^{1}

^{*}

^{1}

^{2}

AC and JJT conceived and designed the experiments and wrote the report. AC and FC performed the experiments and analyzed the data.

The authors have declared that no competing interests exist.

In metabolic networks, metabolites are usually present in great excess over the enzymes that catalyze their interconversion, and describing the rates of these reactions by using the Michaelis–Menten rate law is perfectly valid. This rate law assumes that the concentration of enzyme–substrate complex (_{0}). However, in protein interaction networks, the enzymes and substrates are all proteins in comparable concentrations, and neglecting _{0} is not valid. Borghans, DeBoer, and Segel developed an alternative description of enzyme kinetics that is valid when _{0}. We extend this description, which Borghans et al. call the total quasi-steady state approximation, to networks of coupled enzymatic reactions. First, we analyze an isolated Goldbeter–Koshland switch when enzymes and substrates are present in comparable concentrations. Then, on the basis of a real example of the molecular network governing cell cycle progression, we couple two and three Goldbeter–Koshland switches together to study the effects of feedback in networks of protein kinases and phosphatases. Our analysis shows that the total quasi-steady state approximation provides an excellent kinetic formalism for protein interaction networks, because (1) it unveils the modular structure of the enzymatic reactions, (2) it suggests a simple algorithm to formulate correct kinetic equations, and (3) contrary to classical Michaelis–Menten kinetics, it succeeds in faithfully reproducing the dynamics of the network both qualitatively and quantitatively.

A major goal of molecular systems biology is to build, simulate, and analyze mathematical models of complex molecular regulatory systems comprising genes, proteins, and metabolites [

In principle, the governing equations for any chemical reaction network can be formulated by the law of mass action [

To sidestep these problems, theoreticians often use the quasi-steady state approximation (QSSA) to eliminate the fastest (and the slowest) variables in the system of differential equations

^{−1}

0 < ɛ ≪ 1.

For slow variables, the process is easy: _{T} = constant. For fast variables, it is more subtle: ^{−1}_{T},…). We are left with differential equations for the medium timescale variables only,

The classic example of such timescale analysis is the Michaelis–Menten (MM) theory of an enzyme-catalyzed reaction, E + S ↔ C → E + P [_{1} for binding of enzyme and substrate, _{–1} for dissociation of the complex, and _{2} for the catalytic reaction. We often use a letter other than _{1}, _{–1}, and _{2} for enzyme D; see below.) The total enzyme concentration is a slow variable (_{T} = constant), the substrate concentration is the intermediate variable (

The condition for this QSSA to be valid is _{T} ≪ _{0} + _{m}, where _{0} = [S](0) = initial substrate concentration.

The same sort of analysis can be carried out for networks of enzyme-catalyzed reactions, but modelers sometimes avoid the hard work of separating timescales and simply use the MM rate law to describe enzyme-catalyzed reactions in their differential equations. For protein interaction networks (PINs), such use of MM kinetics is unjustified because the enzymes have multiple substrates, the substrates are acted upon by multiple enzymes, and (worst of all) enzymes and substrates often swap roles (for example, see [_{T} ≪ _{0} + _{m} for both reactions simultaneously. In this report, we show how to formulate the QSSA properly for PINs, and we address some problems in previously published models.

Our approach to PINs relies on a modified QSSA introduced by Borghans, DeBoer, and Segel in 1996 [_{T} and _{0} are comparable numbers, the proper intermediate timescale variable is

Borghans, DeBoer, and Segel called this the total QSSA (tQSSA). A sufficient condition for the uniform validity of the tQSSA was derived by Tzafriri [_{T} = _{T},_{T}) = (_{2}/2_{1}_{T}) · _{T},_{T})), ^{−1/2} − 1, and _{T},_{T}) = 4_{T}_{T}(_{m} + _{T} + _{T})^{−2}. Tzafriri showed that
_{−1} ≫ _{2}; i.e., the dissociation rate of the enzyme–substrate complex is much faster than the catalytic conversion of substrate into product. Notice that _{–1} = 0; so the tQSSA is likely to be an excellent approximation for any ratio of enzyme to substrate and for any ratio of timescales.

