Control analysis of periodic phenomena in biological systems

General rights It is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), other than for strictly personal, individual use, unless the work is under an open content license (like Creative Commons). Disclaimer/Complaints regulations If you believe that digital publication of certain material infringes any of your rights or (privacy) interests, please let the Library know, stating your reasons. In case of a legitimate complaint, the Library will make the material inaccessible and/or remove it from the website. Please Ask the Library: https://uba.uva.nl/en/contact, or a letter to: Library of the University of Amsterdam, Secretariat, Singel 425, 1012 WP Amsterdam, The Netherlands. You will be contacted as soon as possible. Download date: 29 Jun 2019 Control Analysis of Periodic Phenomena in Biological Systems Boris N. Kholodenko,* , † Oleg V. Demin, ‡ and Hans V. Westerhoff §,| Department of Pathology, Anatomy and Cell Biology, Thomas Jefferson UniVersity, 1020 Locust Street, Philadelphia, PennsylVania 19107, A.N. Belozersky Institute of Physico-Chemical Biology, Moscow State UniVersity, 119899 Moscow, Russia, Department of Microbial Physiology, Free UniVersity, De Boelelaan 1087, NL-1081 HV Amsterdam, The Netherlands, and E. C. Slater Institute, Biocentrum, UniVersity of Amsterdam, Plantage Muidergracht 12, The Netherlands ReceiVed: July 31, 1996 In Final Form: January 7, 1997 X Principles of the control and regulation of steady-state metabolic systems have been identified in terms of the concepts and laws of metabolic control analysis (MCA). With respect to the control of periodic phenomena MCA has not been equally successful. This paper shows why in case of autonomous (self-sustained) oscillations for the concentrations and reaction rates, time-dependent control coefficients are not useful to characterize the system: they are neither constant nor periodic and diverge as time progresses. This is because a controlling parameter tends to change the frequency and causes a phase shift that continuously increases with time. This recognition is important in the extension of MCA for periodic phenomena. For oscillations that are enforced with an externally determined frequency, the time-dependent control coefficients over metabolite concentration and fluxes (reaction rates) are shown to have a complete meaning. Two such time-dependent control coefficients are defined for forced oscillations. One, the so-called periodic control coefficient, measures how the stationary periodic movement depends on the activities of one of the enzymes. The other, the socalled transient control coefficient, measures the control over the transition of the system between two stationary oscillations, as induced by a change in one of the enzyme activities. For forced oscillations, the two control coefficients become equal as time tends to infinity. Neither in the case of forced oscillations nor in the case of autonomous oscillations is the sum of the time-dependent control coefficients time-independent, not even in the limit of infinite time. The sums of either type of control coefficients with respect to time-independent characteristics of the oscillations, such as amplitudes and time averages, do fulfill simple laws. These summation laws differ between forced oscillations and autonomous oscillations. The difference in control aspects between autonomous and forced oscillations is illustrated by examples. Introduction Quantitative approaches have led to significant advances in the understanding of the control of metabolic and information pathways under stationary conditions. 1-9 In a biochemical/ biophysical reaction system such as a metabolic pathway in a living cell the control exerted by any enzyme on any steadystate flux (reaction rate) or concentration can be quantified in terms of the corresponding control coefficient defined by metabolic control analysis (MCA). The (stationary) control coefficient is the relative difference between the two steadystates in pathway flux or metabolite concentration, divided by the causative fractional change in the enzyme's activity, extrapolated to infinitesimally small change. Living cells also exhibit various important time-dependent phenomena however. In close vicinity of the (asymptotically) stable steady state the control over a relaxation process has been analyzed by Results A. Definitions of Time-Dependent Control Coefficients. A1. Forced Oscillations. Let us suppose that a system under study is exposed to periodic changes in the environment, resulting in periodic changes of some system parameters, e.g., kinetic constants and the concentration of "external" metabolites. Such a situation can also be described in terms of some periodic external force influencing the system. Here T is the period of the external force and e ) (e 1 , e 2 , ..., e n ) is the vector of the enzyme concentrations. Stationary periodic behavior caused by a periodic external force is called a forced oscillation. During the period of such an oscillation (T), the vector of metabolite concentrations (x) follows a closed trajectory. The system behavior described by eq 1, as well as the system behavior outside that closed trajectory is dictated by chemical processes developing in time according to the kinetic rate equations. Combination of these rates with the map of the chemical network leads to differential equations for all of the independent metabolic variables (x), the so-called chemical kinetics differential equations (eq A1 in Appendix A). The correspondence between the physical system and the mathematical equations allows one to use the work "solution" to refer to "system behavior". We shall assume that the eigenvalues of the Jacobian of this system of differential equations have negative real parts at all the points of the periodic trajectory. Under these conditions, the periodic solution, x i per (t), to eq A1 is unique and asymptotically stable. 