INTRODUCTION

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Cell metabolism is a complex network of biochemical reactions that is continuously interacting with its environment. Thus, it can be viewed as a dynamic system that is able to adapt its behavior to changes in both the internal parameters (kinetic constants or enzyme concentrations) and the boundary constraints (input source of material or concentration of external metabolites). This adaptation occurs in a period of time that depends on the intrinsic properties of the system, mainly the design of the pathway (stoichiometric properties) and kinetic factors. Moreover, this period of time must also depend on both the current state of the systemthe initial state and the boundary constraintsand the kind of perturbation it undergoes.

Getting a wide knowledge of the response time (in a general sense, the time spent to respond to a stimulus) has important implications. Within an evolutionary context, this study may allow us to obtain important clues to how cell metabolism has evolved. Response time is a key feature of living beings that is frequently critical in the struggle for life. It is decisive, for example, for predators to capture prey and for prey to escape from predators. In a more general sense, response time is a variable that determines a kind of behavior, and so it must agree with each particular ecological niche. A logical hypothesis is that every aspect of the macroscopic behavior of a species must have a closely related molecular design behind it. This includes, of course, response time. In effect, Lupiáñez et al. (1996), exploring the transition from aerobic to anaerobic glycolysis, as the metabolic support of the flight promptness in several birds, showed that long-distance flying birdswhich have, however, a slow starthave a long metabolic response time, whereas the sprinterscharacterized by a quick macroscopic startshowed a short metabolic response time. Thus, according to natural selection, the response time of present-day metabolic routes might be strongly adapted to its functionality, and thus macroscopic behaviors must reflect microscopic transition times.

From a biotechnological viewpoint, a suitable knowledge of the response time could allow the control and regulation of cell metabolism (for instance, by changing either the kinetic properties or the design of the pathway). If cells are considered as factories of bioproducts (Bailey, 1991), the main consequence would be the possibility of improving this function. Nevertheless, the complexity of metabolic behavior (Goldbeter, 1996) makes it difficult, in many instances, to measure the response time. There are similar considerations regarding the time for drug action in metabolism.

The theory of the response time has been developed by several researchers for the last 20 years (Heinrich and Rapoport, 1975; Easterby, 1973, 1981, 1986; Meléndez-Hevia et al., 1990, 1996; Torres et al., 1991; Cascante et al., 1995, 1996; Heinrich and Schuster, 1996; Lloréns et al., 1997, among others). It is noteworthy, however, that despite its obvious importance, this subject remains at present virtually unexplored. As far as we know, only a few direct empirical determinations of metabolic response times have been described (Torres et al., 1990; Torres and Meléndez-Hevia, 1992; Lupiáñez et al., 1996). Transition times in human erythrocyte by kinetic modeling have also been assayed (Rapoport and Heinrich, 1975; Werner and Heinrich, 1985). A possible reason for the little attention paid to this key feature could be the lack of a general agreement on the theory. In fact, the proposed definitions of a representative time of transition have differed from each other, depending on the initial and final state, and in general they have been defined for very restrictive boundary constraints and transitions. This aspect clearly causes uncertainty in undertaking experimental work.

Then, a question arises: Is it possible to find a physical magnitude, theoretically well supported and experimentally measurable, that is useful for the study of the time a system variable takes to achieve any transition from a state A to another state B, regardless of what they are, and independently of the boundary constraints? In this work we shall prove that the answer is positive, which leads to a completely general definition for the characteristic time of a transition.

	THEORETICAL FRAMEWORK

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In most models, both experimental and theoretical, time is treated as a parameter. However, to make a theory of temporal transitions (i.e., to investigate how transitions depend on system parameters as kinetic constants or enzyme concentrations), we need to deal with time as a function. The problem of evaluating the transition time and studying its properties is difficult for two main reasons: Mathematically, approaching to the final state is asymptotic, and it requires an infinite time. From an experimental point of view, it is always difficult to decide how close to the steady state the system variable is. To overcome these difficulties, historically the question of how to measure a time representative of the transition in metabolic pathways has been tried through different approaches.

Hess and Wurster (1970) analyzed experimentally an irreversible metabolic system of two reactions under saturating conditions of the first enzyme. They called the intersection point of the asymptote of the progress curve (recording the concentration of the end product with time) with the time axis the transient time, assuming the system was initially empty. They showed that it corresponds to the reciprocal of the eigenvalue from the theoretical model of this system.

