Tài liệu Báo cáo khoa học: What makes biochemical networks tick? A graphical tool for the identification of oscillophores ppt

An autoinfluence path runs from some species k back to species k and travels through positive influence steps [reaction nodes with equally directed arrows, as in Eqn 8] or negative influnce

What makes biochemical networks tick?

A graphical tool for the identification of oscillophores

Boris N Goldstein1, Gennady Ermakov1, Josep J Centelles3, Hans V Westerhoff2and Marta Cascante3 1

Institute of Theoretical and Experimental Biophysics, Russian Academy of Sciences, Pushchino, Moscow Region, Russia;

2

BioCentrum Amsterdam, Departments of Molecular Cell Physiology (IMC, VUA) and Mathematical Biochemistry (SILS, UvA),

Scientific Parc, University of Barcelona, Spain

In view of the increasing number of reported concentration

oscillations in living cells, methods are needed that can

identify the causes of these oscillations These causes always

derive from the influences that concentrations have on

reaction rates The influences reach over many molecular

reaction steps and are defined by the detailed molecular

topology of the network So-called Ôautoinfluence pathsÕ,

which quantify the influence of one molecular species upon

itself through a particular path through the network, can

have positive or negative values The former bring a

ten-dency towards instability In this molecular context a new

graphical approach is presented that enables the

classifica-tion of network topologies into oscillophoretic and

non-oscillophoretic, i.e into ones that can and ones that cannot

induce concentration oscillations The network topologies

are formulated in terms of a set of uni-molecular and

bi-molecular reactions, organized into branched cycles of

directed reactions, and presented as graphs Subgraphs of the network topologies are then classified as negative ones (which can) and positive ones (which cannot) give rise to oscillations A subgraph is oscillophoretic (negative) when it contains more positive than negative autoinfluence paths Whether the former generates oscillations depends on the values of the other subgraphs, which again depend on the kinetic parameters An example shows how this can be established By following the rules of our new approach, various oscillatory kinetic models can be constructed and analyzed, starting from the classified simplest topologies and then working towards desirable complications Realistic biochemical examples are analyzed with the new method, illustrating two new main classes of oscillophore topologies Keywords: graph-theoretic approach; kinetic modelling; oscillations; system identification; systems biology

Oscillatory biochemical networks have regained intensive

interest during the past few years because of the importance

of oscillatory signaling for various biological functions

Oscillations in glycolysis [1,2], oscillations of Ca2+

concen-trations [3,4], and the cell cycle as such [5] are well known

Some of these have been predicted and analyzed by using

mathematical models [6,7] The need for such mathematical

models is appreciated even more when studying biochemical

oscillations and their synchronization [7–13]

The behavior of potential biochemical oscillators may

depend on the kinetic properties of their surroundings,

interacting with the oscillator through common metabolites

(e.g [8,14]) Other systems, such as the cell cycle of tumor

cells may be more autonomous [9] Most intracellular

oscillations involve more than five components that interact

in a nonlinear manner [8] This makes them unsuitable for

intuitive analysis, a phenomenon encountered more fre-quently in Systems Biology [8] New theoretical approaches are needed that streamline the study of such cases of Systems Biology, dissecting the system into various inter-acting kinetic regimes, whilst relating to molecular mecha-nisms

Various types of approach can be helpful here Graph-theoretic approaches can help dissect the dynamics of enzyme reactions [15,16] and this is what made others and ourselves [20,25,26] examine whether these approaches can also do this for networks Earlier we have applied graph theory in order to simplify the King–Altman–Hill [15,16] analysis of steady-state enzyme reactions [17,18] This approach was later extended to presteady-state enzyme kinetics [19], to stability analysis of enzyme systems [20], and to the analysis of concentration oscillations in enzyme cycles [21]

In this paper, the graph-theoretical stability analysis developed by Clarke [22] as modified by Ivanova [21,23,24]

is the starting point for a more comprehensive approach

to the analysis of biochemical networks It enables us to develop a graph-theoretical identification of networks that may, and of networks that cannot, serve as oscillophores (i.e induce oscillations)

In some aspects our approach is similar to that reported previously [25,26] However, we use unimolecular and bimolecular steps and simple catalytic cycles, rather than

Correspondence to M Cascante, Department of Biochemistry and

Molecular Biology, Faculty of Chemistry and CERQT-Parc Scientific

of Barcelona, University of Barcelona, c/Martı´ i Franque`s 1, 08028

Barcelona, Spain Fax: +34 934021219; Tel.: +34 934021593;

E-mail: Marta@bq.ub.es or Hans V Westerhoff, Faculty of Earth and

Life Sciences, Free University, De Boelelaan 1087, NL-1081 HV

Amsterdam, the Netherlands Fax: +31 204447229;

E-mail: hw@bio.vu.nl

(Received 6 July 2004, revised 30 July 2004, accepted 4 August 2004)

quadratic and cubic autocatalytic cycles We identify

subschemes of the specific biochemical network that induce

instability We then consider interconnections between such

a subscheme and other parts of the kinetic scheme with or

without eliminating the instability The procedure of this

paper uses so-called dual graphs with two types of vertices,

i.e for both species and reactions [21] In this way all types

of reactions can be analyzed in a uniform manner The

procedure allows us to estimate the parameter values for

which oscillations occur Presence or absence of steady

states on the border of the phase space [21,23] then suffices

to predict the occurrence of limit-cycle oscillations We

illustrate our method by applying it to two biochemical

systems, which include oscillophores of two different classes

Results

Paths: graphical representation of kinetic influences

We represent kinetic schemes by dual graphs, combining

reaction-centered and substance-centered graphs [21]

