Illustrative example

Chapter II: Polyhedral constraints enable holistic analysis of bioregulation

3.2 Illustrative example

experimentally.

In Section 3.3 we formally define binding reaction networks using chemical reaction network theory, and characterize a class of binding networks that are biologically plausible.

In Section 3.4 we characterize the manifold of equilibrium or detailed balance steady states of binding networks, and introduce log derivatives as a transform between different parameterizations of the manifold. In Section3.6, we focus on a binding network with just one binding reaction and fully analyze the reaction orders (one type of log derivatives) and their biological implications. We observe the polyhedral set bounding the full range of values the reaction orders can take. In Section 3.7, given the central importance of vertices of reaction order polyhedra, we characterize the vertices in terms of minimal support vectors of linear subspaces and develop a computational method to obtain them at scale. In Section3.8, we show that polyhedra arise naturally from decomposition of log derivative operators. Using this, we develop an approach to obtain reaction order polyhedra analytically.

Figure 3.2Illustration of how methods from this chapter can be used to study the enzymatic reaction with product binding. (a). The binding network for enzymatic reaction with product binding. 𝐸 is enzyme,𝑆is substrate, they bind to form complex𝐶𝐸𝑆, which gets catalyzed to complex𝐶𝐸𝑃 which can unbind or bind from𝐸and prouduct molecule𝑃. The catalysis rate of substrate to product conversion is therefore proportional to𝐶𝐸𝑆, which is the target species here (grey cicle). The squiggly arrow denotes catalysis reaction.(b). Simulation of this enzymatic reaction with product binding, converting substrates to products. Blue lines are product fraction, defined as total product over the sum of total product and substrate ^𝑡𝑃

𝑡_𝑆+𝑡_𝑃. Orange lines are the concentration of target species𝐶𝐸𝑆, proportional to catalysis rate. Three different parameter settings are run, with increasing enzyme-product binding strength (i.e. decreasing𝐾𝐸𝑃, graphically represented as increasing opacity). Parameter values are𝑘^cat=𝐾𝐸𝑆 =𝑡𝐸 = 1,𝑡𝑆 +𝑡𝑃 = 10, 𝐾𝐸𝑃 ∈{︀

30⁻¹,1,10}︀

(smaller𝐾𝐸𝑃is less opaque line).(c). The dominance decomposition tree of the binding network, showing how the vertices and rays of𝐶𝐸𝑆’s reaction order polyhedron can be obtained analytically.

Upper right corner lists the definition for totals and the steady state expressions of the target species, to help with keeping track of the decomposition steps. The convex combination of the vertices circled by orange or green corresponds to the orange or green points in (d).(d). The reaction order polyhedron of𝐶_𝐸𝑆, the target species, by computer sampling. The upper left is a 3D view. The other three panels are projection of the 3D polyhedron to different 2D planes. The green and orange points corresponds to the dominance conditions and vertices in the DDT in (c). 10⁵points are taken by log-uniformly sampling(𝐸, 𝑆, 𝑃, 𝐶𝐸𝑆, 𝐶𝐸𝑃)with values in(10⁻⁶,10⁶). Dominance condition is evaluated for 100-fold difference: orange points is ^𝑡𝑆

𝑡_𝐸 ≥100, green points is𝑡_𝐸≫𝐶_𝐸𝑃 defined by ^𝐶𝐸𝑃

𝑡𝐸 ≤0.01.

process with no product, so we see the fraction of product in the total of substrate and product continuously increase over time. However, the rate of production decreases as the product accumulates, with a very nonlinear inhibition effect. While the medium binding strength, a 10-fold increase to weak binding strength, causes only negligible increase in the increase of product fraction, the strong binding strength, a 300-fold increase, causes significant effect. This inhibition effect also unevenly influence production rate at different stage of the process. Here, a 300-fold increase in binding strength causes a 5.8-fold increase in time to reach50%product fraction, but a 20-fold increase in time to reach90%product

fraction. Another way to look at the inhibition effect is to look at the trajectory for amount of𝐶_𝐸𝑆, which is proportional to the rate of production, as shown in orange in Figure3.2(b).

We see𝐶_𝐸𝑆decreases for all three trajectories as the product fraction increases. However, for weak binding, the inhibition does not become significant until product fraction is quite high, while the inhibition kicks in immediately for medium and strong binding. Below, we show that this nonlinearity of inhibition from product binding can be clearly understood by inspecting the full regulatory profile characterized by the reaction order polyhedron.