It is of course possible, using the quadratic formula, to solve _{T} and to substitute this formula into ^{2} ≪ _{T}_{m})/_{T}, we can write the condition ^{2} ≪ _{T}_{T} and _{T}. Per Borghans, DeBoer, and Segel, we call

Recently, Pedersen et al. applied the tQSSA to the case of an enzyme converting two different substrates into products [

In 1981, Goldbeter and Koshland [_{p}) that are interconverted by two enzymes (say, E and D); see _{p} as a phosphorylated form of S and of E and D as a kinase and a phosphatase, respectively.) Assuming the MM conditions, _{T} ≪ _{me} and _{T} ≪ _{p}(0) + _{md}, Goldbeter and Koshland wrote a single dynamical equation for the time evolution of the switch:
_{p}(_{T}, _{T} = _{p}(0), _{e} = _{me}/_{T}, _{d} = _{md}/_{T}, _{e} = _{2e}_{T}/_{T}, _{d} = _{2d}_{T}/_{T}, and the subscripts “e” and “d” refer to the kinase and phosphatase reactions, respectively. Goldbeter and Koshland showed that the steady state solution of

(A) Substrate S is phosphorylated by kinase E and dephosphorylated by phosphatase D.

(B) Unpacked mechanism, including enzyme–substrate (E:S) complexes. The black dots at the tips of a T-shaped arrow indicate the two molecules that come together to form a complex, pointed to by the arrowhead; the dots are meant to indicate that enzyme–substrate binding is a reversible reaction. Formation of the product (E:S → E + P) is indicated by a T with two arrowheads (pointing to E and P); absence of a dot at the foot indicates that the catalytic step is presumed to be irreversible.

(C) Steady state value of _{p} from _{T}/_{T} for _{2d}/_{2e} = 1.7, _{me} = _{md} = 1 nM, _{T} = 50 nM, and for different values of _{T}: 0.5 nM (solid line), 5 nM (dashed line), and 50 nM (dotted line).

(D) Same as (C), but using the exact steady state equations in _{T} = 5 nM, the exact steady state dependence is ultrasensitive, whereas the approximated dependence (C) is not.

Equations for the GK Module (

The steady state response (_{p}^{*}) as a function of stimulus strength (kinase level, _{T}) is simply
_{e} and _{d} ≪ 1, then _{p}^{*} is a steeply sigmoidal function of _{T}. Goldbeter and Koshland called this signal-response curve zero-order ultrasensitivity.

Goldbeter and Koshland's analysis of phosphorylation–dephosphorylation cycles is fine for metabolic control systems, where metabolite concentrations _{T} are orders of magnitude larger than enzyme concentrations, _{T} and _{T}. But for PINs, the condition for the classical MM rate law is not valid, and we must keep track of the enzyme–substrate complexes (

The signal–response curve,
_{e} and _{d} ≪ 1, but this requires that the total enzyme concentrations be small with respect to the total substrate concentration: the standard MM requirement. For PINs, we cannot expect this requirement to be satisfied, which suggests that protein phosphorylation–dephosphorylation cycles are unlikely to be ultrasensitive. In _{p}/_{T}, as a function of _{T}/_{T} given by _{T}. As expected, the response function is ultrasensitive for _{e} and _{d} small, but ultrasensitivity is lost as _{e} and _{d} increase.

This conclusion about ultrasensitivity being lost in PINs, based as it is on the Padé approximant, is not reliable when enzymes and substrates are present in similar concentrations [_{e} and _{d} (see

In the next section we show that the tQSSA, besides giving insights into the steady state behavior of the network, provides a good approximation of its temporal dynamics as well.

In _{p}:S_{p}]) as a function of the relevant slow variable (_{p} for QSSA and _{p} for tQSSA), to highlight the presence of two different timescales. If the timescales are clearly separated, a sudden increase of the complex (displacement along the _{p} forms a complex with D, and before D:S_{p} reaches a maximum, S_{p} starts to be converted into S. As soon as S is produced, E:S is created and converted back into S_{p}, with most of E forming a stable complex with S ([E:S]/_{T} = 0.935 at steady state). Although the fast and slow dynamics are not perfectly separated in either QSSA or tQSSA, the timescale separation is clearly more pronounced in tQSSA (red curve in

Parameter Values, ℓ_{i}, for the Models: ℓ = letter (a,. . . , f) and

The simulation shows the rise of _{p}(0) = _{T}. Equations in _{T} and _{T}.

(A) Exact solution (black line), QSSA solution (blue line). The arrows indicate the direction of time, whereas the distance between two consecutive dots on the lines is 1 min.

(B) Exact solution (black), tQSSA solution (red).