Accordingly, our kinetic schemes for biochemical networks

have two kinds of vertices, i.e one kind for species (here

shown by open circles) and one kind for reactions (shown by

closed circles) The circles are connected by arrows For

example, the reaction xi+ xjfi xmis represented by the

following reaction-centered graph:

where xi, xjand xmare chemical species (substances) and vr

is the rate of the rth reaction Graph 1 shows that two

species xi and xj participate in the same rth reaction as

substrates with corresponding stoichiometric coefficients air

and ajr The species xmis synthesized in this rthreaction at

a stoichiometry bmr Using the mass-action law, in which

molecularity and kinetic order of reaction are equal, we can

write:

vr¼ kr
xair

i
xajr

where kris the kinetic constant This implies that we do not

dissect biochemical networks into the net enzyme-catalyzed

reactions, but into the unidirectional elementary reaction

steps underlying the enzyme kinetics The terms xi, xjand xm

include the concentrations of both metabolites and

enzyme-forms Rates vrare always positive As a consequence of the

dissection down to the molecular processes, the

stoichio-metric coefficients equal one or zero, i.e

air;bir¼ 1 or 0 ð2Þ with 1 for participating and 0 for nonparticipating species

Reactions involving more than one molecule of a single

species are described as a sequence of two independent

reactions Assuming spatial homogeneity and a single compartment, the kinetic equations for the reaction network are then written as follows:

dxi

dt ¼X

r

ðairþ birÞvr; ði ¼ 1; 2; :::; nÞ ð3Þ

where summation is over all r¼ 1, 2,…, R reactions Species xi(i¼ 1, 2,…, n) participate in these reactions as substrates and/or products, as illustrated graphically in the species-centered Graph 2:

ðr  1Þ


!biðr1Þ xi

!air

ðrÞ

Similarly to the procedure developed by Clarke [22], we linearize the system of Eqn (3) in the vicinity of the steady state We do this to investigate the stability of this state In this way we obtain the influence a small change in the concentration of substance j, i.e Dxj, has on the time displacement of the concentration of species i from its steady-state value:

dDxi

dt ¼X

r;j

ðairþ birÞ@vr

@xj

Dxj; ði ¼ 1; 2; :::; mÞ ð4Þ

where the summation is over both all r¼ 1, 2,…, R reactions and all j¼ 1, 2,…, m < n independent concen-trations From the law of mass action (Eqn 1) it follows for the kinetic order of the reactions that:

@vr

@xj

¼ ajr

vr

xj

ð5Þ Therefore, Eqn (4) can be rewritten as:

dDxi

dt ¼X

r;j

ðairþ birÞajr

vr

xj

Dxj¼X

j

bijDxj ð6Þ

Coefficients bijare the elements of the Jacobian (matrix) B representing the direct influences of xjon xi:

bij¼X

r

ðairþ birÞajr

vr

For small deviations from the steady state, the elements of

bij that multiply a b with an a characterize the sum of all reactions that convert xj to xi in a single step, i.e all reactions directed as xjfi xi Any reaction contributing to that overall reaction xjfi xidoes so to the absolute extent

vr/xj, the sign of its contribution depending on the direction

of the reaction, as specified by Eqn (7) The terms that multiply an a and a b therewith represent the positive influence that a substrate of a reaction has on the product of the reaction The terms of bijthat multiply two a’s, represent the negative influences of two substances on each other when both are consumed in that reaction Indeed, each element of the Jacobian corresponds to one or a number of such direct influences of one metabolite on another, direct in the sense that the influence is through single reaction steps

A number of such reaction steps may operate in parallel (but not in series) for each element of the Jacobian In addition, one reaction step may convey more than one influence

Graph 1.

To predict the dynamics of the system it is indeed helpful

to classify these influences into positive and negative ones

Again, depending on whether the reaction stoichiometry

is an a or a b, two types of influence are seen in Eqn (7)

They are shown graphically in Eqns (8) and (9) together

with their corresponding contributions to bij:

xj !ajr vr

!bir

xi; bij¼ ajr
þbir
vr

xj

¼ þvr

xj

ð8Þ

xj !ajr vr

air

xi; bij¼ ajr
air
vr

xj

¼ vr

xj

ð9aÞ

Although they are similar to Graph (1), Eqns (8) and (9a)

have meanings that differ from the meaning of Graph (1):

They do not represent chemical conversions but rather the

influences of one substance on another For this reason we

shall call them one-step influences or one-reaction

(influ-ence) steps: they are not branched and correspond to any

step between two substances in graphs such as Graph (1),

i.e any path that involves a single chemical reaction One

actual chemical reaction may effect a number of such

one-step influences, typically from any substrates onto any of its

products, between its substrates and of a substrate on itself

Equations (8) and (9a) should be read as follows They

indicate the influence the (production rate of the) substance

on the right may experience through reaction r, from the

substance on the left, which is a substrate of that reaction r if

the left hand factor a equals 1 (and not zero, as otherwise)

Such an influence exists and is positive if the substance on

the right is the product of that reaction (then there is a factor

b equal to 1) and the right hand arrows points towards that

substance Such an influence also exists but is negative when

the substance on the right is a substrate of the reaction

Then the arrow points backward, i.e away from the

substance and there is a right-hand factor –a equal to)1

Influences of the type in Eqn (8) contribute positive

values to bijand are called positive one-reaction (influence)

paths, or positive (influence) steps This is the influence that

a substrate has on the product of a reaction If i¼ j, this

positive path becomes a positive loop (see below)