Easterby (1973) extended this analysis to multienzyme sequences under similar constraints (irreversibility and saturating conditions of the first enzyme). He proved that each enzyme has a transient time, and that the overall transient time is given by the sum of the individual transients. Moreover, each transient time can be obtained through the ratio between the stationary concentration of the ith intermediate, _i, and the flux at steady state, , i.e.,

(1)

Later, Hearon (1981a) proved that in linear systems the transient time corresponds to an average time. In subsequent papers, this definition has been extended to 1) reversible reactions, even when the differential equations describing the progress curve are not readily amenable to analytical solution (Easterby, 1981); 2) transitions between steady states (Easterby, 1981); 3) systems in which the input of source material varies with time (Easterby, 1986); and 4) systems evolving under constant affinity constraints (Lloréns et al., 1997). It has also been shown that under constant input flux, the transient time corresponds to the time a molecule needs to cross the reaction chain at steady state, which has been referred to as transit time (Easterby, 1981; Hearon, 1981b; Morán et al., 1997).

An alternative way of calculating the transition time, initially formulated for the concentrations of chemical reactants, was introduced by Heinrich and Rapoport (1975). If _x(t) is the instantaneous deviation of a metabolite concentration, x(t), from the steady-state value, , i.e., _x(t) = x(t)  , then its transition time is given by

(2)

As the authors pointed out, to have a well-defined magnitude, the sign of _x must not change during the transition. This definition takes also into account the overall features of the temporal evolution of the variable by weighting the time with _x(t). Although in particularly simple linear systems (e.g., A  B) the approaches mentioned above yield the same result, in more complex situations they lead to measurements that are clearly divergent. In fact, as will be shown below, none of the referred times are representative of important transitions, and furthermore, these magnitudes are not defined in complex situations.

A third direction in evaluating transition times was suggested by Easterby (1973) and later by Storer and Cornish-Bowden (1974) and Torres et al. (1991). They defined a magnitude t₉₉, which measures the time a variable takes to reach 99% of its steady-state value. This magnitude can be used to compare transition times of different systems (regardless of their nature and constraints). However, its evaluation is prone to high experimental error because of the asymptotic shape (for long times) of the evolution profile. Moreover, t₉₉ could not be really representative of the transition (in fact, it is not difficult to find similar transitions that differ appreciably in the value of t₉₉). Thus an average estimate for the response time seems more convenient.

The previous exposition points out the existence of a broad interest in studying the time biological systems take to undergo transitions. Furthermore, all of these definitions refer to particular transitions and constraints and fail when they are applied to other kinds of transitions, as will be commented on in the following sections. To solve this controversy, here we shall propose a general definition to characterize the response time of any system variable.

	A POSSIBLE SOLUTION: A GEOMETRIC DEFINITION

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A deep characterization of the time associated with any transition requires the definition of a magnitude that can be handled both experimentally and theoretically. Because transitions between states can be described by the evolution profile of the variable of interest (output flux, metabolite concentration, etc.), the characteristic time of the transition must contain some average information of the overall process, also making possible the comparison between different types of transitions.

As shown in Fig. 1, in which three hypothetical transitions appear, valuable information on the time that a variable f takes to reach the stationary regime can be obtained from a quantification of the area between the evolution curve of the variable under study and its final state. However, by the simple consideration of this area, the curve c₁, which is clearly faster than curve c₃, would have a higher response time. Therefore, to get a correct measure of this time, the area must be conveniently normalized. The normalization factor in these curves is the global variation of the variable, i.e.,   f(0). This normalized area will be referred to as the characteristic time of the transition, T_c. Thus the problem is reduced to determining which is the correct area to be computed in each case, as well as finding the criteria of normalization.

View larger version (34K): [in this window] [in a new window]   FIGURE 1   Comparison of the temporal evolution to the steady state of three hypothetical systems. Three evolution curves of a hypothetical variable, f, are shown: c1, c2, and c3. A measure of the time taken by each variable to achieve a steady-state value can be obtained through the quotient of the area enclosed between the final state and the evolution curve, and the overall variation of f in each transition. It can be seen that the hatched area, which corresponds to curve c1, is smaller than that corresponding to c2 (dotted curve), whereas the two transitions lead to the same variation in f. It means that the characteristic time of c1, Tc(1), is lower than that corresponding to c2, Tc(2). With respect to c3, although its area (shadowed area) is smaller than that of the other two, the quotient between this area and the change in f, Tc(3), is lower than that corresponding to c2 and greater than the one corresponding to c1. This reasoning leads to the conclusion It is worth remarking that these kinds of curves are obtained in systems evolving from rest under a constant input flux, Jin, when the output flux, Jout, is analyzed. In this case the area corresponds to the mass accumulated at steady state, , and the overall variation in the variable under study to Jin = Jout()  Jout(0). Then, Tc = /Jin, as defined by Easterby (Eq. 6).

