114 3.8 Graphical illustration of the dominance decomposition tree (DDT) procedure. applied to the bond network of just one bond reaction. a) The binding network of this enzyme allostery example. The vertices surrounded by orange correspond to the orange edge in (b). d) The case where the two substrate molecules bind to the enzyme in one step is considered, with the computational sampling of the reaction order polyhedron of 𝐸2 plotted.
Constraints and hard limits characterize properties of machines
This guarantees that hard limits and laws derived from systems theory actually govern any machine assembled to function in the prescribed manner. From the discussion in this section, we distinguished between constraints from component-level kernel structures and hard limits from machine-level system theories.
Bioactivity as catalysis, bioregulation as binding
Taken together, thermodynamics dictate that the long-term dynamics of the cell must converge toward equilibrium with the bath. In other words, the regulatory profile of the catalysis reactions corresponds to the functions𝐶(𝑡𝐸, 𝑡𝑆, 𝑡𝑃) and𝐶′(𝑡𝐸, 𝑡𝑆, 𝑡𝑃).
Constraints and hard limits from stoichiometry, thermodynamics, and bio-
The background sampling of the reaction order polyhedron (blue dots) is the same as in (d) of Figure 3.2. This can be written as𝑥𝛾𝑗𝑓 =𝑘𝑗, where 𝛾𝑗𝑓 =𝛽𝑗𝑓−𝛼𝑓𝑗 ∈Z𝑛 is the stoichiometry vector of the𝑗th forward (dissociation) reaction.
Polyhedral constraints enable holistic analysis of bioregulation
Reaction order captures binding’s regulation of catalysis
In particular, the bioregulatory profile of the flux is limited to the Michaelis–Menten case (1,1) to (1,0) line segment. Regarding reaction orders on 𝐶 in (𝑡𝑆, 𝑡𝐸), this means in addition to the two regimes of reaction orders(1,1)and(0,1), there is a third regime of order(1,0).
Reaction order polyhedra can be derived and computed at scale
This allows us to sample points in the reaction order polyhedra for each species of interest. For example, in the simple connective network, the order of reaction of 𝐶 to total substrate is 𝜕log𝐶. Then, taking the convex combination of all the resulting constant reaction orders gives us the reaction order polyhedron.
The generic computational complexity for deriving reaction-order polyhedra directly from DDT is therefore exponential with the number of species, 𝑂(𝑒𝑛).
Reaction order polyhedra reveal hidden adaptive regimes
Constraints on reaction orders (𝑎𝑅, 𝑎𝐺) of genes for repressor and plasmid number come from all three parts of the system: binding regulation, catalysis or production-degradation of the repressor, and the plasmid number invariance function. These together with the constraint from the desired plasmid number invariance function result in the constraints on the reaction orders (circled in red).(b) DDT for target species𝐺. The red line is the constraint that𝛼𝐺+𝛼𝑅= 0 from plasmid number invariance and catalysis steady state.
Plasmid number invariance at steady state ⇐⇒ 𝑎𝑅+𝑎𝐺= 0,(𝑎𝑅, 𝑎𝐺)∈ 𝒫bind. 2.16) Visually represented in (d) of Figure 2.5, this is the intersection of the line for 𝑎𝑅+𝑎𝐺 = 0(red) and polyhedron𝒫bind.
Physical and microscopic basis of reaction order
When external variations change the Gibbs free energy of the system, the reaction orders map this change to changes of internal components of the system. The state of the system can therefore be described by the number of molecules of each species, 𝑁1,. To study how the Gibbs free energy of the system would change, we consider that both types of particles can be externally added or removed.
So the Gibbs free energy of the system is described by 𝐺(𝑁1, 𝑁2) and satisfies the equilibrium relation 𝑑𝐺=𝜇1𝑑𝑁1+𝜇2𝑑𝑁2, where 𝜇𝑖 is the chemical potential of the particle 𝑋𝑖. b) A system consisting of two types of particles𝑋1and𝑋2with an internal chemical reaction2𝑋1⇌𝑋2which two𝑋1 dimerize to form𝑋2.
Introduction
This problem amounts to solving a system of polynomial equations, where the magnitude of the problem is greater than the number of binding reactions in general. The MM formula assumed that the substrate concentration is kept much higher than that of the enzyme. To get an idea of the magnitude of the error made, we consider only one binding reaction 𝐺+𝑅 𝑘.
In Section 3.4, we characterize the manifold of equilibrium or detailed equilibrium stable states of bond networks and introduce log derivatives as a transformation between different parameterizations of the manifold.
Illustrative example
In (d) of Figure 3.2 we show the sampling of the polyhedron of the reaction sequence of 𝐶𝐸𝑆 determined by the binding network of this system, enzymatic reaction with product binding. These dominance conditions also correspond in a direct way to faces of the reaction order polyhedron. Comparing (c) and (d) of Figure 3.2, the orange vertices and dominance conditions correspond to the orange points in the polyhedron of the reaction sequence.
Graphically, this corresponds to a line segment, an edge connecting two vertices, in the reaction order polyhedron.
Binding reaction networks
A matrix𝐿∈R×𝑛is a conservation law matrix of the CRN, where𝑑=𝑛−𝑟, if𝐿the full row rank is metrowspan𝐿= ker𝑁 =𝒮⊥. Since the binding network is reversible and the reactant and product vectors have different supports, we can capture all the binding network information in the stoichiometry matrix of the forward reactions only, which we denote Γ𝑓 ∈ R𝑛×𝑚2. To demonstrate this rigorously, we can take all choices of atom types and check that the produced matrix of the conservation laws cannot contain the 3 columns of the atom types.
