Implementation of three subsystems

Figure 5.2 gives a detailed overview on the structure of the MAPK cascade and in-troduces the naming conventions for this chapter. The symbols S_jⁱ denote the concen-trations of proteins and protein complexes which appear as substrates in the model.

The superscript index i indicates the layer of the cascade in which the specific protein is involved, while the subscript index j enumerates the different forms in which the substrate appears in the respective layer. Note that although we avoid the use of aster-isks for notational convenience, in the following all symbols S_jⁱ denote concentrations in the steady state. The free concentrations of the kinases and phosphatases that are not bound to a substrate in a complex are labeled K_Rⁱ and P_Rⁱ, respectively. The total concentrations of the substrate, kinase and phosphatase in each layer are denoted S_Tⁱ, K_Tⁱ and P_Tⁱ. These concentrations are assumed to be conserved quantities in the model and thus constitute constant external parameters. The reason for this assumption is that even though the total concentration can be subject to change in a cell due to changes in the rate of protein synthesis and degradation, these processes are assumed to occur on a slower timescale than the signaling that we want to model.

3 Layers 2 Layers

1 Layer

S⁰₀ S₁⁰

S₂⁰

S₃⁰ K_R⁰

P_R⁰

S¹₀ S₁¹ S₂¹

S₃¹

S₄¹

S₅¹

S₆¹ K¹_R

P_R¹

S₀² S₁² S₂²

S₃²

S₄²

S₅²

S₆²

P_R²

FB a)

FB b)

Figure 5.2: Schematic overview and naming conventions of the MAPK cascade. The three different subsystems that are analyzed separately are indicated by boxes. The solid arrows correspond to biochemical reactions while the dashed arrows denote the binding of an enzyme in the formation of an substrate-enzyme complex. In the two-layer and three-two-layer model, the dotted kinase K_R¹ (1-layer model) is replaced by the activated substrate S₁⁰. The gray dotted lines represent two types of feedback, which represent an extension to the model that is introduced and analyzed in Sec. 5.5.

In the following, we investigate the dynamics in three distinct models with increas-ing size. The first model contains a sincreas-ingle reversible double phosphorylation step with one substrate in isolation, as it occurs in the middle and bottom layer of the cascade.

The second model combines the first and second layer of the cascade but does not include the third layer. Finally, the third model contains all three layers. The three models are highlighted by boxes in Fig. 5.2. A technical detail is that the kinase that catalyzes the phosphorylation steps in the second layer is called K_R¹ in the first model in order to indicate that, in the scope of this model, it is constant and does not act as a substrate. In the two larger models, K_R¹ is replaced by S₁⁰, the activated substrate of the top layer.

Having outlined the large-scale structure of the models under consideration, we now explain the detailed structure of the single phosphorylation steps from which all these models are composed. This is shown explicitly for the phosphorylation in the top layer of the cascade,

S₀⁰+ K_R⁰ ⇋S₂⁰→ S1⁰+ K_R⁰. (5.6) The first step represents the reversible binding of the kinase K to the unphosphorylated substrate S₀⁰, leading to the formation of the complex S₂⁰. The second step describes the actual phosphorylation step catalyzed by the kinase, resulting in the release of the phosphorylated substrate S₁⁰.

There are different possibilities to model the elementary step of a covalent phos-phorylation catalyzed by a kinase. The most widespread approach is to use Michaelis-Menten kinetics. Michelis-Michaelis-Menten theory relies on the quasi-steady-state approxima-tion that the concentraapproxima-tions of the substrate-enzyme complexes do not change in time.

This approximation, which is justified by a separation of timescales, has been found to be not appropriate in many signal transduction networks [119]. Michaelis-Menten kinetics is not a good approximation if sequestration effects of the enzyme play an important role, as it is suspected for the MAPK cascade [120, 117, 100]. Sequestration can affect the dynamics because an enzyme that is bound to a substrate cannot at the same time participate in other reactions, including its own dephosphorylation. In the absence of external feedback, sequestration effects can thereby cause an indirect type of feedback, as will be shown below. Approximation schemes that do not explicitly adopt the complexes as variables, such as Michaelis-Menten kinetics, miss sequestra-tion effects. Although a sequestrasequestra-tion-based approximasequestra-tion scheme has been proposed recently [121], we do not make use of this approximation since the efficiency of GM does not strongly rely on the number of reactions or variables being small. For this reason, we represent the complexes explicitly by dynamic variables, following Ref. [100], and break each phosphorylation step into two separate processes.