In the following subsections, the characteristic time T_c will be calculated for different kinds of transitions that have previously been analyzed in the literature. The variables commonly measured are the output flux, J_out, and the input flux of the pathway, J_in. It is assumed that both variables are monotonous functions, which means that the signs of their derivatives do not change during the evolution. In all of these cases the area and the normalization factor can be straightforwardly found. However, as will be discussed later, when flux is not monotonous, the relationship between the area and the normalization factor is not so obvious (see, for instance, Fig. 5).

Transition from rest under constant input flux

Many biochemical systems can be supposed to work under a constant J_in. In this situation, it is interesting to analyze the transition time of the output flux of the pathway, J_out. Under special conditions, and assuming that initially the concentration of every intermediate is null (transitions from rest) (Hess and Wurster, 1970; Easterby, 1973), the temporal evolution of J_out has a shape similar to those depicted in Fig. 1.

To illustrate the evaluation of the characteristic time of the transition, consider the curve labeled c₁. As commented on in the previous paragraph, the area to be taken into account for the estimation of T_c should be the hatched one, i.e.,

(3)

But, at any time t from the initiation of the transition, mass conservation requires

(4)

and integrating over the time,

(5)

From this expression it becomes clear that, if the system achieves a stationary regime, the area A corresponds to the mass accumulated at steady state,  = lim_{t i=1}ⁿ x_i(t). If A is normalized by the variation of flux as a consequence of the transition, i.e., J_in = J_out()  J_out(0), the characteristic time that results is

(6)

Therefore, in systems under this kind of constraint, the characteristic time of the output flux corresponds to the transient time defined by Easterby, ^E. In addition, as was already pointed out by this author, the transient time (and so, the characteristic time) is given by the intersection of the asymptote to the progress curve (i.e., the integral of the output flux) and the time axis (Easterby, 1973).

Transition between steady states under constant input flux

Under physiological conditions, transitions from rest rarely occur. In practical situations, metabolism responds to variations in the environment by changing its steady state, characterized by particular values of the intermediate concentrations and fluxes, to another state (that, in the case of temporal perturbations, could be the previous one). In Fig. 2 A, curve 1 represents a transition between a steady state a, characterized by a stationary flux _a, and a state b, in which the final flux is _b. The area representative of the transition is the hatched one, which, as before, is given by

(7)

Again using mass balance equations, and assuming that at time t = 0 the system is at state a, it is possible to prove that this area coincides with the difference between the mass accumulated at the steady state b (_b) and that corresponding to the state a (_a). In this case, as can clearly be seen in Fig. 2 A, the normalization factor is the difference in flux between the two states, which yields the following expression for the characteristic time:

(8)

It is important to remark that now this time differs from the one proposed by Easterby for transitions between steady states (Easterby, 1981), _ab = (_b  a)/_b. A careful inspection of Fig. 2 A shows that, in this case, considering _b as the height of the transition leads to an underestimation of the response time (but the result could also be an overestimation, when _b < |_b  a|). Actually, the higher _b is, the lower the value obtained for _ab. To illustrate this fact, let us compare transition 1 with transition 2 (Fig. 2 A). They represent qualitatively similar transitions, but 2 starts from rest and 1 starts from the stationary state a. Strikingly, the resulting _0b (i.e., ^E for transition 2) would be much higher than _ab for transition 1, whereas their characteristic times evaluated through Eq. 8 are equal.