To understand how the production rate varies with the total concentrations as the catalysis process evolves, we need to characterize the space of regulation on the active complex 𝐶_𝐸𝑆governed by the two binding reactions. As discussed in the introduction (Eqn (3.3)), solving for 𝐶_𝐸𝑆 in terms of the totals here is solving a degree-3 polynomial equation.

This quickly becomes intractable to scan for all possible solutions since the polynomial degree increases with the number of binding reactions. Therefore, we instead focus on characterizing the reaction orders of𝐶_𝐸𝑆in the totals. Reaction orders correspond to log derivatives, capturing infinitesimal fold-change variation rather than additive difference in linear derivatives. For monomials, log derivatives yield the exponents, such as𝑓(𝑥) = 𝑘𝑥^𝑎 with ^𝜕log𝑓

𝜕log𝑥 =𝑎. Reaction orders therefore capture how the catalysis flux varies with total concentrations in fold-change, determining the flux magnitude up to a multiplicative constant.

In Section 3.3and3.4we formally define the binding networks studied, and develop a formula for reaction orders, allowing efficient computational sampling at scale. In (d) of Figure3.2, we show the sampling of the reaction order polyhedron of𝐶_𝐸𝑆 determined by the binding network of this system, enzymatic reaction with product binding. We see that the set of all possible reaction orders indeed form a polyhedral set. Furthermore, we see the points condense around edges and vertices, implying those are the reaction orders for most concentration values. This motivates the idea of structural regimes corresponding to the vertices. For a large range of concentrations, the reaction orders are kept at one vertex, therefore the regulatory behavior of the binding network governing catalysis fluxes is the same. When very large concentration changes happen, the system would quickly move from one structural regime to another, since the reaction orders would quickly go from one vertex to another. Therefore, we can consider the catalytic process as evolving in the reaction order polyhedron, with different stages of the process corresponding to different regulatory modes in different structural regimes.

To apply the reaction order polyhedron to clearly understand the regulation of the catalysis process, we also want an explicit correspondence between the structural regimes. We

Figure 3.3Trajectories in reaction order space of the three catalytic processes in (b) of Figure3.2. The background sampling of the reaction order polyhedron (blue dots) are the same as in (d) of Figure3.2. The trajectory of the strong enzyme-product binding strength case (most opaque in (b) of Figure3.2) is orange color, that of the medium binding strength case (medium opacity in (b) of Figure3.2) is in green color, and that of the weak binding strength (most transparent in (b) of Figure3.2) is in red. For each trajectory, the triangle end denotes initial point, and the end with a circle denotes end point.

can ask for which part of the reaction order polyhedron corresponds to which part of the concentration space via computer sampling. Indeed, the reaction order polyhedron can be considered as an empowering tool for the classical approach that numerically solves polynomial equations to study bioregulation. Instead of scanning through parameters in the numerical solutions for ad-hoc performance criteria, reaction order polyhedra serves as a structural intermediate between parameters and bioregulatory performance. All effects of parameter variations show up in reaction order polyhedra before influencing bioregulation, and conversely, desired behaviors are definable through reaction orders that can be mapped to parameters.

While visual inspection or further computation on the sampled points in reaction order polyhedra can yield fruitful analysis for bioregulation, we also want to directly tackle the vertices and structural regimes which define the polyhedra and the great majority of bioregulatory behavior. In Section3.8, we develop a analytical method called dominance decomposition tree (DDT) based on fundamental rules of calculus for positive variables that can directly obtain the vertices of reaction order polyhedra, with corresponding asymptotic conditions on the concentrations. In (c) of Figure3.2, we show the DDT for this example of enzymatic reaction with product binding. Each vertex corresponds to a dominance condition for the total concentrations, and a structural regime of regulatory behavior. This allows us to go back and force between at a given concentration, what is the structural regime, and for a desirable structural regime, what is the concentration to reach it. These dominance conditions also correspond to faces of the reaction order polyhedron in a direct way. Comparing (c) and (d) of Figure3.2, the orange vertices and dominance conditions correspond to the orange points in the reaction order polyhedron.

This is the classical Michaelis-Menten approximation, where substrate is assumed much higher than the enzyme and product binding is assumed negligible. Indeed, there are two structural regimes contained, the(1,1,0)regime that is linearly proportional to total substrate, and the(1,0,0)regime that has the enzymes saturated therefore independent of total substrate. Graphically, this corresponds to a line segment, an edge connecting two vertices, in the reaction order polyhedron. The Michaelis-Menten assumption that total substrate is much higher than the enzyme corresponds to the first branching in the DDT,𝑡_𝑆 ≈𝑆, that free substrate dominates total substrates. Relaxing this assumption to generalize Michaelis-Menten, with only the negligible product binding assumption, we obtain the green region with three structural regimes. These three vertices together form a triangle, with the new regime(0,1,0) corresponding to the case where enzymes are overabundant. Lastly, when making no assumption at all, we obtain the full reaction order polyhedron, with the two new regimes, vertex(1,1,−1)and ray(1,0,−1), corresponding to inhibition by product binding. Indeed, the reaction order in total production is the negative 3rd entry. Interestingly, this inhibition can be hyper-sensitive in the regime corresponding to the ray towards(1,0,−1), in the sense that the inhibition effect is more than linear. This happens under the dominance condition𝑡_𝑃 ≈𝐶_𝐸𝑃, when most enzymes and products are bound together.

