typically yields

precisely m input channels. Since each enzyme (typ-

ically) catalyses a single reaction, this results in a

set of reaction-specific parameters, i.e. each reaction

rate vj(s, p) depends on exactly one parameter: vj =vj(s, pj). In this case the system of equations in (5)and (6) takes on a simple uncoupled form.

Analysis of the complete authority control ob-

jective of (5) and (6) is a purely academic exercise

our ability to manipulate biochemical systems falls

far short of the requirements determined above. We

next consider a special case of this open-loop tracking

problem which is of more significance in a metabolicsetting. Keeping in mind the application of metabolic

redesign, consider the problem of asymptotic tracking

of a constant target r = (r1, r2) Rm. We will

neglect issues of stability and address the problem of

choosing a constant parameter input which introduces

a steady state satisfying vi = r1, si = r2. Specializingthe proceeding analysis to this case, we see that the

parameter vector p must satisfy

v(Lr2 + T, p) = Kr1. (7)

This steady-state tracking problem has been addressed

by Kholodenko et al. in [13], [14]. Under the as-

sumption that the parameters are reaction-specific and

appear multiplicatively, the solution is immediate, as

pointed out in [13]. A discussion of biochemically

relevant cases where those assumptions are relaxed

can be found in [14].

While the system of m equations in (7) cannot be

treated in general, a local description of the solution

can be investigated regardless of the form of the

reaction rates, as follows. Let p0 be a nominal pa-

rameter value corresponding to a steady state species


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concentration s0. Provided the matrix vp|(s0,p0) is

invertible, equation (7) can be used to locally define

the parameter input as a function of r, i.e. p = p(r).

In some cases it may be possible to solve for the

function p(r). More generally, the function can be

characterized locally by differentiating (7) at (s0

, p0

),which gives

v

p

p

r1= K and

v

sL +

v

p

p

r2= 0.

In terms of the vector r = (r1, r2), this can be writtenas

v

p

p

r= [K

v

sL]. (8)

The condition can be put in a more standard linear

setting by redefining r to describe the displacement of

the set point from the nominal position. Equation (8)

then gives a local description of the parameter p(r)required to track the point (vi(s0, p0), s0i ) + r. Wheninterpreted in this manner, (8) provides a link between

the results in this section and the main results of

MCA.

IV. METABOLIC CONTROL ANALYSIS

The field of Metabolic Control Analysis was born in

the mid-70s out of the work of Kacser and Burns [11]

and Heinrich and Rapoport [5], [6]. These two groups

independently arrived at an analytical framework for

addressing questions of control and regulation of

metabolic networks. Specifically, these papers out-

lined a parametric sensitivity analysis around a steadystate and also presented relationships between the

sensitivity coefficients. These relations, known as the

Summation and Connectivity Theorems, have been

used to provide valuable insights into the behaviour

of metabolic pathways.

Since its inception MCA has been used successfully

in the study of a great many metabolic systems. In

addition to elucidating these biochemical mechanisms,

this sensitivity analysis allows prediction of the effects

of intervention. As such, it is a powerful design tool

which has been adopted by the metabolic engineering

community (cf. e.g. [2], [15], [20]) and has been used

in rational drug design (cf. e.g. [1], [2]).

We follow the formalism developed in [17] (see

also [7], [9]). Beginning with system (1), follow the

decomposition described in Section III-A to arrive at

a row reduced system of the form (3). As mentioned

above, MCA is primarily concerned with local be-

haviour around an asymptotically stable steady state.

While this is the simplest behaviour to address ana-

lytically, the primary motivation for this narrow range

of focus is the difficulty of experimentally analysing

time-varying behaviour. In particular, at steady state,

an assumption of asymptotic stability is not restrictive,

since unstable steady states are not observed in the lab.