View larger version (15K): [in this window] [in a new window]   FIGURE 2   (A) Transition between two steady states under a constant input flux restriction. Curve 1 shows the output flux of a system that evolves from a steady state a, characterized by a stationary flux a, to a steady state b, with a final flux b. In this case b > a, but the calculation of Tc is equally valid for the opposite situation, i.e., when a > b. As in Fig. 1, time Tc is obtained through the ratio between the hatched area and the difference of fluxes b  a. As was discussed in the text, this time must correspond to the characteristic time of curve 2, which is qualitatively similar to curve 1. It can be seen that the transient time defined by Easterby, ab, is higher for curve 2 than for curve 1, which seems to be contradictory. (B) Geometric determination of Tc from the progress curve of a transition between steady states under constant input flux. Mass that enters (Sin = Jint) and leaves the system (Pout = 0t Joutdt) is plotted versus time. Tc is obtained as the time at which the asymptote to the progress curve of state a, ra: at, intersects that corresponding to state b, rb: bt  (b  a). Notice the difference between Tc and Easterby's transient time, ab.

It should be noted that in those situations in which _a = _b (as occurs with temporal perturbations), Eq. 8 is not valid, because in those cases the derivative of J_out changes its sign (i.e., J_out is not monotonous). This situation will be analyzed later.

As in the previous subsection, the characteristic time can be geometrically obtained from the progress curve shown in Fig. 2 B. In fact, T_c corresponds to the solution of the following equation (see Eq. 8):

(9)

where _at is the straight line asymptotic to the progress curve of state a, and _bt  (_b  a) is that corresponding to state b. Therefore, the characteristic time can be geometrically obtained as the intersection point of the two asymptotes.

Transition from rest under constant affinity constraints

When a metabolic pathway, with a given equilibrium constant for the conversion of the initial substrate into the final product, evolves under a constant concentration of these metabolites, is said that it is constrained to work at constant affinity. In this case, both the input and output fluxes, J_in and J_out, respectively, are reversible and variable with time. Fig. 3 A shows their typical profile. Because initially the system is empty, a negative local affinity appears in the last reaction and mass enters from the product, which is transduced in a negative value of J_out at the beginning (Lloréns et al., 1997). The evolution of the input flux has already been discussed for systems under variable input of material and irreversible output (the special case of infinite affinity) (Hearon, 1981b; Easterby, 1986; Torres et al., 1991). Again, the areas enclosed between each curve and the final state will be chosen to estimate the corresponding characteristic time:

(10)

where is the steady-state flux. It has been proved that these areas, A_in and A_out, can be associated with the stationary masses accumulated because of the variable input, _in, and the variable output, _out, respectively (Torres et al., 1991; Cascante et al., 1995). The overall mass at steady state is given by  = _in + _out. The normalization factor corresponds to the difference between the steady-state flux and the value of the fluxes at time zero. Thus the expressions for the characteristic times of J_in and J_out are, respectively,

(11)

It must be stressed that, in systems evolving under this kind of constraint, other transient times associated with both the input and output flux were previously defined (Easterby, 1986; Torres et al., 1991; Cascante et al., 1995), _in = _in/ and _out = _out/. However, it is clear from Fig. 3 A that neither _in nor _out informs us about the average time of the corresponding transitions, because the normalization factor used in both cases () is not adequate.

View larger version (15K): [in this window] [in a new window]   FIGURE 3   (A) Temporal evolution of a system evolving from rest under a constant affinity constraint. Input and output fluxes (Jin and Jout) are plotted versus time. Because both velocities are variable, it is possible to define a characteristic time for each one, Tc(in) and Tc(out). For the estimation of Tc(in), the area to be considered is labeled Ain (which corresponds to the mass accumulated at steady state because of the variable input, in), and the normalization factor is Jin(0)   (the overall variation in the input flux). Then, Tc(in) = in/(Jin(0)  ). Similarly, Tc(out) is obtained as the ratio between the area labeled Aout (which corresponds to out, i.e., the mass accumulated in the steady state because of the variable output) and   Jout(0). Therefore, Tc(out) = out/(  Jout(0)). Notice the difference between these magnitudes and previously defined transition times, in = in/ and out = out/. (B) Geometric determination of Tc from the progress curve of a system evolving under a constant affinity restriction. Mass entering (Sin = 0t Jint) and leaving (Pout = 0t Joutdt) the system is plotted versus time. As can be seen, Tc(in) is the time at which the asymptote to Sin, rs: int + in, intersects the straight line with a slope equal to the initial value of the input flux, r0s: Jin(0)t. In a similar way, Tc(out) is given by the intersection of the asymptote to Pout, rp: outt  out, and the straight line r0p: Jout(0)t.