Influences of the type in Eqn (9a) are designated as

negative one-reaction (influence) paths or negative influence

steps [21] because they contribute negative values to bij

They correspond to the influence of a substrate on another

substrate of the same reaction r If i¼ j, Eqn (9a) defines a

so-called negative half-step instead of a negative step (we

omit ÔinfluenceÕ for brevity):

xi !air



v r

air

xi; bii¼ air
air
vr

xi

¼ vr

xi

ð9bÞ which could also have been symbolized as:

xi !ðairÞ

2



v r

; bii¼ air
air
vr

xi¼ vr

xi ð9cÞ hence its name Ôhalf-stepÕ This is the (negative) influence a

substrate has on its own removal It is obtained for all

substrates of any elementary reaction

The main point of the present section is that any Jacobian

matrix element equals the sum of a number of direct parallel

influence steps (one-step influence paths) in the kinetic

scheme, i.e the sum of paths through reaction-centered

graphs of the type of Graph 1 (these paths may contain parallel and antiparallel arrows) The sign of that element therefore depends on the both the sign and the magnitudes

of these influence paths (see below) If all its influence paths are positive, the Jacobian matrix element will be positive and for the Jacobian matrix element to be negative at least one influence path must be negative These are properties that we shall use below

How graphical structures relate to instability For the linear system given in Eqn (6) the so-called characteristic polynomial p(k) is:

pðkÞ ¼ detðB  kIÞ ¼ 0 ð10Þ Here B is again the Jacobian with elements bijand I is the unit matrix The polynomial Eqn (10) can be expanded as follows

pðkÞ ¼ kmþ a1km1þ a2km2þ ::: þ am¼ 0 ð11Þ where m < n continues to refer to the number of inde-pendent concentration variables The coefficient aiis related

to the element bijof the Jacobian by:

a1¼ ð1Þ1
X

i

bii; a2¼ ð1Þ2
 X

i;j

biibjjX

i;j

bijbji



;

a3¼ ð1Þ3
 X

i;j;k

biibjjbkkX

i;j;k

biibjkbkj

i;j;k

bijbjkbki



; ; am¼ ð1Þm
det B ð12Þ

Each coefficient of the characteristic equation hereby is a ÔsumÕ of products (with various signs, see below) of elements

of the Jacobian

In graphical terms the coefficient apequals the sum of all possible Ôpth order simplest combinations of minus auto-influence pathsÕ An autoauto-influence path (or, shorter, a cycle)

is defined as a cyclic path of any length through the diagram, such that any reaction and any species occurs only once on that path Autoinfluence paths of lengths 1, 2, 3, etc correspond to the terms bii, bijbji, bijbjkbki, etc., respectively,

in Eqn (12) They contain 1, 2, 3, etc species and 1, 2, 3, etc influence steps, respectively Autoinfluence paths of length 1 are half-steps, graphically represented as in Eqn (9c) An autoinfluence path runs from some species k back to species

k and travels through positive influence steps [reaction nodes with equally directed arrows, as in Eqn (8)] or negative influnce steps [reaction nodes with oppositely directed arrows, as in Eqn (9a)] Consequently, a Ôminus autoinfluence pathÕ is negative (Ôeven cycleÕ) if the number of its negative steps as shown in Eqn (9a) is even and positive (Ôodd cycleÕ) if the number of its negative steps is odd

A Ôminus combined autoinfluence path of order pÕ is defined as a set of minus ÔcyclesÕ such that (a) each of a set of

pspecies is involved precisely once in that set of cycles, and (b) the various cycles in this combination have no reactions

or species in common (i.e the cycles in such a combination

do not touch each other) The value (the sign) of a cycle is the product of the values (the signs) of its steps The sign of the magnitude of a Ôminus combined autoinfluence path

of order pÕ equal those of the arithmetic product of its

component Ôminus cyclesÕ Accordingly, the influence

(positive or negative) of a combined autoinfluence path

depends only on the number of its Ôeven cyclesÕ, having an

even number of negative one-step influences (Eqn 9a) and

any number of positive one-step influences (Eqn 8) In other

words, if the number of Ôeven cyclesÕ in the combination is

even, this combination contributes to the coefficient of

characteristic equation a positive term Thus, a single Ôeven

cycleÕ gives rise to a negative in the characteristic equation

(a positive autoinfluence) The absolute magnitude of a

combined autoinfluence path is the product of all the rates

divided by the product of all the concentrations of its

species Therewith the coefficient of order p of the

characteristic equation equals the sum of all simplest

combined Ôminus autoinfluence paths of order pÕ in the

network That each coefficient of characteristic equation

therewith corresponds to a sum of minus paths in the

network, is the basis of the graphical analyses of the

characteristic equation and of the method we develop here

Inspection of Eqn 12 shows that in all coefficients apthe

term consisting of negative half-steps (Eqn 9c) only, which

corresponds to products of Jacobian elements bii only, is

always positive: of the term of order p the sign is ()1)p

multiplied by ()1)p Indeed, all these terms always constitute

negative combined autoinfluence paths

This graphical procedure allows us to determine all

coefficients of the characteristic polynomial for systems of

simple reactions The graphical determination of

character-istic polynomial coefficients for complex stoichiometries has

been elaborated by Ivanova [23]

The concentrations are restricted by balance constraints

(conserved sum concentrations, such as NADH + NAD)

and by the requirement that they be positive These

restrictions define upper and lower limits for the values

the concentrations can assume (i.e borders of the so-called

phase space) Any negative aicoefficient implies that the

system is unstable [22] Such instability could lead to infinite

growth (explosion) of some concentrations, unless the

highest-order coefficient am is or becomes (the reference

state may shift) positive If the system does not have steady

states on its border, all phase trajectories lead inward

Steady states on the border are readily identified ([21] and an

example below)

