Metabolic regulation as control of flux exponents

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

4.3 Metabolic regulation as control of flux exponents

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.

The constraint-based approach, on the other hand, was motivated by sparse data available to characterize metabolism dynamics for whole-cell metabolism modeling, such as for bacterial growth and human metabolism. Therefore the questions that the constraint-based approach focused on was how to effectively use the sparse data available to constrain reality, i.e. constrain the space of plausible metabolic fluxes. This is like a “sufficient” mapping from a point in mechanism space to a feasible set in phenotype space. For example, given the known stoichiometry of the glycolysis pathway and some upper and lower bounds on the fluxes based on rough maximum numbers of enzymes per cell, then flux balance analysis (FBA), a constraint-based approach, yields a set of possible steady state fluxes the cell can have, such as ATP production rate, that are constrained by this mechanism. For a point in the set of feasible fluxes, the glycolysis mechanism is sufficient to allow it to happen, although whether it can actually happen depends on the details not included in the constraint-based model, such as actual enzyme numbers and catalysis rates.

Although these two approaches were developed from different motivations, they yield complimentary insights for the same problem, therefore should be integrated in a cohesive fashion. In the following, we take the constraint-based approach as the main perspective to develop our formulation for modeling metabolism dynamics. But once a control system is formulated, we compliment the story with the hard-limits or laws perspective.

Since constraint-based modeling is fundamentally rooted in the layered architecture of metabolism, we introduce this next.

Stoichiometry-flux split in layered architecture of metabolism enables flux control as a constraint-based approach

Layered architecture of metabolism and the stoichiometry-flux split. Constraint-based modeling splits the mechanisms of a metabolic system into two parts: a slow-varying known part of which we have solid knowledge, and a fast-varying unknown part of which we have little knowledge. Then the known part is taken as constraints, and the unknown part is allowed to vary freely. All feasible behaviors of the system are then the set of behaviors that the system can achieve by varying the unknown parts, with the known parts held fixed. Fundamentally, for this split to be effective conceptually and mathematically, the system of concern needs to have a natural layered architecture so that the lower layer, the layer that already exists and is to be controlled, is known and fixed, and the higher layer, the layer controlling the lower layer, is unknown and varies. This split requires both a time-scale separation of dynamics at each layer, and a corresponding structural split in the organization of the system into layers. Such layered architectures may be generally viewed

Figure 4.4Diagram showing the bowtie shape of metabolite stoichiometry in microbial metabolism. Adapted from Figure 1 of [13], also reproduced as Figure 2.1 in [59].

as a result of adaptation to achieve optimal system performance at diverse timescales using components that are limited by severe tradeoffs such as ones on speed-accuracy [38,79,80].

Cells use an architecture built on metabolic reactions to achieve diverse behaviors across several timescales, from consumption and secretion of molecules and response to envi- ronmental signals, to growth and death, to differentiation and cell-cell communications.

This motivates us to view the metabolic architecture of a cell as interconnected layers with separating time scales. For the purpose of this work on constraint-based modeling of metabolism dynamics, we split metabolic regulation into three layers: the bottom metabolic stoichiometry layer, the middle enzyme regulation layer, and the top gene expression layer.

For the bottom layer, the stoichiometry of metabolic reactions describe the number of metabolite molecules consumed and produced in each reaction (see Figure 4.4). The reaction fluxes involved are catalyzed by enzymes with millisecond to second timescale for catalysis rate. In contrast, the metabolism stoichiometry, i.e. what chemical reactions can happen in metabolism, varies on a much slower timescale, therefore can be considered fixed. Indeed, since metabolic reactions in cells happen largely because of the existence of corresponding enzymes, the timescale for modifying the stoichiometry is the timescale