Following geometric considerations, now each characteristic time can be obtained from the progress curve of the respective velocity as the time at which its asymptote intersects the straight line whose slope equals the initial value of the velocity (see Fig. 3 B).

A mathematical expression for the normalized area

Although Eqs. 6, 8, and 11 for the characteristic time may look different, a careful inspection shows that all of them can be deduced from the general definition,

(12)

In fact, when the function f analyzed corresponds to the output flux, J_out, and the system evolves from rest under a constant input, integration by parts of Eq. 12 leads to Eq. 6. Similarly, when transition starts from a state a, Eq. 12 reduces to Eq. 8. And finally, when the constraint imposed on the system is of constant affinity, both T_c(in) and T_c(out) (Eq. 11) are obtained if f is considered to be J_in and J_out in Eq. 12, respectively.

It is worth mentioning that this definition can be used to estimate the characteristic time for any variable of the system (individual reaction velocities, concentration of metabolites, etc.). The only requirement for T_c to give accurate information on the transition time is that f must be a monotonous (increasing or decreasing) function of time. In addition, df/dt must be integrable over the interval [0, ], for all  > 0. The question of the convergence of the improper integrals involved in the definition of T_c can be solved by attending to the mass conservation law. For instance, when f is a reaction rate, the convergence of Eq. 12 is ensured, because its numerator is always a fraction of the mass accumulated in the system at steady state, which must be finite to accomplish the mass balance at the stationary state. On the contrary, an infinite value of T_c would indicate that the system does not reach a steady regime.

The characteristic time so defined is the subject of three complementary interpretations:

1. For systems with linear kinetics evolving from rest under constant input of substrate, Hearon proved that the transient time  is given by a linear combination of the reciprocal of the eigenvalues of the system (all of them are strictly negative real numbers, because chemical (or biochemical) reaction models are considered; Hearon, 1981b). Under these assumptions the characteristic time coincides with the transient time, and then T_c = _i=1ⁿ 1/_i.

In general, if f is a monotonous function that can be expressed by a linear combination of real exponential functions, f(t) = _i=1ⁿ a_ie^_it, with _k < 0 for all k, then it can easily be shown that the characteristic time reads

(13)

Therefore, the characteristic time has the meaning of a preexponentially weighted average time. This magnitude has already been used to measure the features of other types of transitions, e.g., decay of excited states (Carraway et al., 1991).

2. If f is viewed as the distribution of mass of a line of infinite length, then Eq. 12 is formally identical to the expression commonly used to calculate the center of mass of the line, with a density distribution given by (x) = dm/dx, which is
Accordingly, the characteristic time has the meaning of the hypothetical time at which the whole transition is concentrated.

3. Complementarily, if h(t) = f(t)/(  f(0)) is considered as a distribution function, which is typical in statistics, the characteristic time
may be interpreted as the first moment or average of the probability distribution, thus sharing the meaning of mean time of the transition.

	GENERALIZATION TO COMPLEX TRANSITIONS

TOP ABSTRACT INTRODUCTION THEORETICAL FRAMEWORK A POSSIBLE SOLUTION: A... GENERALIZATION TO COMPLEX... EVALUATION OF THE... DISCUSSION APPENDIX REFERENCES

Contrary to the profiles shown in Figs. 1-3, metabolic systems often present more complex dynamics. In fact, transitions involving critical damping, damped oscillations, or even sustained oscillations can be found under any kind of external constraint (Chance et al., 1964; Pye and Chance, 1966; for a review see Goldbeter, 1996). In these cases, the sign of df/dt changes during the transition, and then the previous definition of the characteristic time is no longer valid, because negative weighted times appear. Nevertheless, as will be proved in the next subsections, there is a straightforward way of generalizing the previous definition (Eq. 12) to these complex transitions.

Critically damped transitions

The simplest complex transition involving changes in the sign of the derivative is the critically damped transition. The curve labeled 1 in Fig. 4 A shows the evolution of a variable with this sort of dynamics. To find an explicit expression for its characteristic time, let us consider the hypothetical transition represented by curve 2. Until t = t₁, its evolution profile is identical to the critically damped transition (curve 1). From this time to infinity, both transitions are mirror images with respect to the axis, M, parallel to the time axis. It can be assumed that if two transitions have a symmetry axis parallel to the time axis, then their characteristic times must be equal. Therefore, the evaluation of T_c for curve 2 yields a straightforward way of defining the characteristic time for critically damped transitions.