(C) Time evolution of the same simulation, with the same color scheme. The enzyme–substrate complexes in this simulation are not negligible. In particular, at steady state, E is almost completely bound to S, whereas at the beginning of the time course, D:S_{p} accounts for roughly half of all substrate molecules.

(A) A simplified diagram shows all the actors of the network: two kinases S and E, and two phosphatases D and F. In grey, the additional reaction whereby S_{p} retains some catalytic activity.

(B) The unpacked diagram, keeping track of all enzyme–substrate complexes. Diagram conventions as in

(C) Upper panels: bifurcation diagrams for the exact model, equations in _{p} has no catalytic activity, the system does not show hysteresis. Hysteresis is recovered if S_{p} has some residual activity—grey lines in (A) and (B). The background activity has the following parameter values: _{1}= 0.05 nM min^{−1}, _{−1}= 0.005 min^{−1}, and _{2}= 0.0001 min^{−1}.

(C) Lower panels: phase plane diagrams for the tQSSA model with _{T}=12. _{p} has some catalytic activity, the system is bistable. Right: when S_{p} has no catalytic activity, the system is monostable.

(A) A simplified diagram shows all the actors of the network: two kinases S and E, and three phosphatases C, D, and F.

(B) The unpacked diagram, keeping track of all enzyme–substrate complexes. Diagram conventions as in

(C) The time course of the network shown in (B); equations in _{p} = 200 nM,

Equations for the S/E Module (

Equations for the Network of

In this section we study the steady state behavior of a system of two coupled GK switches (_{p}) and unphosphorylated (E) forms. Suppose that E_{p} is a less active form, and E → E_{p} is catalyzed by S. S and E are antagonists since they phosphorylate and inactivate each other. F is a phosphatase that converts E_{p} back to E. (In our notation for complexes, A:B, the enzyme comes first and substrate follows; for example, for the reaction whereby S phosphorylates E, the enzyme–substrate complex is denoted S:E.)

This is no arbitrary example; it describes exactly the interactions between two regulators of the G_{2}-to-mitosis (G_{2}/M) transition in the eukaryotic cell cycle [_{2}/M transition during early embryonic cell cycles. The equations of the model were derived from an implicit application of the QSSA, but to our knowledge an explicit derivation has never been done: this observation prompted us to investigate the matters addressed in this paper.

The antagonism between E and S (_{2}/M transition in the eukaryotic cell cycle [

In Novak and Tyson's original model, both S_{p} and E_{p} had some residual activity, a feature that in their model was not essential to generate hysteresis and that we have not taken into account so far. This residual activity becomes essential when the network is reduced to elementary steps: we find that bistability is recovered when we add the reactions E + S_{p} ↔ S_{p}:E → E_{p} + S_{p}, i.e., when S_{p} retains some limited kinase activity. (In _{p}:E complexes, thus helping E to inactivate S. Indeed, hysteresis is possible (unpublished data) with the association–dissociation reactions alone (E + S_{p} ↔ S_{p}:E) without the catalysis step (S_{p}:E → E_{p} + S_{p}). Of course, bistability can also be restored by allowing E_{p} to phosphorylate S.

To apply the tQSSA to such networks, we first define “hat” variables to include a single free molecular species plus all the complexes in which this species appears. Defined thus, the association–dissociation reactions and all the reactions where a chemical species serve as a catalyst cancel out. Here we define

Our definition of hat variables extends what was originally proposed for a single enzymatic reaction by Borghans, DeBoer, and Segel. In terms of the hat variables, we can describe each catalytic reaction in the network with equations similar to _{2}_{2} [E:S] and with the concentration of enzyme–substrate complex given by

Concerning this network: we perform no numerical simulations to compare tQSSA and QSSA; we will do that in the next section for a larger and biologically more significant network. Rather, we use the model reduced with tQSSA to perform phase plane analysis (

In the G_{2}/M network, the antagonism between Wee1 and MPF is aided by a second positive feedback loop, involving MPF and Cdc25, a phosphatase that removes the inactivating phosphate group from Cdc2 [_{p} is a more active form, S and D_{p} activate each other. Finally, we have C, an unregulated phosphatase that converts D_{p} back to D.