At steady state, the (reduced) system dynamics give

0 = NRv(Lsi + T, p). (9)

Suppose given a nominal parameter value p0 and a

corresponding steady state s0. Under the assumption

of asymptotic stability, (9) yields a local implicit

description ofsi as a function ofp, with si(p0) = s0i .

To determine the effect of small parameter changes on

this steady state si(p), differentiate (9) with respectto p at the nominal point to yield

0 = NR

v

sL

dsi

dp+

v

p

.

The sensitivities in species concentration, referred

to as unscaled independent concentration response

coefficientsare the components of the matrix

Rsip :=d

dpsi(p) =

NR

v

sL

1

NRv

p. (10)

This calculation can be extended to the complete

concentration vector s by defining Rsp := LRsip . Per-

turbations in the initial conditions can be treated along

similar lines (see, e.g. [9]) but will not be addressed

here. The use of the specialized terminology response

for the coefficients in (10) was introduced to distin-

guish these system sensitivities (total derivatives) from

component sensitivities (partial derivatives) which will

be introduced below as elasticities.

In addition to this sensitivity in the state variables,the sensitivities of the reaction rates, referred to as the

unscaled flux response coefficients, are also of interest:

Rvp :=d

dpv(s(p), p)

= v

sL

NR

v

sL

1

NRv

p+

v

p.

The first m r rows of this matrix constitute the in-dependentunscaled flux response coefficients Rvip :=ddp

vi(s(p), p).

These response coefficients represent absolute sen-

sitivities. In application it is the relative sensitivities,

reached through scaling by the values of the related

variables, that provide useful measures of system be-

haviour. In the biochemical literature (and especially

in addressing experimental data) the scaled versions

are preferred. On the other hand, from a mathematical

point of view, the unscaled sensitivities can be seen as

more fundamental. For that reason, we will deal with

unscaled coefficients in what follows.

The response coefficients describe the asymptotic

response of the linearized system to (step) changes in


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the parameter vector p. As such, they can be used

to predict the asymptotic effect of small changes in

the parameter values. However, in using sensitivity

analysis to address the inherent behaviour of a net-

work, it is often more useful to ignore the details of

the actuation and identify the reaction rates directlywith the parameters. In addressing absolute (unscaled)

sensitivities, this amounts to supposing that the reac-

tion rates depend on the parameters specifically and

directly: vp

= I. Under this assumption the responsesdefined above are referred to as the unscaled control

coefficients of the system. These are the primary

objects of interest in MCA as they provide a means to

quantify the dependence of system behaviour on the

individual reactions in the network. They are defined

by

Cs := L

NR

v

sL

1

NR (11)

Cv := v

sL

NR

v

sL

1

NR + Im. (12)

To make the distinction between component and sys-

tem sensitivities explicit, the partial derivatives of

v are referred to as the elasticities of the system.

Specifically, define the unscaled substrate elasticity

s :=vs

and the parameter elasticity p :=vp

.

A. Sensitivity Invariants: The Theorems of MCA

The stoichiometric nature of a biochemical network

imposes invariants on the sensitivity coefficients. De-

scriptions of these invariants originally appeared in

the papers by Kacser and Burns [11] and Heinrichand Rapoport [5], [6] and have since been generalized

and extended. These results, known as the Summation

Theorem and the Connectivity Theorem will be stated

next.

The Summation Theorem was stated in general

form by Reder in [17] in which it is observed that if

the matrix K is chosen as in Section III so that the

columns of K lie in the nullspace of N, then

CsK = 0 CvK = K, (13)

which follows directly from the definitions of the

control coefficients (11), (12).

Reder [17] generalized the original form of the

Connectivity Theorem to the algebraic statements

CssL = L CvsL = 0, (14)

which again follow directly from (11), (12).

These results can be stated simultaneously in the

Control Matrix Equation as given in [8]: combin-

ing (13) and (14) givesCv

Cs

K sL

=

K 0

0 L

.

In terms of the independent and dependent variables,

this statement can be written asCvi

Csi

=

K sL1

.