Assuming that the copy numbers of proteins are sufficiently high for a description with a continuous variable, we capture the dynamics of the cascade by a system of coupled ODEs. For the phosphorylation step in the first layer of the cascade this leads to

d

dtS₀⁰ = −f1(S₀⁰, K_R⁰) + f₂(S₂⁰) d

dtS₂⁰ = f₁(S₀⁰, K_R⁰) − f2(S₂⁰) d

dtS₁⁰ = f3(S₂⁰)

(5.7)

where f1, f2, and f3 are general functions, which are not yet restricted to specific functional forms.

In an analogous way, the system of ODEs for the complete one-layer model can be constructed. The corresponding system of ODEs is

d

dtS₁¹ = f₉(S₃¹) − f10(S₁¹, P_R¹) + f₁₁(S₄¹)

− f¹³(S₁¹, K_R¹) + f14(S₅¹) + f18(S₆¹) d

dtS₂¹ = f₁₅(S₅¹) − f16(S₂¹, P_R¹) + f₁₇(S₆¹) d

dtS₃¹ = f7(S₀¹, K_R¹) − f⁸(S₃¹) − f⁹(S₃¹) d

dtS₄¹ = f10(S₁¹, P_R¹) − f¹¹(S₄¹) − f¹²(S₄¹) d

dtS₅¹ = f13(S₁¹, K_R¹) − f¹⁴(S₅¹) − f¹⁵(S₅¹) d

dtS₆¹ = f16(S₂¹, P_R¹) − f¹⁷(S₆¹) − f¹⁸(S₆¹)

(5.8)

From these equations, the stoichiometric matrix N is derived according to the rule that Ni,j = 1 if fj+6 appears in the equation for S_i¹ with a positive sign, Nij = −1 if f^j+6 appears in the equation with a negative sign and Nij = 0 otherwise. Since the top layer of the MAPK cascade is not included in the one-layer model, the enumeration of the fluxes starts with f₇ here.

In the next step, Λij = Nijfj+6/S_i¹ is constructed from the stoichiometric matrix.

Since the model explicitly describes the complexes formed by enzymes with their sub-strates, it is reasonable to assume that the remaining processes are governed by mass action kinetics, thus assuming that all functions depend in a linear way on all of their arguments. In this study, we thus use the ability of GM to deal with general nonlin-ear functions only at a later stage when we consider the effect of external feedback loops, while the ability to explore a large part of the parameter space efficiently is used throughout the whole study. With this choice, our model corresponds to the conven-tional models considered in [96, 100].

Note that even after it is assumed that all functional dependencies are linear, the system of equations is still nonlinear as it includes bilinear terms. Because of mass conservation, one of the variables including the substrate can be expressed as a function of the remaining variables and the total substrate concentration, changing some of the bilinear terms into quadratic terms as shown below. Assuming mass action, the matrix

of derivatives is given by

The entries of θ that are equal to 1 indicate linear dependencies. The remaining nonzero entries arise due to indirect effects of mass conservation. Since their expressions are derived in a very similar way, we only show the calculation for

θ_x^µ11 = ∂µ1

in which we explicitly use asterisks to denote steady-state concentrations.

Assuming mass action kinetics, f7(S₀¹, K_R¹) = α7S₀¹K_R¹ with a rate constant α7. It follows that

µ1 = f7

f₇^∗ = S₀¹K_R¹

S₀^1,∗K_R^1,∗ (5.10)

Because of mass conservation of the substrate and the kinase, S₀¹ = S_T¹ − S1¹ − S2¹ − S3¹ − S4¹ − S5¹ − S6¹ and K_R¹ = K_T¹ − S3¹ − S5¹. Therefore,

The Jacobian J = Λθ^µx is obtained by matrix multiplication. The result can be written as a product of the two matrices:

J = For the two-layer model and the three-layer model, the derivation of the Jacobian works in an analogous way. Since the two-layer model consists of 9 and the three-layer model of 15 dynamic variables, we do not show the large terms here. The system of ODEs for the three-layered cascade can be found in Appendix A.