Glycolysis as an illustrative example of flux exponent control

Chapter IV: Flux exponent control in metabolism: biological regulation as control of

4.2 Glycolysis as an illustrative example of flux exponent control

regulations done on a metabolic system. Section4.5poses regulation of metabolic fluxes as an optimal control problem as a way to explore metabolic dynamics. It discusses how to computationally solve such optimal control problems using a popular method in control theory called model predictive control (MPC). Section4.6uses optimal controllers solved by MPC to investigate rules of metabolic regulation in specific examples of metabolic networks, capturing important biological behaviors such as glycolytic oscillation and cell growth arrest under stress.

4.2 Glycolysis as an illustrative example of flux exponent control

In this section, we walk through how to apply flux exponent control (FEC) to a lumped model of glycolysis to illustrate the main features of FEC.

A simple model of glycolysis is introduced in Section1.4of Chapter1. Also see Figure1.13.

Instead of considering the detailed steps of reactions in glycolysis, we consider two lumped reactions that capture the structure of autocatalysis. This yields the following dynamics for the concentrations of metabolites:

𝑑 𝑑𝑡

⎡

⎣

𝑥1

𝑥₂

⎤

⎦=

⎡

⎣

1 −1

−𝑞 1 +𝑞

⎤

⎦

⎡

⎣

𝑣1

𝑣₂

⎤

⎦+

⎡

⎣

0

−1

⎤

⎦𝑤. (4.1)

This is of the form ^𝑑

𝑑𝑡𝑥=𝑆𝑣+𝑆^𝑤𝑤, where𝑥is the vector of metabolite concentrations, 𝑆 is the metabolic stoichiometry matrix,𝑣 is the vector of internal metabolic fluxes, and 𝑆^𝑤 is the metabolic stoichiometry matrix of external fluxes𝑤. Here𝑥₂is ATP, or energy charge, and𝑥₁ is a lumped intermediate of the glycolysis pathway, such as fructose 1,6- bisphosphate. The first reaction with flux𝑣₁ consumes𝑞 units of ATP and produce one unit of intermediate. We can take 𝑞 to be 1as a value that match the stoichiometry of ATP production in glycolysis. This reaction represents the activation of glucose by ATP to produce glycolysis intermediates. We assume glucose is overabundant, and therefore does not influence the fluxes. The second reaction with flux𝑣₂ consumes one unit of intermediate and produces1 +𝑞units of ATP. This reaction represents the net production part of the glycolysis pathway. Together, looping through the two reactions once results in a net production of one unit of ATP. We also include an external disturbance with flux𝑤 that consumes one unit of ATP. This corresponds to the maintenance energy cost of the cell, which can increase under environmental disturbances such as heat shocks.

When the regulatory mechanisms of the fluxes are known in detail, we can specify exactly how 𝑣₁ and 𝑣₂ are regulated, as static functions or dynamic processes that depend on (𝑥₁, 𝑥2). Although we do know some of the mechanisms in this case because glycolysis has been studied for decades, let us assume we do not have this information to illustrate how to deal with a generic metabolic network.

One constraint-based approach to make progress is flux control. Since we do not know how any of the fluxes are regulated, flux control considers all the fluxes as arbitrarily regulated by the cell. This gives the following formulation:

𝑑 𝑑𝑡

⎡

⎣

𝑥₁ 𝑥₂

⎤

⎦=

⎡

⎣

1 −1

−𝑞 1 +𝑞

⎤

⎦

⎡

⎣

𝑢₁ 𝑢₂

⎤

⎦+

⎡

⎣

0

−1

⎤

⎦𝑤. (4.2)

Here we simply substituted the flux variables(𝑣₁, 𝑣2)with control variables(𝑢₁, 𝑢2)that the cell can adjust. Flux control as a constraint-based approach then says that(𝑢₁, 𝑢₂)can take arbitrary trajectories as controlled by the cell, if no further information about the fluxes is given. This often results in systems with trivial behavior. For example, here we can do a change of variable by defining new control variables𝑢^′₁ =𝑢₁−𝑢₂, and𝑢^′₂ = (1 +𝑞)𝑢₂−𝑞𝑢₁ to achieve a trivial dynamics:

𝑑 𝑑𝑡

⎡

⎣

𝑥₁ 𝑥2

⎤

⎦=

⎡

⎣

𝑢^′₁ 𝑢^′₂

⎤

⎦+

⎡

⎣

0

−1

⎤

⎦𝑤. (4.3)

Since the cell can adjust𝑢^′₁ and𝑢^′₂ arbitrarily, the metabolite concentrations(𝑥₁, 𝑥₂)can also be made arbitrary. In other words, flux control is often underconstrained to capture metabolic dynamics.