17 So-called "conservative" systems (often considered in physics) lack the dissipation of free energy. Such systems usually have an infinitely large number (continuum) of periodic solutions determined by the initial conditions and will not be considered here. Here we consider isothermal, isobaric systems that continuously dissipate free energy, as found in chemical reaction systems Considering (fractional) changes in a steady-state periodic solution caused by a change in a particular enzyme concentration (e j ), one can define (steady-state) "periodic" control coefficients over metabolite concentrations and reaction rates (fluxes) as follows: Since the period T does not depend on system parameters, it follows from eq 1 that the control coefficients, C j x (t), are periodic functions of T. If the periodic solutions for the reaction rates can assume zero values at some time values, one should consider the non-normalized flux control coefficients 13 in eq 3. In eq 3 periodic control coefficients are defined as formal derivatives of the asymptotically stable periodic solution (eq 1) with respect to a parameter of choice (e.g., e j ) (cf. ref 43). This definition corresponds to the comparison of two steadystate periodic solutions (closed trajectories) that differ in e j by an infinitesimal change, ∆e j . Most importantly, these two solutions are synchronized by the periodic external force. In fact, a one-to-one correspondence exists between any point of either closed trajectory and a value of the periodic force. Hence, also between pairs of the points of the two different trajectories, a one-to-one correspondence exists. This synchronization makes it possible to assign an operational meaning to the (steady-state) periodic control coefficients in terms of (infinitesimal) perturbations (see below and Appendix A). Alternativley, let us consider the periodic solution x per -(t,e;t*,x*) and the other solution x tr (t,e+∆e;t*,x*) that occurs when a particular enzyme concentration (e j ) is perturbed by ∆e j at the moment t* (here and below the superscript "tr" specifies the transition process). The function x tr (t,e+∆e;t*,x*) is not periodic. It describes the transition process from the periodic solution corresponding to the value e j to the periodic solution corresponding to the value e j + ∆e j . Initially (t ) t*), the two Control Analysis of Periodic Phenomena J. Phys. Chem. B, Vol. 101, No. 11, 1997 2071 solutions coincide. At any time t > t*, the relative difference between x tr and x per divided by ∆e j /e j shows how the particular enzyme e j affects the metabolite concentration or flux during the transition. The resulting function, obtained in the limit of infinitesimally small ∆e j , is called a transient (time-dependent) control coefficient: Contrary to periodic control coefficients transient control coefficients do not depend on time periodically, although they do depend on time. Appendix A shows that in the case of forced oscillations the transient control coefficients (eq 4) tend to the corresponding periodic control coefficients (eq 3) as time tends to infinity: This clarifies the operational meaning of the formal periodic control coefficients defined by eq 3. They quantify the control when a system has already relaxed to a new oscillation pattern after a change in the activity of a particular enzyme. Appendix B shows that the periodic and the transient control coefficients satisfy the same variation equation. Periodic control coefficients are given by the unique periodic solution of the variation equation, whereas transient control coefficients are determined as a time-dependent solution, assuming the initial conditions equal to zero. To come to grips with this result, one may revisit the definitions given by eqs 3 and 4 and note that both the periodic and the transient solutions for metabolite concentrations and fluxes must satisfy the same kinetic equations (see Appendix A, eq A1). For explicit expressions for the periodic and transient control coefficients see Appendix B (cf. ref 38). It is instructive to compare these periodic and transient control coefficients, which describe the control of forced oscillations, to the corresponding control coefficients defined for perturbations near asymptotically stable steady states. 