Altogether, the network consists of three GK modules: the first (C/D/S) controls D's phosphorylation, the second (D/S/E) affects the phosphorylation state of S, and the last (S/E/F) controls the activity of E (

Bifurcation analysis and CRNT show that the network is bistable even with no residual activity for S_{p} or E_{p} (unpublished data). Indeed, the positive feedback between D_{p} and S suffices, by itself, to generate hysteresis, as confirmed by CRNT. Interestingly, given the parameter values in

Since the model performs satisfactorily concerning its steady state behavior, we move on to compare the dynamics of both QSSA and tQSSA to the exact solution (

To compare further the QSSA and tQSSA with the exact solution, we plot in _{p}:S_{p}] and [E:S]) as functions of the slow variables (_{p} and _{p} and _{p}:S_{p}]—due to phosphorylation of D by S followed by dephosphorylation of D_{p} by C—(black curves in _{p}, [F:E_{p}], and

The exact solution (black lines) is compared with the QSSA, blue lines in (A), and to the tQSSA, red lines in B. Arrows indicate the direction of time, whereas the distance between consecutive dots on the lines is 1 min. Equations in

Coupled enzymatic reactions (e.g., interacting kinases and phosphatases) are common features of PINs. We analyze one of these networks, composed of three phosphorylation and three dephosphorylation reactions, altogether comprising 14 chemical species. The network was originally proposed by Novak and Tyson [

The complexes play an important role because of the topology of the network. In the cell cycle model, some molecular species are at the same time enzymes and substrates of each other. If for one reaction enzyme concentration is negligible compared with substrate concentration, then the opposite must be true when the roles are exchanged, allowing for a significant fraction of the substrate to be sequestered in the complex. Similarly, when some molecules form complexes with several enzymes, even if each complex is not present in a large amount, their sum may not be negligible.

The role played by enzyme–substrate complexes in PINs could be more important than currently appreciated. In the cell cycle network, Wee1 and MPF are antagonists that phosphorylate and inhibit each other. In our simulations, the concentrations of Wee1:MPF and MPF:Wee1 are not negligible. The presence of high concentrations of such complexes can be interpreted as a second way for Wee1 to inhibit MPF, by sequestration. In this sense, Wee1 behaves as both an inhibitory kinase and a stoichiometric cyclin dependent kinase (CDK) inhibitor. In our simulation we notice that MPF:Cdc25 is also present in high concentration (40% of total MPF, 10% of total Cdc25). If confirmed, that would suggest an intrinsic way to attenuate the positive feedback loop between MPF and Cdc25. Finally, trimeric complexes might form as well (e.g., Wee1:MPF:Wee1), and such molecular species may have great effects on the qualitative behavior of PINs (Sabouri-Ghomi, Ciliberto, Novak, and Tyson, unpublished data).

Of course, one can follow the exact dynamics of a control system by solving the full system of ordinary differential equations (ODEs). However, a full description of the system comes with many equations, making any qualitative analysis difficult. A typical way to reduce the number of equations is the QSSA originally formulated for an isolated enzyme-catalyzed reaction, when substrate is in great excess over enzyme. When applying the QSSA to a network of coupled catalytic reactions, with realistic values of rate constants and total protein concentrations, we found that the reduced system of ODEs obtained by the QSSA does not faithfully reproduce the dynamics of the full system of ODEs.

The tQSSA works for a larger range of parameter values than does the QSSA; in particular, it is valid even when the enzyme is in excess compared with the substrate. Such an approximation is particularly appealing in PINs where, as mentioned above, enzymes and substrates often exchange roles. Given its appeal, we applied tQSSA to a full model of the PIN that regulates the G_{2}/M transition in the cell cycle. We found by numerical simulations that the tQSSA does a good job describing coupled catalytic reactions. Moreover, applying tQSSA to PINs generates a set of differential–algebraic equations of standard format,

Summarizing, we propose that large networks of coupled enzymatic reactions should first be written in full and then reduced by applying the tQSSA. This way it will be possible to reduce the number of dynamic equations while maintaining the complexity of the network (i.e., including enzyme–substrate complexes) and simultaneously to achieve reliable approximate solutions for the transient dynamics of the network.

All calculations have been made using XPPAUT, software developed by Ermentrout [

We provide .ode files, readable by XPPAUT, for reproducing the results in this paper.

(6 KB ZIP)

This paper is dedicated to the memory of Lee Segel, a friend and inspiration to many theoretical biologists, including the authors. We thank Bela Novak for discussions. AC was supported by the Associazione Italiana Ricerca sul Cancro, and JJT by the James S. McDonnell Foundation.

Chemical Reaction Network Toolbox

Goldbeter–Koshland

Michaelis–Menten

M-phase promoting factor

ordinary differential equation

protein interaction network

quasi-steady state approximation

total quasi-steady state approximation