That am be positive and ai (i < m) be negative in a

(unstable) steady state [25] (together with the border

conditions mentioned above [21]) is what we shall here call

the ÔoscillophoreticÕ condition, i.e the condition for a stable

limit cycle around the unstable steady-state point All phase

trajectories (i.e all time evolution of the system through the

space of the concentrations) should then approach a cyclic

trajectory more and more closely as time proceeds,

approaching that stable limit cycle either from the outside

or from the inside

In this paper we focus on this aspect of instability We

shall ask when the above instability condition, i.e at least

one ai being negative, is met We shall not consider the

condition that ambe positive The formalism described in

the preceding paragraphs will help us find the graphical

structures, i.e ÔsubgraphsÕ (see below), in the kinetic scheme

that contribute terms to the coefficients of the characteristic

polynomial of a predictable sign and that hence help

determine the stability properties of the system The aim of

this paper is to identify ÔnegativeÕ subgraphs, because they can induce instability; their positive combined autoinfluence bestows them with oscillophoretic potential

The instability condition that ap be negative for some

p< m translates to the condition that the positive combined autoinfluence paths of order p should outweigh the negative combined autoinfluence paths of that same order From this, an Ôinstability ruleÕ follows This is stated

as ÔInstability is promoted (counteracted) by positive autoinfluence paths.Õ This connotes with instability being generated by positive feedback loops

Subgraphs favoring instability

As mentioned above we deal here with the formulation that decomposes biochemical networks into truly elementary reactions A network then consists of a great many such reactions (each represented as a black node in our reaction equations), each of which connects a number of species (represented as white nodes) The entire network may become unstable when part of it would by itself be unstable Consequently it can be useful to identify parts of the larger network that are unstable

The graphical representation of a subnetwork with equal numbers of species and reactions is here called a subgraph

It is useful to consider subgraphs because all combined autoinfluence paths that visit all reactions and species within such a subgraph have equal absolute magnitudes (i.e the product of the rates divided by the products of all the species concentrations) but may differ in sign By considering all such combined autoinfluence paths of a subgraph together, one can therefore decide whether the subnetwork as a whole promotes or counteracts instability: one simply determines whether more positive than negative combined autoinflu-ence paths occur in that subnetwork We shall speak of a negative subgraph in this case We define the value of a subgraph that contains p reactions and p concentrations, as minus the sum of all its combined autoinfluence paths of order p Note therefore that negativity of a subgraph and positivity of autoinfluence connote with instability

We shall now determine the signs and hence the stability properties of a number of subgraphs We shall do this first for subnetworks that consist of a single reaction, then for subnetworks of two reactions, then for subnetworks of three reactions Finally we shall consider subnetworks of arbi-trary size

One-reaction subgraphs For the elements bii,r,1 that correspond to the reactions r in which xi drives its own production (i.e xifi xi), there is both a negative half step (because, as usual, xistimulates its own removal; Eqn 9c) and a positive loop [because xinow also stimulates its own production; compare Eqn (8) with i¼ j ] Adding these two, Eqn (7) shows that they cancel each other:

a13 bii;r;1¼ ðairairþ birairÞ
vr

xi¼ 0; ð13Þ where the symbol’ means ÔcontainsÕ The value of zero is obtained because all stoichiometric coefficients equal one (i.e the reaction xifi xi cannot lead to a net increase in xi because of the restrictions we here impose on the stoichio-metries; we can only have such a reaction produce a single

molecule of xi) These reactions are thus without any

influ-ence, as the negative influence is balanced by the positive one

In terms of autoinfluence paths, the former negative

half-step is one cycle with one negative one-half-step influence, hence

a negative autoinfluence and stabilizing, whilst the latter

positive loop is one cycle with no negative one-step

influence, hence a positive autoinfluence and destabilizing

but of equal magnitude (because it belongs to the same

subgraph): the two cancel Autocatalytic processes such as

xifi 2xi are not described as single reactions in our

formalism, as stoichiometry b would exceed one

What remains for a1 is all the reactions that have xias

the substrate and not as the product Therefore, a1 is

constructed only from the corresponding negative half-steps

with the values [compare Eqns (7) and (9c)]:

a13 bii;2¼ airair
vr

xi

¼ vr

xi

ð14Þ This corresponds to a single cycle (from xiback onto itself)

with a single negative influence step, i.e it is negative in

terms of autoinfluence and promotes stability

The sum total for one-reaction subgraphs is thereby

always positive (their total autoinfluence is negative)

Indeed, it follows from Eqns (12) and (14) that the

coefficient a1 is always positive, favoring stability We

conclude that one-reaction subgraphs cannot give rise to

instability These need not be analyzed therefore for

deciding on the potential instability of large networks

Two-reactions subgraphs We here consider examples of

subgraphs with two species and two reactions, such as the

branched cycle:

According to Eqns (12) and (7) this subgraph contributes

to minus the a2coefficient of the characteristic polynomial

the two terms on the right-hand side of the following

equation:

a2¼ biibjjþ bijbji ð15Þ where

biibjj¼ ðairairÞ
ða jsajsÞ þ ajsbjs

vrvs

xixj

¼ ðairairajsajs airairajsbjsÞ
vrvs

xixj

¼ 0 ð16Þ

bijbji¼ ðajsbisÞ
ðairbjrÞ
vrvs

xixj

¼vrvs

xixj

ð17Þ

In the first factor of Eqn (16), which corresponds to the

direct self-influence term b, one recognizes the influence

that xihas on itself through its own degradation vr(the path

of influence –airair) In the second factor of Eqn (16), which corresponds to the direct influence of j on itself (bjj), one recognizes the two direct influences xjhas on itself, i.e one through its degradation (–ajsajs) and a second influence through its autocatalytic feedback (+ajsbjs) through the same reaction, vs The latter influences cancel each other, again as all stoichiometries are equal to one