One way to make progress despite that flux control is underconstrained is by focusing on the steady state fluxes. This results in the constraint-based approach of flux balance analysis (FBA). In this case, we obtain the following at steady state:

0 =

⎡

⎣

1 −1

−𝑞 1 +𝑞

⎤

⎦

⎡

⎣

𝑢^*₁ 𝑢^*₂

⎤

⎦+

⎡

⎣

0

−1

⎤

⎦𝑤^*. (4.4)

Here(𝑢^*₁, 𝑢^*₂)and𝑤^*are steady state fluxes. We see that the steady state condition results in a constraint on the steady state fluxes by metabolic stoichiometry. In this case, the stoichiometry matrix is invertible, and therefore determines the internal fluxes in terms of the external flux uniquely at steady state.

This may seem magical, that by looking at steady state fluxes, we have gone from having no constraints to work with to having strong constraints. The deeper reason is that when FBA looks at steady state fluxes, it is implicitly assuming that the steady state fluxes are achieved, which requires the cell to maintain a stable homeostasis. This is not guaranteed at all when the fluxes are dynamic. For example, the system can be oscillatory without ever reaching a steady state. It can also crash into a disaster state that the cell dies.

Fundamentally, flux control cannot answer questions about stability because it eliminates the intrinsic dynamics of metabolism by assuming all fluxes are adjustable. One evidence of this in the glycolysis case is that glycolytic intermediates oscillate under stress, from yeast to mouse muscle cells [16,49,51]. This is due to the intrinsic instability of autocatalytic stoichiometry (also see Section 1.4). This implies that metabolic fluxes have intrinsic dynamics that are not modifiable by cells’ regulation. Mathematically, if the fluxes are static functions, then this implies the fluxes are not fully controlled (𝑣 = 𝑢), but rather partially controlled (𝑣 =𝑣(𝑥,𝑢)).

Flux exponent control (FEC) exactly formalizes which part is controlled and which part is not. FEC is based on our previous study on binding’s regulation of catalysis in Chapter2and Chapter3. Since cells regulate metabolic fluxes through binding reactions, while binding reactions adjust fluxes’ exponents, or reaction orders, within a constrained polyhedral set, we conclude that cells regulate fluxes’ exponents. This is the content of the FEC rule.

Mathematically, this means we split each metabolic flux in the following way:

𝑣𝑗 =𝑣⁰_𝑗𝑥^𝐻

𝐴 𝑗1

1 · · ·𝑥^𝐻

𝐴

𝑛𝑗𝑛𝑢^𝐻

𝐵 𝑗1

1 · · ·𝑢^𝐻

𝐵

𝑛𝑢𝑗𝑛𝑢 =:𝑣_𝑗⁰𝑥^𝐻𝑗𝐴∘𝑢^𝐻𝑗𝐵 =𝑣⁰_𝑗exp^{︁𝐻_𝑗^𝐴log𝑥+𝐻_𝑗^𝐵log𝑢^}︁. (4.5) The above equation writes the decomposition of the flux of reaction𝑗 in several different notations. The first notation writes each term explicitly. The second notation is the most succinct. The third notation makes it clear that the control actions are on the flux exponents.

The flux is decomposed into three parts: (1) a constant reference flux magnitude𝑣⁰_𝑗; (2) the passive dependence on metabolite concentrations𝑥^𝐻

𝐴 𝑗1

1 · · ·𝑥^𝐻

𝐴 𝑗𝑛

𝑛 =:𝑥^𝐻𝑗𝐴 =: exp^{︁𝐻_𝑗^𝐴log𝑥^}︁, where𝐻_𝑗^𝐴is the𝑗th row vector of a matrix𝐻^𝐴; (3) the active regulation depending on control actions𝑢^𝐻

𝐵 𝑗1

1 · · ·𝑢^𝐻

𝐵

𝑛𝑢𝑗𝑛𝑢 =:𝑢^𝐻𝑗𝐵 =: exp^{︁𝐻_𝑗^𝐵log𝑢^}︁. Here∘denotes component-wise product between two vectors, and exponentialexpis applied component-wise. The matrix 𝐻^𝐴specify the passive exponent or reaction order, capturing how the metabolic reactions would proceed if there is no regulatory mechanisms in place. The matrix 𝐻^𝐵 specify which control variables𝑢influence which fluxes.

Back to the glycolysis example, we may propose the following natural split of the fluxes:

𝑣₁ =𝑣₁⁰𝑥₂𝑢₁, 𝑣₂ =𝑣⁰₂𝑥₁𝑢₂. (4.6) This corresponds to having 𝐻^𝐵as identity matrix so that there is one exponent control variable for each flux, and having𝐻^𝐴with𝐻₁₂^𝐴 =𝐻₂₁^𝐴 = 1and other entries zero. This is because reaction𝑣₁consumes𝑥₂, therefore higher𝑥₂should naturally increase the flux𝑣₁ without active regulation. Similarly, reaction𝑣₂ consumes𝑥₁, therefore higher𝑥₁ should naturally increase the flux𝑣₂. We then choose the exponent of this increase to be1as a default choice, simply because first order dependence is the most common. The entries 𝐻₁₁^𝐴 and𝐻₂₂^𝐴 are zero here because these two reactions are almost always irreversible from thermodynamic considerations, therefore not inhibited by product molecules.