38,44 The periodic control coefficients (eq 3) defined by the formal differentiation of steady-state periodic solution correspond to the traditional steady-state control coefficients. Indeed, in standard MCA, eq 3 will define the usual control coefficients if the steady-state concentrations and fluxes are substituted in this equation for the periodic ones. The transient control coefficients of an oscillating system (eq 4) correspond to the time-dependent control coefficients, as introduced by Acerenza et al. A2. Autonomous Oscillations (Limit Cycles). The forced oscillations considered above arose from the influence of a periodic external force on a system that exhibited asymptotically stable steady states in the absence of that force. By contrast, autonomous oscillations occur at time-independent (internal and external) parameter values in systems in which the corresponding steady states are unstable. For autonomous oscillations one can define formally the control coefficients, C j x (t) and C j J (t), analogously to eq 3, i.e., as the log-log derivatives of a unique periodic solution with respect to the enzyme concentrations (or as the non-normalized derivatives if periodic reaction rates assume zero values at some time moments). However, in contrast to the case of forced oscillations these control coefficients do not depend on time strictly periodically. Moreover, they do not exist when time tends to infinity. To illustrate this, let us present a periodic solution (x k per ) by its Fourier series. We emphasize that both the Fourier coefficients and the frequency (ω) depend on systemic parameters, i.e., on enzyme concentrations (e): Here x k h (e) denotes hth Fourier coefficient and i is the imaginary unity. Differentiating the Fourier series (6) with respect to a particular enzyme concentration, e j (see eq 3), one obtains for the control coefficients, C j x k (t), Because of the second term on the right-hand side of this x i tr (t,e j +∆e j ;t*,x*) -x i per (t,e j ;t*,x*) J k tr (t,e j +∆e j ;t*,x*) -J k per (t,e j ;t*,x*) 2072 J. Phys. Chem. B, Vol. 101, No. 11, 1997 Kholodenko et al. equation, which is proportional to t, the control coefficients, C j x k (t), do not depend on time periodically. Since this term becomes unlimited with time, the coefficients C j x k (t) cannot be defined when time tends to infinity. The transient control coefficients, tr C j x k (t), are defined according to eq 4. Hence, they can be found by solving the variation equation (cf. ref 38 and Appendix B). mor the case of autonomous oscillations it has been proved 46 that no solution of the variation equation exists at time tending to infinity. Therefore, also the transient control coefficients with respect to metabolic concentrations or fluxes during the transition from the initial to a perturbed closed trajectory can only be defined for limited time intervals. In the limit of infinite time this control coefficient does not exist either. The question arises why both control coefficients, C j x (t) and tr C j x (t), fail to exist when the time of the observation of the periodic or transition process tends to infinity. For the control coefficients, C j x (t), this is explained by the phase difference between the original and perturbed oscillations, which continues to increase with time. For the control coefficients tr C j x (t), the time-dependent phase difference between the transition and the periodic movements is the culprit. Although the initial and the perturbed trajectories are very close in concentration space, due to the dependence of oscillation frequency on a perturbed parameter, e j , the phase difference does not vanish with vanishing of ∆e j when time tends to infinity; the infinitesimal difference in phase is amplified infinitely. From the reasoning above one may conclude that in the case of autonomous oscillations due to divergence of the initial and the perturbed movements, the control coefficients determined by either eq 3 or eq 4 cannot describe the control exerted by enzymes over periodic values of metabolic concentrations and reaction rates. However, the same reasoning shows that the control over those characteristics of self-sustained oscillations that do not depend on the phase of the movement can be defined (at any time of observation). The log-log derivatives of these time characteristics (Y) with respect to enzyme concentrations determine the coefficients, C j Y , that describe the control of (stationary) self-sustained oscillations. For instance, the control coefficients over the amplitude and period of oscillations and over various mean values do exist. B. Properties of Time-Dependent Control Coefficients in an Example of Forced Oscillations. We shall consider a simple example where a periodic solution (eq 1) and, hence, periodic control coefficients (eq 3) can be found analytically, as functions of time and parameters. This will make it possible to illustrate a number of the control properties, such as the variations of the control distribution with time, and to test whether the summation theorem that is true for steady-state control coefficients continues to apply in the case of forced oscillations. Scheme 1 shows a metabolic chain of two reactions, We shall suppose that the substrate concentration (S) changes periodically, and the product concentration (P) is kept zero: Here S 0 is the substrate concentration in the absence of (external) periodic force, ω 0 is the frequency of the periodic force, and a < 1 is the amplitude of the oscillation of the substrate concentration. Let us assume that the reaction rates, V 1 and V 2 , are linear functions of metabolite concentrations: Here k (i , i ) 1, 2, are the kinetic constants; e 1 , e 2 are the total enzyme concentrations; and x is the concentration of the intermediate. Thanks to the linearity of eq 9 with respect to x, its periodic solution (under the influence of the periodic force described by eq 8) can be found readily (see Appendix