Figure 4.5Diagram showing the layered architecture of microbial metabolism. Metabolism can be roughly viewed as consisting of three layers, from bottom up. The bottom layer is the stoichiometry of metabolic reactions, capturing how metabolite amounts are regulated by reaction fluxes. The middle layer is the proteins’ regulation of metabolic enzymes, capturing how reaction fluxes are regulated by protein binding (squares) The top layer is transcription translation (TXTL), capturing how protein concentrations are regulated by production and degradation. The gene expression layer viewed as a controller for the lower layers has an hourglass shape, connecting diverse genes with diverse proteins via a thin waist of transcription and translation machinery, which has building blocks such as amino acids and nucleotides supplied by the bottom layer.

for generating new enzymes catalyzing new reactions or deleting existing enzymes. This largely come from mutations and evolution. While loss of function can happen on a timescale of tens of hours in bacteria, gain of function to catalyze new reactions often takes much longer, limited either by the generation of the right collection of mutations or by selection pressure to propagate the new function. Exceptions might be horizontal gene transfers, which can happen on tens of hours timescale similar to loss of function. So we conclude that the timescale for modifying the stoichiometry layer is more than tens of hours, with loss of function and horizontal gene transfers on the faster end, but overall much slower than the timescale for metabolic fluxes. Therefore the stoichiometry layer can be viewed as having fixed stoichiometry, with quickly varying fluxes regulated by higher layers (see Figure4.5and Figure4.6).

This separation of timescales between the stoichiometry of bottom layer and fluxes regulated by higher layers is at the heart of existing constraint-based approaches to metabolism.

Namely, instead of arbitrary dynamics of varying metabolite concentrations in a metabolic network, this layered architecture motivates a stoichiometry-flux split. Mathematically, the metabolite concentration dynamics without any further knowledge can be written as a generic nonlinear dynamical system,

𝑑

𝑑𝑡𝑥(𝑡) =𝑓(𝑥(𝑡),𝑤(𝑡)), (4.11)

Figure 4.6Illustration of how the two constraint-based approaches, flux control and flux exponent control, relate to the split in the layered architecture of metabolism. For simplicity, the terms for external exchange fluxes𝑤are omitted.

where𝑥(𝑡) ∈ R^𝑛_>0 is the concentration of metabolite in the cell, 𝑤(𝑡) ∈ R^𝑚_>0^𝑤 is exchange fluxes with external environments, and𝑓 :R^𝑛_>0 ×R^𝑛_>0^𝑤 → R^𝑛 is the change in metabolite concentrations. We do not write the system as an autonomous one like ^𝑑

𝑑𝑡𝑥 = 𝑓(𝑥) because exchange with external environments is often essential to maintain any steady state of a metabolic system. As an example, for glucose fermentation in bacteria,𝑥may include internal metabolites such as ATP, fructose biphosphate, pyruvate, and internal glucose, while 𝑤 may include glucose import flux and lactate export flux. See Figure 4.7for the control diagram describing this general unstructured formulation. With the stoichiometry-flux split, we can further write

𝑑

𝑑𝑡𝑥(𝑡) =𝑓(𝑥(𝑡),𝑤(𝑡)) =𝑆𝑣(𝑥(𝑡)) +𝑆^𝑤𝑤(𝑡), (4.12) where𝑆 ∈R^𝑛×𝑚is the cell internal metabolism stoichiometry matrix,𝑚is the number of internal metabolic reactions, and𝑣(𝑥(𝑡))∈ R^𝑚_>0 is the fluxes of the 𝑚internal reactions, varying with metabolite concentrations through regulatory mechanisms such as enzyme allostery and gene regulation. Here 𝑆^𝑤 ∈ R^{𝑛×𝑚𝑤} is the stoichiometry for the external exchange fluxes. Note that we required the internal and external fluxes 𝑣 and 𝑤 to be positive, so for reversible reactions, the forward and reverse fluxes are represented separately. This is important to fully capture the dynamics in fluxes such as futile cycles, which cannot be distinguished by just the net fluxes.