Thus, we have the following characterization of isomeric atomic CRNs as a consequence of the theorem for stoichiometric atomic CRNs.
Detailed balance steady states of binding networks
What variables can these degrees of freedom represent for the detailed balance solution log? A natural choice is the conserved amounts. We will see this explicitly as alternate coordinate maps for the many detailed stable equilibrium states. The manifold is then defined by the constraint of these variables imposed by the detailed equilibrium state of a given binding network.
Then is the manifold of detailed equilibrium stable states of the binding network, also called the equilibrium manifold of the binding network. 3.12).
Log derivative as transform between two coordinate charts
So,𝑓 : log𝑥 ↦→ (log𝑡,log𝑘) is a diffeomorphism, and the log derivative 𝜕log𝑥. log𝑡,log𝑘) is well defined at all points inℳ. We note that the atomic log derivative formula in Eq (3.23) highlights an internal symmetric structure that is not clearly seen in the more general log derivative formula in Eq (3.19). Namely, we can first just calculate the log derivative of 𝑥𝑎, and then use it to obtain the log derivative of 𝑥𝑐.
If we are only interested in the log derivative of 𝑥 with respect to𝑡, we only need to calculate 𝜕log𝑥.
Polyhedral shape of log derivatives in one binding reaction
These calculations gave an explicit formula for the reaction order of one binding reaction (Eq (3.34)). First, we look at the overall form of all the possible values that reaction commands can take. Now that it is fixed and normalized, the reaction order of interest is the reaction order of the complex 𝐶 per common enzyme and common substrate 𝜕log𝐶.
Another option is to include the variable𝐾 in the reaction order to see where the vertices are.
Vertices of binding polyhedra as minimal representations
The minimal representations of 𝑥𝑗* in terms of all species then amounts to finding the vectors of minimal support in the affine subspace𝑒𝑗*+𝒮. If 𝑢 is not of minimal support in𝒮, then there exists non-zero vector𝑢′ ∈ 𝒮 with smaller support than𝑣. Since we consider vectors in a linear subspace, the correspondence between vectors of minimal sign and vectors of minimal support is straightforward.
A vector𝑣 in a linear subspace𝒱 ⊂R𝑛 is of minimal sign if it has minimal support.
Polyhedra from decomposition of log derivative operators
In the above case, we want to write a log-derivative with respect to𝑡𝑋 =𝑥1+ 2𝑥2 in a convex combination of log-derivative for 𝑥1 and log-derivative for𝑥2. In other words, the goal of log-derivative operator decomposition is to reveal the intrinsic polyhedral structure of reaction orders. More generally, Theorem 3.8.2 enables a procedure for obtaining the reaction order polyhedra through decomposition of log-derivative operators.
While the DDT procedure, based on Theorem 3.8.2, guarantees that the polyhedron obtained from log-derived decompositions always contains the set of reaction orders. a) The binding network of this enzyme allostery example. (b) Arithmetic sampling of the reaction order polyhedron of𝐸2.
Summary
Back to the glycolysis example, we can propose the following natural division of the fluxes: This is the description of the lower stoichiometry flux layer when the glycolysis pathway is considered as a layered architecture. Note that the chemical potential𝜇ext𝑖 is the same as the molar Gibbs free energy of the external metabolite species [27].
Controller refers to how the control variable𝑢 is determined based on the state variable𝑥 of the plant, which is the metabolite concentration 𝑥 in our case.
Flux exponent control in metabolism: biological regulation as control of
Introduction
Therefore, mechanisms can be used to build models that sufficiently demonstrate certain phenotypes, linking mechanisms to phenotypes in the forward direction. To do this, the scientific rules about these mechanisms can be summarized by formalizing the essential structures involved, often in a mathematical form. These scientific rules can then be seen as the waist of a clockwork that systematically converts knowledge about mechanisms at the top into models at the bottom.
On the other hand, FEC can be seen as a system-theoretic formalization that scales up the hard limit approach invented in [25] to constitute the regulation of any metabolic network.
Glycolysis as an illustrative example of flux exponent control
When the regulating mechanisms of the fluxes are known in detail, we can specify exactly how 𝑣1 and 𝑣2 are regulated, as static functions or dynamic processes depending on (𝑥1, 𝑥2). Since we do not know how any of the fluxes are regulated, the flow control considers all fluxes to be arbitrarily regulated by the cell. 4.5) The above equation writes the flux decomposition of the reaction 𝑗 in several different notations.
For example, [25] proposed the following simple flux models based on the knowledge of catalyzing allosteric feedback of enzymes:.
Metabolic regulation as control of flux exponents
Then the entropy production of the external environment due to the input and output of external metabolites by the system is −(𝑠ext)⊺(𝑆ext𝑣+𝑆ext,𝑤𝑤). While equation (4.22) can derive a value for the entropy production rate𝜎 based on the free energy dissipation, quantitatively relating𝜎 to the actual rate of the reaction streams is difficult. Therefore, for a non-negligible reaction rate, equation (4.23) relates the free energy of the reaction to its direction.
Regarding the restriction set𝒰ss, the irreversibility of the𝑗th reaction restricts the flux𝑢𝑗 to be non-negative,𝑢𝑗 ≥0.
Tools, hard limits, and laws from control theory
Explore behaviors of interest via optimization
Case studies of computational exploration
Summary and future directions
Dynamics and control of production and degradation via reaction orders 216
Dissipativity in positive scalar birth death systems
Multiplicative networks of scalar birth-death systems
Summary and future directions
Appendix. Proof for interconnection of two component systems
Appendix. Background on storage functions and dissipativity with fixed