With all this together, we can fully understand the various production dynamics as shown in (b) of Figure3.2. For weak enzyme-product binding strength, we expect the enzyme- product complex𝐶𝐸𝑃 to not dominate total enzymes, therefore mostly restricted to the green triangular region with the three vertices(1,0,0),(0,1,0)and(1,1,0). Because we

simulated the case with total substrate 10-fold more than total enzyme, the(0,1,0)vertex cannot be reached, with just(1,0,0)and(0,1,0)left. The system begins with abundant substrate, therefore close to the(1,0,0)vertex, then as the substrate gets converted into product, moves towards the(1,1,0)vertex. This means the flux would proceed mostly at a constant speed, and then when the substrate level finally becomes lower than the binding constant𝐾_𝐸𝑆, the inhibition due to low substrate starts to kick in, and the flux decreases proportional to the decrease in substrate concentration. Throughout, the process can be largely explained by the classical Michaelis-Menten approximation. Indeed, as shown in the red trajectory in Figure3.3, we see although the trajectory goes slightly negative in the order in𝑡𝑃, and goes slightly below the(1,1,0)vertex, the behavior is not far off the Michaelis-Menten edge.

As for the case of medium binding strength, where the enzyme has similar binding affinity to substrate and product molecules, we expect the enzyme to be mostly bound throughout, simply substituting the substrate with product molecule as the production progresses. In detail, we begin again with overabundant substrate, so we are at the(1,0,0)vertex where most enzymes are in the form of enzyme-substrate complex 𝐶_𝐸𝑆. As time progresses, substrate concentration decreases while product concentration increases, but the total of metabolites, substrate plus product, is the same. This means most enzymes are still bound, but instead of enzyme-substrate complex 𝐶_𝐸𝑆, more and more enzymes are bound in enzyme-product complex𝐶_𝐸𝑃. This means going from the𝑡_𝐸 ≈𝐶_𝐸𝑆dominance condition for the (1,0,0) regime, to the 𝑡_𝐸 ≈ 𝐶_𝐸𝑃 dominance condition for the (1,1,−1) regime.

Therefore, as the time goes on, we begin with a constant speed of production to decreasing speed like𝑡⁻¹_𝑃 and𝑡¹_𝑆. Indeed, this is what we observe in the green trajectory plotted in Figure3.3.

Lastly, we consider the strong binding case. When transitioning from substrate-saturating regime(1,0,0)to the product-binding regime(1,1,−1), because of strong enzyme-product binding, the inhibition by product binding requires much less product to be synthesized, therefore kicks in much earlier. In other words, enzyme-product complexes𝐶_𝐸𝑃 would occupy most product molecules formed, therefore causing𝐶_𝐸𝑃 to dominate total product 𝑡_𝑃, pushing the system in the regime of ray towards(1,0,−1). This causes hyper-sensitive inhibition of the catalysis flux, stronger than first order inhibition. As a result, we see when binding strength increases, the production rate does not decrease in first order, but faster than first order. Indeed, as shown in the orange trajectory in Figure3.3, we see although the same end-point is reached in this case as the medium binding strength case, the trajectory reaches into the hypersensitive regime of ray to(1,0,−1), causing much stronger product

inhibition.

In summary, we see how reaction order polyhedra can holistically capture the regulatory profile of binding’s regulation of catalysis fluxes. Hence, we can analyze bioregulatory dynamics in the space of reaction orders, with different regulatory behavior corresponding to shifting between structural regimes as vertices in the reaction order polyhedra. The computational sampling of reaction order polyhedra based on log derivative formula enables large-scale analysis of polyhedra, connecting concentrations and bioregulatory behaviors via reaction orders. Complementing this, the dominance decomposition tree (DDT) relate vertices, or structural regimes of bioregulation, with dominance conditions in concentration space, enabling intuitive understanding that maps dynamic regulations with trajectories through structural regimes.

Below, we begin our formulation and analysis of binding networks, detailed balanced steady states, and their parameterization via reaction orders.