Together, we have the following FEC formulation of glycolysis fluxes:

𝑑 𝑑𝑡

⎡

⎣

𝑥₁ 𝑥₂

⎤

⎦=

⎡

⎣

1 −1

−𝑞 1 +𝑞

⎤

⎦

⎡

⎣

𝑣⁰₁𝑥₂𝑢₁ 𝑣⁰₂𝑥₁𝑢₂

⎤

⎦+

⎡

⎣

0

−1

⎤

⎦𝑤. (4.7)

We see that when the control variables (𝑢₁, 𝑢₂) are constant, the system has a passive dynamics that is linear and unstable, since𝑥₁ and𝑥₂positively influence each other (also see Section1.4). So FEC retains the intrinsic dynamics of metabolism by placing control variables on the exponents. On top of this formulation, further constraints can be added if more information is known. For example, we may upper bound𝑢₁ by1if we know the maximum flux of𝑣₁and set it to𝑣₁⁰. We may also constrain the rate of change in the control variables, if we believe the regulatory mechanisms are slow. We could even require the control actions to keep𝑥₂ above a certain level, if we consider there exists a minimum level of ATP concentration needed to keep cells alive.

Given the FEC formulation of a metabolic network, we can perform several types of analysis.

First, since FEC formulates a metabolic network as a control system, we can use control theory tools to study the hard limits on system performance. This is exemplified by the

work [25], where Bode’s theorem on conservation of robustness is used to explain glycolytic oscillations as inevitable side effects of the tradeoff between steady-state error and system fragility (which causes oscillations). Second, we may be interested in finding particular regulatory behaviors of the system when the regulation is optimal for a certain objective.

This can help us interpret various regulatory strategies in different scenarios. For example, we may consider optimizing for keeping the ATP concentration steady at a reference level.

Once an objective is formulated, solving for the optimal control variables and the system behavior becomes an optimal control problem. This is investigated in Section4.5and some simulation result is discussed in Section4.6.

A third way to use FEC is to relate control actions with underlying binding networks, since control of flux exponents biologically correspond to binding networks’ regulation of catalysis fluxes. We can explicitly illustrate this here since mechanisms of the flux regulations in glycolysis is well studied, so plausible models of the underlying binding networks exist. For example, [25] proposed the following simple models of the fluxes based on knowledge about catalyzing enzymes’ allosteric feedback:

𝑣₁ =𝑣₁⁰𝑥₂𝑢₁ =𝑣₁⁰𝑥₂ 1

1 +𝑥^2ℎ₂ , 𝑣₂ =𝑣₂⁰𝑥₁𝑢₂ =𝑣₂⁰𝑥₁ 1

1 +𝑥^2𝑔₂ . (4.8) In other words, both control variables𝑢₁ and𝑢₂implement negative feedback based on the ATP concentration𝑥₂, with allosteric coefficients2ℎand2𝑔, respectively. These rational function forms of𝑢_𝑖 could have mechanistic origins from binding networks. For example, 𝑢1 may come from the following binding network:

2𝑋₂+𝐸₁ ⇌𝐸₁^′, (4.9)

where𝐸₁is the enzyme catalyzing flux𝑣₁, and𝐸₁^′ is an inactive form of the enzyme. When 𝑋₂is overabundant compared to𝐸₁, we have the following expression for the flux of the first reaction:

𝑣₁ =𝑣₁⁰𝑥₂ 1 1 + ^𝑥_𝑘²

1

2, (4.10)

where𝑘₁is the dissociation constant in the above binding reaction of𝐸₁ with two𝑋₂. This binding network implements the control action in [25] withℎ= 1. This simple example illustrates how we can relate control variables with biological mechanisms of binding networks.

This concludes our illustration of FEC through the glycolysis example. In the next section, we begin our study of constraint-based methods from the layered architecture of metabolism.

Figure 4.3Cartoon illustrating constraint-based models and hard-limits or laws as two approaches in mapping between mechanisms and phenotypes. For a class of phenotypes, laws or hard limits can specify a criteria that many mechanisms can be used to satisfy it. As examples, many codes and hardware can implement certain channel capacity in communication networks, and many electronic circuits can implement certain performance criteria of a signal processing input-output map. For a class of mechanisms, described as a constraint-based model where some mechanisms are fixed and other mechanisms are free to vary, many phenotypes or system behaviors can be achieved. As an example, cars have some common features fixed as constraints, with the rest left to vary, resulting in a wide range of mileage, speed, safety, and comfort.