View larger version (16K): [in this window] [in a new window]   FIGURE 4   (A) Temporal evolution of a system variable with critical damping. Curve 1 shows the output signal (f) of a system that evolves toward the stationary state with critical damping. From t = 0 until t = t1,the derivative of f is positive. From t = t1 to infinity, this derivative is negative. To estimate the characteristic time of this observable, curve 2 will be considered. This curve is identical to 1 until t = t1. From this time to infinity, it is its mirror image with respect to the axis M, parallel to the time axis. Thus we have obtained a curve whose characteristic time will be equal to that corresponding to curve 1, but now the derivative of profile 2 is always positive, a necessary condition for Eq. 12 to be applied (see text). The areas to be taken into account in the calculation of Tc will be the hatched ones, A1 + A2, whereas the normalization factor will be the asymptotic state of curve 2, f*. (B) Decomposition of the evolution profile depicted as curve 1 in A. The sum of the nondecreasing functions g and h, and yields function f. Because both g and h are monotonous functions, their respective characteristic times can be obtained through Eq. 12 as Tcg = B1/f(t1) and Tch = B2/(f(t1)  ). Therefore, the characteristic time of f can be defined as the weighted average of Tcg and Tch (see Eq. 20). (C) Geometric evaluation of Tc and E for the transition described in A, for f = Jout. Progress curves corresponding to transitions 1 and 2 of part A when f = Jout are depicted. The difference between Tc and E is due to the process of inversion of those parts of the curve with negative derivative, as discussed in the text. To evaluate the characteristic time, the absolute value of df/dt must be taken into account, which yields an asymptotic behavior with greater (as in the example) or equal slope. In general, Tc can be greater than, equal to, or lower than E.

Because the sign of df/dt does not change during the transition of curve 2, Eq. 12 can be applied to evaluate its characteristic time. In this case, the area that informs us about the transition is that enclosed between the asymptotic state (which will be referred to as f*) and the evolution of curve 2 (hatched area in Fig. 4 A). This total area has two contributions: A₁, the area enclosed from 0 to the time t₁, at which df/dt(t) = 0 (which corresponds with the maximum in the curve), i.e.,

(14)

and the rest of the area, A₂, from t₁ to infinity, which can be easily calculated as

(15)

In both expressions, f* = 2f(t₁)  , where is the steady-state value achieved by the function f. Then, the characteristic time of damped transitions from rest responds to the expression

(16)

Notice that now the normalization factor is not given by the real change in the variable under study, . Using integration by parts, the numerator of this equation can be rewritten as

(17)

In a similar way, the denominator can be expressed as

(18)

resulting in the following equation for the characteristic time:

(19)

Therefore, a mathematical expression for the characteristic time of critically damped transitions can be found simply by considering the absolute value of the derivative of the function under study. The consideration of the absolute value is not unexpected, because the evaluation of T_c requires the estimation of all periods of time, independently of the sign of df/dt. In other words, the weight function must always be positive. The situation described here is similar to the problem of calculating the time a mechanical pendulum takes to reach the equilibrium state (independently of the direction of its movement, the time always increases). Obviously, the use of Eq. 12 would yield to an underestimated T_c.

As occurs with monotonous transitions, the convergence of the improper integrals involved in the definition of T_c (Eq. 19) is automatically ensured because of the mass conservation law. If f is a velocity, it can be shown that, because the mass accumulated in the steady state, , is always finite when the system achieves a stationary regime, then the area A₂ is also finite. Therefore, T_c must be finite.

Another alternative interpretation of Eq. 19 comes from the theory of distribution functions. The variable f can be always decomposed as a sum of nondecreasing functions, i.e., f = g  h, where

Because both g and h are increasing functions of time, their characteristic times can be calculated by using Eq. 12. As Fig. 4 B shows, T_c^g = B₁/f(t₁) and T_c^h = B₂/(f(t₁)  ). Now, the characteristic time of f can be defined as the weighted average of the characteristic times of the increasing g and h, T_c^g and T_c^h:

(20)

As can easily be checked, this expression coincides with Eq. 16 and, therefore, with the definition given in Eq. 19.