The second phrasing of Eqn (16) corresponds to the sum

of two autoinfluence paths of order two Both of these contain the negative half step of xiback onto itself which is negative One of them multiplies with the negative half-step

of xjback onto itself and constitutes negative autoinfluence (even number of cycles and even number of negative influences) The other multiplies with the positive loop of xj back onto itself through vs: two cycles with one negative influence constituting a positive (destabilizing) autoinflu-ence These two autoinfluence paths of order two cancel each other

In Eqn (17) one recognizes the influence that xjhas on xi (+ajsbis) because the former is the substrate of the reaction

vsthat produces the latter, as well as the analogous influence

xihas on xj(+airbjr) Together they constitute the positive influence that xi has on itself through the negative path constituted by the sequel of reactions vrand vs Equation (17) corresponds to a single even cycle with two positive influence steps, i.e promoting positive autoinfluence and hence instability The net sign of the three autoinfluence paths is)1 + (+1) + (+1) ¼ +1, i.e Graph (3) contri-butes the negative (instability) term [Eqn (17)] to a2 Graph (3) is negative

As Graph (3) will be part of a larger network, whether it actually will be able to cause instability (oscillations) depends on whether the precise kinetic parameter values make its positive autoinfluence dominate the negative autoinfluence in the other subgraphs of order two of the network It may be noted that the corresponding subgraph that lacks the loop, i.e in which reaction s does not reproduce its substrate xj, cannot be negative, and hence cannot cause instability Then only the negative autoinflu-ence path of Eqn (16) remains, which then cancels the positive autoinfluence of Eqn (17) Revolving around the cycle then does not lead to an increase in the number of molecules Elementary reactions producing more types of product than types of substrate are essential for the occurrence of instability, due to the restrictions on stoichiometries that derive from our descent to the molecular level

Another branched cycle, Graph (4), having two bran-ches, contributes to –a2the same positive, destabilizing term:

Graph 3.

Graph 4.

Negative Graphs (3) and (4) involve positive paths with

branching that can be interpreted as positive feedback

interactions (autocatalysis) They can also be interpreted as

product activation in some enzyme reactions, because a

reaction product here stimulates the same reaction Another

example of such positive paths of influence occurs in the

case of the antiport of two ligands by a protein molecule

through the membrane [24]

Three-reaction subgraphs In the same way we identify

the negative (instability generating) subgraphs with three

species and three reactions, by pointing out that they

have positive paths of influence We divide these graphs

into two classes, i.e those with positive influence steps

only, and those including an even number (two) of

negative steps in the cycle The former class is shown in

Graph (5):

The latter class of graphs, involving two negative steps in

the cycle with three species and three reactions, is presented

in Graph (6)

All of the subgraphs in Graphs (5) and (6) contribute the

following 3! terms to the coefficient a3:

a3¼ ð1Þ3ðb11b22b33þ b21b13b32þ b31b12b23 b11b23b32

 b22b13b31 b33b12b21Þ

The first term here corresponds to all half-steps multiplied,

the second and the third terms correspond to the circular

paths running through all three reactions and all three

species The other three terms correspond each to a single

half-step, multiplied by a circular path running through two

reactions and two species

Using the graphical rules mentioned above, we now

consider the left hand subgraph in Graph (6) as an example

This subgraph contains the following three simplest

com-bined autoinfluence paths (simplest subgraphs):

1 One combined autoinfluence path is the positive cycle

going through all three species and all three reactions,

from species one to species two to species three

(corres-ponding to b12b23b31) This one cycle involves two

negative steps such as shown in Eqn (9a) and then a

positive step [as in Eqn (8)] The rule then makes for odd

(number of cycles), even (number of steps), hence positive combined autoinfluence: + 1 The second such auto-influence path, i.e the reverse of this cycle is absent here

2 A second combined autoinfluence path consists of one cycle going through the two species and two reactions on the left hand side of the subgraph, and one negative half step on the right hand side (as indicated in bold;

b12b32b33) The arithmetic product of these two positive autoinfluence paths is positive (once cycle, no negative influences): +1 The mirror image combined autoinflu-ence path (b12b12b33) would run through the two reactions on the right hand side (negative), and one negative half-step on the left (positive) However, the latter would touch the cycle, hence this one does not count The third term, i.e b11b23b32is also ÔemptyÕ for this diagram

3 The third type of combined autoinfluence is negative: a combined influence of three separate anti half-steps:)1 The sum of these two positive and one negative combined autoinfluences contributes a negative term into the coeffi-cient a3, and therewith promotes instability The subgraph

on the left of Graph (6) is therewith negative

In fact all of the subgraphs in Graphs (5) and (6) contribute the positive termv1 v2v3

x1x2x3to minus the a3coefficient

of the characteristic polynomial (the negative term to the coefficient a3), where subscripts 1, 2 and 3 refer to the different species and reactions in the subgraph Consequently they promote instability All of these subgraphs are negative All subgraphs represented in Graphs (5) and (6) have a single branched reaction, constituting two so-called Ôeven cyclesÕ An Ôeven cycleÕ involves an even number (here 0 or 2)

of equally directed influence steps and any number of positive influence steps Such cycles cause graphs to become negative

In full reaction schemes some reactions start from species

in a negative subgraph These efflux reactions can eliminate the negative subgraph, when the system reaches its steady state In the steady state the equality of rates for internal and external opposite reactions eliminates the negative term, induced by the negative subgraph Then only damped oscillations can be observed (see below for examples) To obtain sustained oscillations, the efflux reaction should be reversible, leading, for example, to an inhibitory enzyme complex (see below for an example) In the latter case the reversible steady-state efflux equals zero and the negative subgraph is upheld

n-Reaction subgraphs We can now formulate the proper-ties of any negative subgraph that contains an arbitrary equal number of species and reactions Such a negative graph should be constructed at least of two even cycles, formed by a branched reaction Moreover, species of the negative subgraphs should not be connected with other parts of the full scheme by outgoing irreversible reactions The outgoing irreversible reactions cause the correspond-ing opposite stationary fluxes to be equal, cancelcorrespond-ing the negative subgraph with a positive graph of the same absolute value (see below for an example) Therefore, only damped oscillations can be obtained in such a case Additional reversible reactions, leading through the species of the negative graph to dead-end species, do not eliminate the negative graph

Graph 5.