B. The Theorems of MCA: an Input-Output Interpre-

tation

Several researchers have investigated the conse-

quences of the Theorems for metabolic redesign.

The Universal Method of Kacser and Acerenza [12]

describes coordinated parameter changes which lead

to increased flux without affecting metabolite con-

centrations. The complementary problem of altering

species concentrations without affecting steady state

fluxes was addressed in [19] using the Connectivity

Theorem. A local treatment of the general output

tracking problem was presented in [21] and will be

highlighted below.

To determine the consequences for system be-haviour of the Theorems (13), (14), one can observe

that these equations describe response coefficients

corresponding to parameter perturbations of a par-

ticular form. Such statements are not directly useful

in addressing issues of controlling system (1) since

the parameters (i.e. the input channels) are specified

directly in the model. It will be helpful introduce

a new input variable and generalize the notion of

response coefficient to allow description of the effect

of simultaneous changes in multiple parameters.

To that end we introduce a q-dimensional input

space and call a function p : Rq Rm a coordinatedparameter input. The definition of system response

can be extended to inputs u Rq as the sensitivity ofthe system to perturbations of the form p = p(u):

Rsu :=d

dus(p(u))

Rvu :=d

duv(s(p(u)), p(u))

When q = m and p() is the identity map, thesereduce to the responses defined earlier.

When stated in terms of coordinated parameter

inputs, the Theorems of MCA can be given direct

interpretations as tools for network design since they

specify which choice of coordinated parameter input

p(u) will produce a desired effect.

Summation Theorem: Suppose a coordinated param-

eter input p(u) is chosen such that vu

lies in the

nullspace of N, i.e. vu

= KA1 for some (mr)qmatrix A1. Then

Rsiu = 0, Rviu = A1.

Connectivity Theorem: Suppose a coordinated pa-

rameter input p(u) is chosen such that vu

lies in the


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column space of sL, i.e.vu

= sLA2 for somer q matrix A2. Then

Rsiu = A2, Rviu = 0.

Together, these statements provide an alternative

version of the control matrix equation.

Control Matrix Equation: The response to any co-

ordinated parameter input p(u) can be characterizedas follows: Since the columns of [K sL] forma basis for Rm, there exists a unique m q matrixA = [AT1 , A

T2 ]

T such that vu

= [K sL]A, and

Rviu = A1, Rsiu = A2. (15)

A geometric interpretation of this result was presented

in [16].

In this design form, the Theorems provide a

transparent solution to the local tracking problem. To

make an explicit connection to Section III, considera coordinated parameter input p() which is linearand satisfies p(0) = p0. Then, if the parametricsensitivities satisfy (15), a constant input u in the

linearization of (1) leads to a steady state (vi, si) =(vi(s

0, p0), s0i )+Au. It was determined in Section IIIthat to track a steady state satisfying (vi, si) =(vi(s

0, p0), s0i ) + r we must take p = p(r) satisfying

v

p

p

r= [K sL]. (16)

This condition coincides with (15) when q = m,A = Im and u = r. (An alternative derivation ofthese sensitivity results can be reached by addressing

the linearization of system (1) directly as carried out

in [10].) We conclude that the discussion in Section III

can be seen as providing a generalization of the

Theorems of MCA to tracking of time-varying targets

in the original nonlinear system.

V. CONCLUSION

The discussion in this paper provides a control-

theoretic interpretation of the mathematical basis of

MCA, but addresses only a small part of the work

in that area. Once the mathematical preliminaries are

in place, the bulk of the analysis comes in applica-

tion: developing useful models, identifying important

questions, and interpreting the results.

Acknowledgments

The author would like to thank Herbert Sauro,

Christopher Rao, and Matthew Scott for helpful dis-

cussions. This work was funded by a Discovery Grant

from the Canadian Natural Sciences and Engineering

Research Council.

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