Fig. 4 C illustrates the difference between the characteristic time T_c and Easterby's definition, ^E. In this example the value obtained for ^E is lower than that corresponding to T_c, but the opposite situation can also be found. This fact can be understood by noting that the characteristic time takes into account the mass mis au jeu during the transition, whereas ^E only considers the mass accumulated at steady state, which is always lower than or equal to the former. Nevertheless, the normalization factor that enters the expression for ^E is lower than the respective value for T_c (i.e.,   f*), which explains why the quotient between them can be lower than, equal to, or greater than the characteristic time.

It is important to remark that the same definition (Eq. 19) also applies to transitions from any state a to any state b. In particular, temporal perturbations can be treated as a particular case of critically damped transitions in which the final state is the original one, and thus T_c can be obtained from Eq. 19.

Damped oscillations

The evolution of a system variable that reaches the steady state through damped oscillations is shown by curve 1 of Fig. 5. To look for the correct expression for the characteristic time, we follow the same reasoning as that applied in the previous section. Now the curve must be inverted in every place where df/dt is negative, that is, between each maximum and its next minimum. This yields curve 2 in Fig. 5. Because the curve oscillates infinitely approaching the steady state, the number of terms in which the overall area must be decomposed tends to infinity. As discussed in the previous subsection, both transitions 1 and 2, have identical characteristic times. Thus we define

(21)

where, again, f* is the asymptotic state obtained after inversion of the evolution curve, and A* is the area enclosed between this asymptotic state and the inverted curve. It can be deduced that (see Appendix)

(22)

and then, the characteristic time responds again to Eq. 19.

View larger version (38K): [in this window] [in a new window]   FIGURE 5   Temporal evolution of a system variable with damping. Curve 1 represents the evolution of a variable f toward the steady state through damped oscillations. As the estimation of Tc requires the monotonicity of f, its profile must be inverted in every place in which df/dt is, for instance, negative. This process leads to curve 2. Following the same reasoning as that applied in Fig. 4, the characteristic time of this system can be obtained through the ratio between the hatched area and f*, Tc = A*/f* (see Appendix).

For these complex situations, the problem of showing the range of convergence of Eq. 21 cannot be straightforwardly solved. Now, in the determination of T_c an infinite sum of areas (a series of real numbers) is involved. Then the improper integrals that appear in Eq. 22 converge if and only if this series converges. In general, it can be stated that to have a finite characteristic time, df/dt must tend to zero faster than 1/t² as t approaches infinity. This condition is always satisfied when f is a combination of negative exponentials, as are the solutions of linear ordinary differential equations. It is worth remarking that an infinite value of T_c would mean either that the system does not reach a steady regime or that this approximation is tremendously slow (see section Evaluation of the Characteristic Time in a Reaction Model Involving an Allosteric Enzyme, below).

Sustained oscillations

The evolution of an observable f that evolves toward a limit cycle is represented in Fig. 6 A. Although the final state is nonstationary (d/dt  0), it is still possible to compute the characteristic time of the transition of any variable, f, by analyzing the evolution of f  . The resulting curve is represented in Fig. 6 B, and, as can be seen, it is completely analogous to the curve that evolves under damped oscillations to a steady state. Therefore, Eq. 19 can be extended to these kinds of systems only by taking into account the function f   instead of f:

(23)

However, contrary to Eq. 19, now the function of time appears in the expression of T_c.

View larger version (20K): [in this window] [in a new window]   FIGURE 6   (A) Temporal evolution toward a limit cycle. The thicker curve shows the temporal evolution of an output signal, f, to a limit cycle, , represented by the thinner curve. (B) Damped convergence of the function f  . The thinner curve shows the evolution of f   versus time. This evolution profile represents the approximation of the output signal, f, to the limit cycle, , and is qualitatively identical to that shown in Fig. 5. This makes it possible to follow the same reasoning and to obtain Tc as the ratio between the hatched area A* and the normalization factor (f  )*.

It must be remarked that, in one-step linear reaction schemes forced with a periodic input, the characteristic time corresponds to the reciprocal of the real part of the eigenvalue. This fact shows the existence of a relationship between T_c and the eigenvalues of the system, even when the absolute value of df/dt is used.

	EVALUATION OF THE CHARACTERISTIC TIME IN A REACTION MODEL INVOLVING AN ALLOSTERIC ENZYME

TOP ABSTRACT INTRODUCTION THEORETICAL FRAMEWORK A POSSIBLE SOLUTION: A... GENERALIZATION TO COMPLEX... EVALUATION OF THE... DISCUSSION APPENDIX REFERENCES