Graph 6.

Examples of biochemical oscillations

Two interacting enzymes Here we discuss one of the

simplest biochemical oscillators Its kinetic scheme (Fig 1)

contains a negative graph [Graph (4)] of second order (two

species and two reactions) This suggests that the system of

Fig 1 may oscillate A more detailed analysis should then

be undertaken to determine whether it will actually oscillate

We shall now do this

In Fig 1, two enzymes, E1 and E2, modify each other

by releasing group P in the reactions E2-P fi P + E2

and E1-Pfi P + E1 The former reaction is catalyzed by

E1, the latter by E2 The arrows indicate the preferential

reaction orientation Various biochemical systems can be

recognized in Fig 1, for example, mutual

dephosphory-lation (or phosphorydephosphory-lation) if P represents phosphate P is

not considered to be a variable; it should be present in

excess

The scheme in Fig 1 is open for the fluxes through E1

(e.g synthesis, degradation) but the total concentration of

E2is conserved:

½E2-P þ ½E2 ¼ E2or x4þ x2¼ 1 ð18Þ

The reaction participants and their normalized

concentra-tions xi (i.e their concentrations divided by the total

concentrations of E2) are shown in Fig 1 The following

kinetic equations correspond to Fig 1:

dx1

dt ¼ k2x2x3 k4x1

dx2

dt ¼ k1x1x4 k3x2

dx3

dt ¼ k5 k2x2x3

ð19Þ

Taking into account the constraint in Eqn (18), Eqns (19)

involve three independent variables, i.e x1, x2, x3

We shall now analyze the characteristic polynomial of the third order for Eqns (19) We know that the coefficient a1is always positive, and it is readily seen that the coefficient a3 is also positive here The coefficient a2

contains a negative term, which corresponds to the negative graph highlighted in Fig 1 by the heavy lines This negative term equals –v1v2/x1x2 At steady state v2¼

v4and v1¼ v3 Then the positive term +v4v3/x1x2, which

is also present in the coefficient a2 for Fig 1, cancels the term –v1v2/x1x2 of the negative graph Therefore, in the steady state a2 is positive Consequently, only damped oscillations can be observed in Fig 1

Considering the structure of the reaction scheme in Fig 1, one recognizes that irreversible effluxes of species

of the negative graph must be present in order for the steady state condition to be satisfied For the same reason, all the biochemical schemes involving a negative graph of second order can only induce damped oscilla-tions Damped oscillations calculated for Fig 1 are shown in Fig 2

Substrate inhibited bifunctional enzyme Many kinetic graphs that generate oscillations with only positive auto-influence paths are known from the literature Some of them have been classified [25] Although the graphs with positive paths implemented here e.g Graphs (3–5) are simpler than the kinetic graphs in [25], they can represent biochemical reality For example, the second of the subgraphs in Graph (5) has been used to analyze the network topological basis for oscillatory antiport of two different ions across the cell membrane [24]

Less studied are the graphs that include negative paths, such as those in Graph (6) Two negative paths in the cycle

of subgraphs here reflect the competition of two reactions for a single species For example, the competition of protein

X and the enzyme E2 for the acetyl group in pyruvate dehydrogenase complex has been shown to be important for the prediction of oscillatory behavior [28]

The phenomenon of substrate inhibition is often associ-ated with the potential for oscillations [33] An earlier

graph-Fig 2 Calculated time dependence of the normalized E 1 concentration (X 1 ) for Fig 1 Time scale in relative units The following parameter values were used for these calculations: k 1 ¼ 0.1, k 2 ¼ 1, k 3 ¼ 2.2, k 4 ¼

5, k 5 ¼ 1 The dimensions of these parameters values are not specified because they depend on the time scale and can be different for different actual systems Species concentrations were normalized The initial values of the normalized concentrations were: x 1 (0) ¼ 1, x 2 (0) ¼ 0.4,

x 3 (0) ¼ 0.2, x 4 (0) ¼ 0.6 Calculations used the computer program

DBSOLVE (created by I I Goryanin, Institute of Theoretical and Experimental Biophysics, RAS, Moscow Region, Russia).

Fig 1 Reaction scheme of two enzymes demodifying each other Filled

circles represent reactions, open circles represent substances The rate

of the reaction that combines E 2 with P to yield E 2 -P is given as v 3 If P

represents a phosphate group, reaction number 3 could be a protein

kinase, and v 1 should then represent the dephosphorylation of E 2 -P, as

catalyzed by E 1 This reaction releases P and E 2 In this reaction E 1 is

used but immediately released as it is a catalyst The rate of reaction 2 is

v 2 , which is catalyzed by E 2 (which then functions as a protein

phos-phatase) and dephosphorylates E 1 -P Reaction 4 degrades E 1

Reac-tion 5 synthesizes E -P de novo (i.e not from E ).