Based on timescale separation, this stoichiometry-flux split captures an essential structure of metabolism that makes experimental measurements and modelling much easier. Without this split, to write down the metabolite dynamics requires detailed knowledge about𝑓(𝑥), i.e. how the change in metabolite concentrations depend on metabolite concentrations.

Without further structures, this require measurements for transient changes with the metabolic system initiating at all possible metabolite concentrations𝑥∈R^𝑛_>0. This is too

costly to obtain experimentally. In contrast, with this split, we can obtain knowledge for the stoichiometry𝑆 and the flux regulation𝑣(𝑥)separately. In particular, the stoichiometry 𝑆 can be easily determined based on bulk end-point measurements. The regulation of flux𝑣_𝑖(𝑥)can also be measured independent of the other fluxes, if this metabolic reaction can be perturbed and measured in an isolated fashion. Therefore, the stoichiometry-flux split greatly facilitates mechanistic measurements and modelling by capturing an essential structure of metabolism, which corresponds to a timescale separation in the layered architecture of metabolism.

Flux control from stoichiometry-flux split. We can develop a constraint-based modeling approach based on this stoichiometry-flux split. Naturally, the stoichiometry, which varies slowly, is taken as constraints, while the fluxes, which vary quickly, are allowed to vary.

So we call this constraint-based approachflux control, since the free-to-vary fluxes can be considered as control knobs on this metabolic system (see Figure4.7). Indeed, existing constraint-based approaches can all be considered as flux control. While further constraints and relations, such as ones based on thermodynamics or kinetic measurements, may be added to the fluxes, the variables to vary are always the fluxes. A generic description of flux control can therefore be written as follows:

𝑑

𝑑𝑡𝑥(𝑡) = 𝑆𝑢(𝑡) +𝑆^𝑤𝑤(𝑡), 𝑢(0 : 𝑇)∈ 𝒰_(0:𝑇). (4.13) Here 𝑢(𝑡) ∈ R^𝑚_>0 is the control variable, or the variable to vary, representing metabolic fluxes. Since the fluxes can vary with time, any constraints we put on the fluxes should be constraints on time trajectories of fluxes. Here𝑢(0 :𝑇)denotes the time trajectory of fluxes in the time interval of concern[0, 𝑇], and𝒰_(0:𝑇) denotes the constraint on the flux trajectories, or the set of allowed flux trajectories. Without any knowledge constraining the fluxes, we would take𝒰_0:𝑇 to be any function mapping[0, 𝑇]→R^𝑚_≥0, or some trajectory space with regularities such as ones with finite𝐿2 norm.

One powerful aspect of constraint-based approaches is that they allow investigation into the set of all possible behaviors. We notice that although this flux control formulation incorporated the stoichiometry-flux split, if we do not have severe constraints on flux trajectories from 𝒰_(0:𝑇), then the metabolite trajectories 𝑥(0 : 𝑇) can be quite arbitrary.

This is because the stoichiometry matrix𝑆 ∈R^𝑛×𝑚 is often of wide rectangular shape, i.e.

𝑛 < 𝑚, or the number of metabolite species less than the number of reactions. As a result, 𝑆 almost always has full row rank, which allows the control by fluxes𝑢to make arbitrary trajectories of𝑥achievable. More explicitly, we can view Eqn (4.13) as a control system with trivial plant dynamics, so the system is controllable, i.e. 𝑥can be controlled by𝑢

Figure 4.7Control diagrams for several formulations of metabolism dynamics. The left is the unstructured general description of metabolite concentration dynamics, with input as external exchange fluxes𝑤and output as metabolite concentrations𝑥. The middle one is the flux control formulation, with stoichiometry explicitly represented, and internal fluxes considered as control variable𝑢. The state variable𝑥has trivial plant dynamics that is a direct integration of inputs. The right one is the flux exponent control formulation, with exponents of internal fluxes as control variable𝑢. The state variable𝑥has nontrivial plant dynamics representing the uncontrolled internal fluxes.

to achieve arbitrary values at arbitrary time, if𝑆𝑆^⊺is full rank. This is very likely since 𝑆 is wide rectangular. Here plant is a control theory term referring to the uncontrolled dynamics, or the process to be controlled.