C): Substituting eq 10 into the rate equations (9), the periodic solution for the fluxes through the first (J 1 ) and the second (J 2 ) reactions are obtained: Here J 0 is the steady-state flux. A 1 and A 2 are the amplitudes of the oscillations of the reaction rates, and is the initial phase of oscillations of the J 1 (the explicit expressions for A 1 , A 2 , and are given in Appendix C). From eqs 11 it follows that the fluxes through the first and the second reactions differ at most times. Only their averages are equal. Consequently, in contrast to the case of systems at steady states, the control coefficients over the time-dependent fluxes through sequential reactions in oscillating systems will differ (see (10) S f X f P (scheme 1) Control Analysis of Periodic Phenomena J. Phys. Chem. B, Vol. 101, No. 11, 1997 2073 tude of the substrate oscillation increases further, even the direction of reaction rates changes during the period and such that they equal zero at some moments. In this case the control can become infinite. If the amplitude of the oscillation of the pathway substrate is small (a , S 0 ), the periodic control coefficients do not cross, but oscillate near the corresponding steady-state values (not shown). Using the explicit expressions for periodic control coefficients (see Appendix C), one can show readily that the summation theorem, which governs the steady-state control coefficients 4 and time-dependent control coefficients for the relaxation near steady states 38 (it requires the sum of these coefficients to be equal to 1), is not valid for periodic control coefficients: C. Summation Theorems. C1. Summations in the Case of a Forced Oscillation. Since the reactions rates depend linearly on the enzyme concentrations (activities), simultaneous transformation of these concentrations, of the time, and of the frequency of a periodic external force, leads to a new equation system that coincides with the initial system after eliminating the superscript (*). Therefore, if the initial conditions are the same, metabolite concentrations of the transformed system at the moment t/λ will coincide with concentrations of the initial system at the moment t, whereas the fluxes will increase by factor λ (proportional to the new enzyme activities): Applying to eq 14 Euler's theorem on homogeneous functions, one arrives at Although for forced oscillations the control coefficients with respect to the frequency of the external force (C ω 0 x , C ω 0 J ) can be defined formally as the derivative of the periodic solution with respect to ω 0 , they become unlimited as time tends to infinity. This is explained by the phase divergence of the initial periodic movement with the frequency ω 0 and the perturbed one with the frequency ω 0 + dω 0 , corresponding to the infinitesimal change in ω 0 . This phase difference does not remain infinitesimal at infinite times (cf. the case of autonomous oscillations above). Moreover, also the transient control coefficients defined by eq 4, in which the derivatives should be taken with respect to ω 0 (instead of e j ), become unlimited at infinitely large time. Hence, in a general case the summation theorems given by eqs 15 and 16 have no operational meaning as t tends to infinity. Note, however, that the sums given by the first terms in eqs 15 and 16 do exist at infinitely large times. For the above example of forced oscillations, eq 12 shows that the sum of flux control coefficients (the first term in eq 16) depends periodically on time. One may note that if the form (amplitude) of the oscillation in x were independent of the forcing frequency ω 0 , x could be written as In this case, the second and third term of eqs 16 disappear and the classical summation theorems, but now for periodic control coefficients, are retrieved. This condition holds in electrical networks without capacitances, and in linear chemical networks where the variable metabolites occur in such small volumes that the corresponding relaxation times are much smaller than the period of the applied oscillation. As illustrated by eq 10, Dependencies of periodic and steady-state enzyme control coefficients of the linear pathway of scheme 1 on time. Lines 1 and 3 refer to the periodic control coefficients over flux J1 with respect to the first (1) and the second (3) enzyme. Lines 2 and 4 refer to steadystate control coefficients over flux J1 with respect to the first (2) and the second (4) enzyme. The magnitudes of the parameters were k1 ) 35, k2 ) 30, k-1 ) 25, k-2 ) 1, S0 ) 20, e1 ) 0.1, e2 ) 0.05, a ) 0.1, and ω ) 1. Phys. Chem. B, Vol. 101, No. 11, 1997 Kholodenko et al. through the frequency dependence of the amplitude of the oscillation in x per , the form of oscillations in a metabolic network often depends on the frequency of the applied oscillations. The following summation theorems hold for the control coefficients of enzymes over the amplitude of stationary oscillations of metabolic concentrations (A x ) or fluxes (A J ) and their average values over the period (x j, J h) in the case of oscillations forced at a frequency ω 0 : In the example considered above (section B) the control coefficients over the amplitudes of metabolic concentrations and fluxes with respect to the frequency of the external force and of the enzyme concentrations can be calculated readily (see Appendix C). One can se

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