theoretic analysis, however, [21] showed that some kinetic

schemes with substrate inhibition cannot induce sustained

oscillations We shall here discuss an example of subgraphs

with substrate inhibition, in the context of oscillations

observed for

phosphofructo-2-kinase:fructose-2,6-biphos-phatase [21,27] Nonlinear oscillations or bistable switches

in this bifunctional enzyme could be highly important for a

switching mechanism between the opposing fluxes in

glycolysis/gluconeogenesis This case has been analyzed

before (e.g in [21,27]), but this paper will now present the

detailed analysis of conditions necessary for the negative

graph to induce sustained oscillations We shall analyze the

steady states, because the presence of steady states on the

phase-space border and the irreversible steady-state efflux

from the species of the negative graph can eliminate

oscillations

The following kinetic graph (compare [21,27]) is drawn to

illustrate the analysis of the steady states

Graph (7) shows the substrate cycle, S1fi S2fi S1, as

catalyzed by the bifunctional enzyme (E1/E2; E1and E2are

two states of a single protein) Here the arrows between

symbols correspond to the preferential reaction orientation

In reaction 1, E1 catalyzes the forward reaction S1fi S2

and E2 catalyzes reaction 2, which runs in the opposite

direction, S2fi S1 Reaction 2 may be coupled to a source

of external free energy Alternations between two enzyme

activities are caused by conformational transitions, induced

by the modifying enzymes, E3(catalyzing reaction 3) and

E4(E4is not shown in the scheme) The reaction S1fi S2is

merely catalyzed by E1alone, but during the reaction S2fi

S1the enzyme undergoes cyclic conformational transitions

E2fi E1fi E2, where the latter transition is catalyzed by

E3 Graph (7) does not contain irreversible effluxes from

the species (E2, E3 and S2) of the negative graph that is

shown by heavy arrows, and contains only the influx to S2,

i.e 1fi S2(from reaction 1 to S2) Therefore it retains its

oscillophoretic potential Inhibitory reversible reactions,

added to the negative graph, do not interfere with that

potential

The subgraph highlighted by the heavy arrows in the full Graph (7), is one of the negative graphs identified

in this paper [i.e the second left of the subgraphs in Graph (6)] This negative graph is the branched cycle with one positive loop, representing the E3 catalyzed reaction that makes E2 out of E1, and one longer cycle, involving two negative influence steps, E3fi 4 ‹ S2and

S2fi 2 ‹ E2 These two negative influence steps corres-pond to the competitive interactions of S2 with E3 and with E2 The reaction E3fi 4 ‹ S2 is the forward reaction for E3 inhibition by S2 For simplicity, the reverse reaction (not participating in the negative graph) is not shown

Oscillations can be expected if we add reversible inhibi-tion of E3 by substrate S1 to Graph (7) The reversible inhibitions of E3by both S2 and S1 do not eliminate the negativity of the negative subgraph, because these reversible steps do not contribute additional terms to the terms of the negative graph Their contributed effluxes are equal to influxes However, the number of species of reactions becomes larger with this new inhibition Accordingly, a positive graph with four species and four reactions, as well

as a negative graph with three species and three reactions, are obtained in Graph (7) This is a sufficient condition for oscillations to arise

We shall now show how a necessary condition for oscillations to occur follows from the absence of steady states on the border of the phase space The full system contains seven species variables:

x1¼ ½E1; x2¼ ½S1; x3¼ ½E2; x4¼ ½S2;

x5¼ ½E3; x6¼ ½E3S2; x7¼ ½E3S1 These species are interdependent through the following three balance constraints:

x2þ x4þ x6þ x7¼ S ¼ constant

x1þ x3¼ E ¼ constant ð20Þ

x5þ x6þ x7¼ E0¼ constant These constraints reflect the conserved total concentra-tions of the substrates (we shall use S¼ 3.3 relative units) of the bifunctional enzyme (E¼ 0.2 relative units), and of the modifying enzyme (E¢ ¼ 0.31 relative units) The following four equalities for the steady state reaction rates are deduced from the structure of Graph (7):

v1¼ v2¼ v3; v4¼ v5; v6¼ v7 ð21Þ where indices 1,2,3,… relate to various reactions, v4and v5 relate to the reversible reaction E3+ S2b« E3S2, v6and v7 relate to the reversible reaction E3+ S1b « E3S1 The equalities [Eqn (21)] together with the constraints [Eqn (20)] allow us to obtain all seven concentration values for one of the steady states in the phase space [E1]¼ 0, [S2]¼ 0, [E3S2]¼ 0, [E2]¼ E, and [S1], [E3], [E3S1] inside of the phase space No steady states exist on the borders of the phase space This is the necessary condition for sustained oscillations to be observed in this system

The stability of the steady states was analyzed by using four differential equations for four independent species variables:

Graph 7.

dt ¼ k2x3x4 k3x1x5

dx2

dt ¼ k1x1x2þ k2x3x4 k6x2x5þ k7x7

dx3

dt ¼ k3x1x5 k2x3x4

dx4

dt ¼ k1x1x2 k2x3x4 k4x4x5þ k5x6

ð22Þ

In addition to referring to the absence of steady states on the

border of the phase space, the procedure by Clarke [22]

enables us to identify qualitatively phase trajectories that

lead to a stable limit cycle The characteristic polynomial of

the system in Eqn (22) reads:

k4þ a1k3þ a2k2þ a3kþ a4¼ 0 ð23Þ

If in this polynomial a4> 0 for all concentration values

and a3< 0 in the unstable steady state, oscillations can be

obtained Analysis of negative and positive subgraphs and

comparison of their values gives rise to the estimation of the

kinetic parameters that enable such oscillations The main

result of such an analysis is that oscillations arise if the

parameter k3is the largest and the parameters k6and k7are

the smallest in the system Oscillations in this system can

indeed be observed [27]