One approach,flux balance analysis (FBA), makes the flux control formulation more useful by restricting our attention tosteady-state fluxes. This assumes steady states exist and are achieved, which is highly plausible for metabolism as homeostasis is one of the hallmarks of biological systems. This steady state assumption is applicable whenever the phenomenon of concern is much slower than cell metabolism, i.e. approximately hours or longer. By making this steady state assumption, Eqn (4.13) becomes

0=𝑆𝑢+𝑆^𝑤𝑤, 𝑢∈ 𝒰_ss. (4.14)

This equation comes from Eqn (4.13) simply by setting ^𝑑

𝑑𝑡𝑥 = 0 at steady state. Here 𝑢∈R^𝑚_≥0 is the steady state fluxes,𝑤is steady state exchange fluxes, and the constraint set has become static as well,𝒰_ss ⊂R^𝑚_≥0. Another simplification is often used in this steady state case. In metabolism, often both the forward and reverse directions of a reaction can happen with nontrivial rates. For dynamics, it is important to keep the forward and reverse directions separate, as a net flux could correspond to different pairs of forward and reverse fluxes. This is also important to capture “futile cycles” and energy consumption. However, focusing on steady state, these issues can no longer be captured, and thus a pair of forward and reverse reactions can be combined to one reaction without loss of generality, and their fluxes can be combined to one net flux. This can be captured by writing only one direction of each forward-reverse pair in the stoichiometry matrix𝑆, and allowing the fluxes𝑢∈R^𝑚 to be negative. This also makes the constraint set allowing negative fluxes by default, 𝒰 ⊂R^𝑚.

Now we look at how the FBA formulation in Eqn4.14 constrains the set of all possible steady state fluxes. Since FBA focuses on steady states, we can no longer capture metabolite concentrations. This is because𝑥disappears in Eqn (4.14), as steady state concentrations are set by flux regulations𝑣(𝑥), which are assumed to be unknown, or set as control variables 𝑢, in flux control. Instead, the steady state assumption enables a natural constraint relating steady state internal and external fluxes𝑢 and𝑤, that 𝑆𝑢= −𝑆^𝑤𝑤. There are several ways to view this relation. For example, we can view exchange fluxes𝑤as measurable, therefore fixed and used as constraints on internal fluxes𝑢. This reduces𝑢’s degrees of freedom from𝑚, the number of reactions, to𝑚−𝑟, where𝑟is the rank of𝑆. Indeed, if we only focus on the linear equation in Eqn4.14and ignore the constraint set𝒰ssfor a moment, we can write the general solution of the linear equation as

𝑢=−𝑆^†𝑆^𝑤𝑤+𝑢^⊥, for any𝑢^⊥∈ker𝑆.

Here𝑆^†is the pseudo-inverse of𝑆, andker𝑆 is the kernel of𝑆, i.e. the space of vectors𝑢 such that𝑆𝑢= 0. If𝑟is the rank of𝑆, thenker𝑆 has dimension𝑚−𝑟. So𝑢is constrained to have the specific solution part𝑆^†𝑆^𝑤𝑤fixed, and only vary the𝑢^⊥part inker𝑆, which is 𝑚−𝑟dimensional rather than𝑚dimensional that𝑢began with. If𝑚 =𝑟, which could happen for small or simplified metabolic systems, then𝑢is uniquely determined.