Discussion

Oscillatory phenomena in biochemical systems are being

studied more and more intensively All known kinetic

models for calcium oscillations have been reviewed

recently [4] Models for other oscillatory phenomena

continue to appear [33–39] and many more will appear in

the future with the increasing possibilities for inspecting

the dynamics inside single cells [40,41] Our classification

of kinetic schemes (or ÔmotifsÕ [40]) into ones that may

and ones that cannot exhibit oscillations may be useful

for the analysis of the existing models that are responsible

for the oscillations Such an analysis may help to

understand the mechanisms underlying the oscillations,

and perhaps even suggest ways to influence the dynamics

of such systems Our method is based on the molecular

mechanisms without any preliminary simplifications and

without using phenomenological equations It may

there-fore be suitable, especially now that functional genomics

is unraveling more and more of the molecular specifics

that underlie cell function

Our method to classify potential biochemical oscillators is

based on the graphical analysis of the kinetic schemes Our

approach is similar in some aspects to the procedure

described previously [25,26] However, the representation of

the kinetic schemes in terms of dual graphs [21] is different,

and has enabled us to simplify the identification and the

classification of oscillophoretic networks

Because above we were most concerned with

demon-strating the basis of our method, we here summarize how

the approach may be implemented in the context of a

known reaction network First the network kinetics

should be drawn out in a detailed molecular scheme

making all molecular interactions, such as the binding of

a ligand to an enzyme, explicit Then one should try to recognize subgraphs of known sign in that scheme Here one may resort to the subgraphs identified in this paper,

or to subgraphs that may appear in future work analyzing networks more extensively Alternatively, one may use the method of making an inventory of the autoinfluences within each subgraph and determine whether there are more positive ones than negative ones,

in which case the subgraph is negative (unstable) Having identified the (negative) subgraphs with oscillophoretic potential, one may then analyze their effect quantitatively and compare the results to those obtained for through analysis of all other subgraphs of the same order in the same network, as was illustrated for the two examples in this paper The network outside the former subgraph may do away with the oscillophoretic potential of the subgraph or maintain it by contributing subgraphs of equal order but of different or equal sign and magnitude: the dynamic development of a system is ultimately dictated by the influences the concentrations of its substances have on each others (and their own) develop-ment in time [7,35,42]

When systems are analyzed on a more coarse-grained level than we do here, the influences are not defined in terms

of rates, concentrations and reaction stoichiometries only

In such analyses, other properties such as elasticity coeffi-cients [31], Michaelis constants, and kinetic powers [29,30] also determine the dynamics and stability of the system [12]

To the extent that these analyses deal with the origin of dynamics in terms of network topology, then that topology

is the topology of influences This type of more coarse-grained analysis is useful when the systems are not yet understood to molecular kinetic detail, or when the systems are so large that a detailed molecular analysis is beyond reach and modularization is required

Because we here analyze at the level of complete molecular detail (i.e only reactions with zero and first order kinetics), the topology of the influences coincides with the topology of the network stoichiometries Our method has this as an important advantage, which comes with its other advantage of being completely molecular This very advant-age can of course become a disadvantadvant-age in cases where molecular detail is not known or required The dynamics of cellular systems are determined at many different levels of the cellular control hierarchy For different levels of the hierarchy, different methods for the analysis of the dynam-ics are needed

We demonstrated how reaction networks that are formulated down to the detail of simple unimolecular and bimolecular reactions can be organized into topologies The latter can then be examined for their potential to induce oscillations Oscillophoretic topologies involve branched directed cycles, constructed of an even number of negative paths and any number of positive paths Our approach has the advantage that it considers positive and negative interactions in a unified manner

The implication of the identification of an oscillophoretic subgraph is that if such a subgraph is found in a large network, then that network may be unstable and give rise to oscillations; the presence of an oscillophoretic subgraph is a necessary condition for the network to engage in the oscillations However, it is not a sufficient condition

Whether the overall network actually engages in an

oscillation when an oscillophoretic subgraph is present

depends on the precise parameter values To estimate the

parameter domain where oscillatory phenomena can be

observed, the numerical value of the negative graph should

be compared with the values of other graphs of the same

order in the system In practice this means that to produce

oscillations, reactions involved in the negative graph should

be rapid enough as compared with their surrounding

reactions We here performed such an analysis for two

examples, one with positive and one with negative

inter-actions

We classified graphs of different topologies with two

species and two reactions as well as with three species and

three reactions, which can induce oscillations, if they are

connected with other parts of the system Sustained

oscillations can be induced if these connections are

irreversible influxes or reversible dead-end reactions All

considered topologies involved a single branched reaction

More complicated topologies with additional reaction

branching do not eliminate oscillations Graphs of similar

topologies but with different numbers of species and

reactions (the number of species and reactions in the

analyzed graphs is the same) retain the oscillophoretic

property

On the basis of their network topologies, our approach

can predict a number of new biochemical oscillators that

fit the classification developed here, but were not included

in former classifications [25] It turned out that not only

well-known substrate inhibition and product activation

induce oscillations Any competition of a single, channeled

intermediate for multiple active sites in multienzyme

com-plexes can also induce oscillatory kinetics [28]

Our approach to Ôoscillophore topologiesÕ can be

com-bined with other known theoretical approaches [29–32] to

simplify the study of complex biochemical systems It

contributes to the recognition that biochemical networks are

more subtle than hitherto realized Not only the control of

flux but also the control of the occurrence of oscillations is a

subtle function of network topology and (in the more

coarse-grained approaches) enzyme elasticities There may

not be a single oscillophore, but rather a number of

component properties that contribute to the tendency of a

system to engage in more complex behavior such as

limit-cycle oscillations Actual and subtle interactions of the

components then determine whether or not the oscillations

actually occur

Acknowledgements

This work was supported by a grant from Ministry of Science and

Technology of the Spanish Government (SAF 2002–02785), INTAS

grant (97–1504), and the Netherlands’ Organization for Scientific

Research We thank T Sukhomlin for discussions.

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