Further constraints on𝑢that can be implemented through𝒰_sswould require knowledge beyond stoichiometry. One example of a general constraint is upper and lower bounds on the steady state fluxes based on estimates on the maximum number of enzymes per gram of cell weight, and the maximum enzyme catalysis activity. In fact, 1000 milli-molar per gram of cell dry weight per hour (mmg⁻¹h⁻¹) is widely used as a generic upper bound on flux magnitude [85]. Similar to this, any knowledge about the fluxes from experimental measurements, such as upper bounds on rate of certain nutrient uptake, can be incorporated as constraints through𝒰_ss. Another class of general constraints on fluxes𝑢 are based on energy dissipation and thermodynamics of the metabolic reactions involved.

This is discussed in more detail in the next subsection. An important class of qualitative constraints from thermodynamic arguments is the irreversibility or direction of fluxes.

Thermodynamics dictates that only reactions with a negative Gibbs free energy can happen.

The Gibbs free energy of the𝑗th reaction can be written asΔ𝐺_𝑗 = Δ𝐺⁰_𝑗 +𝑅𝑇log𝑄. Δ𝐺⁰_𝑗 is the standard free energy change of this reaction under some standard condition such as pH 7, 1 atmospheric pressure, and 1 molar concentrations, so this part is independent of concentrations. 𝑄is the reaction quotient, calculated as the ratio between a numerator from multiplying product concentrations to exponents of their stoichiometric coefficients, divided by a denominator from the same procedure for reactants. 𝑅is molar gas constant,

𝑇 is temperature, andlogis natural log. SoΔ𝐺_𝑗is the sum of the standard part independent of concentrations, and the reaction quotient part that increases with product concentrations and decreases with reaction concentrations. Therefore, for a given reaction, if Δ𝐺_𝑗 is always negative based on standard free energy and the range of reactant and product concentrations for scenarios of concern, then we can claim the flux𝑣_𝑗 always flow in the forward direction, i.e. 𝑣_𝑗 > 0. Since the reaction quotient is under log, the influence of concentrations on the reaction free energy is limited. So if the standard free energy is very negative, then this usually guarantees the flux is irreversible. Therefore irreversibility of reactions serve as a general class of easy to obtain constraints on the fluxes.

Altogether, the set of all possible steady state fluxes in FBA formulation (Eqn (4.14)) can be written as

{︁𝑢∈ 𝒰_ss ⊂R^𝑚 :𝑢=𝑆^†𝑆^𝑤𝑤+𝑢^⊥, 𝑢^⊥ ∈ker𝑆^}︁. (4.15) Then steady state behaviors of interest can be further explored through optimization, such as fluxes under maximum biomass production in aerobic or anaerobic environments with different nutrients as carbon sources, or limits on ATP or redox potential regeneration.

With a good curated model of stoichiometry𝑆 and constraint set𝒰_ss, the FBA formulation has been successfully applied to whole-metabolism models of a wide range of microbes.

Flux control fails to capture internal dynamics of metabolism. However, there is one severe restriction in the FBA or the flux control formulation in general: the lack of internal dynamics. Many important metabolic behaviors and interactions are dynamic or due to transient dynamics, such as microbial growth arrest under nutrient or environmental shock and Crabtree or Warburg effects [3]. In turn, essential features of these dynamics are due to limits from internal dynamics. One essential aspect of steady state behavior is whether a given steady state is stable, i.e. adapts back from small perturbations. Stability, however, can only be determined from dynamics. Now looking back at FBA, it is unable to capture dynamics since the founding assumption of FBA is metabolism has reached a stable steady state. This is not remedied by going back to the flux control formulation, however. The reason is that the flux control formulation splits all metabolic fluxes into just two types, external exchanges and internal controls (see Figure4.7). In other words, all internal fluxes are assumed to be controllable, and the metabolite dynamics is simply integrating the controlled fluxes. As a result, the “plant”, or the metabolic process to be controlled, is trivial. Therefore metabolism has no internal dynamics, and all dynamics is determined solely by the controller. If we make an analogy from mechanics, mechanical objects have internal dynamics which we often call “inertia”. This is because mechanical control actions are limited to forces, or accelerations, which directly alters speed or moments, but only