Physical and microscopic basis of reaction order

Chapter II: Polyhedral constraints enable holistic analysis of bioregulation

2.4 Physical and microscopic basis of reaction order

indeed scenarios where the number of repressors expressed per plasmid is low, while the binding is tight.

All this together, we have shown via one example how the application of holistic bioregula- tion analysis based on reaction order polyhedra can be fruitful in biocircuit design. The power of holistic analysis fundamentally come from its capability to derive necessary and sufficient conditions, rather than just sufficient ones. This allows it to state all possible scenarios a desired function can be achieved. Although this capability is only used here to find a hidden regime, its power in bioengineering is much more profound. Debugging when a design is not performing as desired is part of the foundation for design-build-test cycles in engineering. With only sufficient conditions, debugging can only be done with trial and error. With necessary and sufficient condition for a function in a given circuit design, we can either find how to improve, or conclude that the current circuit design have to be modified.

transformations from external chemical potentials to internal chemical potentials. 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. Since chemical potentials measure the tendency to react for species, reaction orders translate external changes into their effect on internal species. In fact, since chemical potentials and Gibbs free energy are quantities more generally defined than log derivatives, e.g. for discrete molecule counts, this physical interpretation of reaction orders can be used as the definition to begin with, and log derivatives emerge as the approximate expression in bulk scale.

We first give a general description of how reaction orders and chemical potentials are related, and then give a concrete example of dimer-monomer mixture to show how the relation works in calculation.

The generic scenario we are considering is a binding network, or any system in reaction equilibrium, that exchanges molecules with external environments. We belabor here that the exchange with the environment is not in equilibrium in general. So in the language of statistical mechanics, this is a canonical ensemble rather than a grand canonical ensemble.

Although we allow exchange with the environment to change molecule numbers in the system, this change is slow compared to the system’s equilibration. This is akin to tuning temperature when obtaining a phase diagram in thermodynamics. For each point, or each experiment, the temperature is fixed, although the temperature is varied overall.

With this setting of a system that quickly equilibrates while exchanging molecules with external environments in mind, we can have two perspectives on the system. Internally, we see 𝑛 molecular species 𝑋₁, . . . , 𝑋_𝑛 with distinct chemical properties. So the state of the system can be described by the number of molecules of each species, 𝑁₁, . . . , 𝑁_𝑛. The Gibbs free energy is therefore 𝐺(𝑁₁, . . . , 𝑁_𝑛). Since our system quickly reaches equilibrium internally, when these numbers change, the Gibbs free energy satisfy𝑑𝐺= 𝜇1𝑑𝑁1+· · ·+𝜇𝑛𝑑𝑁𝑛, where𝜇𝑖is the chemical potential of𝑋𝑖. Note that although there are reactions happening inside the system, we do not need to describe them explicitly, since chemical reactions simply cause changes in the numbers𝑁₁, . . . , 𝑁_𝑛, hence accounted for in our above description.

Externally, the environment can only distinguish the molecules pushed in or pulled out, but does not have access to the fast reactions inside the system. Therefore, the external description of the system consists of only the exchangeable molecular species, which we denote𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡, so there are𝑑of them,𝑑≤𝑛. Since an exchangeable species can be in multiple forms inside the system through chemical reactions, these𝑁_𝑖^𝑡are total numbers of a species that accounts for different forms the species is in. For example, a species in

monomer or dimer form are not distinguished and summed together in the total.

Just knowing the totals cannot determine the state of the system, since reactions inside the system can change how a given species vary in its various forms. So, to relate the external variables(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡)to the internal variables(𝑁₁, . . . , 𝑁_𝑛), we need to find other degrees of freedom to extend(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡)so it can describe the state of the system. The degrees of freedom that cannot be tuned externally are the chemical reaction equilibria inside the system, captured by reaction equilibrium constants(𝐾₁, . . . , 𝐾𝑟), where𝑟=𝑛−𝑑. There might be more reactions, but there is always exactly𝑟reactions with linearly independent stoichiometry, so that they determine how the number of species is distributed in its various forms when the total is held fixed. Hence, the Gibbs free energy described from the external view is𝐺(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡, 𝐾₁, . . . , 𝐾_𝑟). Note that although we included𝐾₁, . . . , 𝐾_𝑟in the description, they cannot be modified externally, and only here to parameterize the states of the system. So the equilibrium relation of Gibbs free energy to external changes in these coordinates is𝑑𝐺=𝜇^𝑡₁𝑑𝑁₁^𝑡+· · ·+𝜇^𝑡_𝑑𝑑𝑁_𝑑^𝑡+𝜅₁𝑑𝐾₁+· · ·+𝜅_𝑟𝑑𝐾_𝑟, where𝜇^𝑡_𝑖 is the chemical potential of the𝑖th total species𝑁_𝑖^𝑡, and𝜅_𝑖 is the partial derivative of𝐺with respect to𝐾_𝑖. This nicely splits the part of Gibbs free energy that is adjustable externally, namely the 𝑑𝑁_𝑖^𝑡’s, and the part that is not adjustable internally, namely the𝑑𝐾𝑖’s determining internal reactions’ equilibrium.

Now we have two view of the Gibbs free energy, or more generally the state of the system, via internal variables(𝑁₁, . . . , 𝑁_𝑛)and external variables(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡, 𝐾₁, . . . , 𝐾_𝑟). How the Gibbs free energy responds to changes in these variables is also characterized differently: internally by chemical potentials(𝜇₁, . . . , 𝜇_𝑛), and externally by total chemical potentials (𝜇^𝑡₁, . . . , 𝜇^𝑡_𝑑) and (𝜅₁, . . . , 𝜅_𝑟). Since external exchanges happen through the external variables, while system properties are expressed in internal variables, we would like to map external changes to internal changes. We claim that reaction orders does exactly this, mapping Gibbs free energy changes from external adjustments to internal effects.

We derive this mathematically. Since chemical potentials are partial derivatives of Gibbs free energy to molecule numbers, we have

[︁𝜇₁ · · · 𝜇_𝑛^]︁= 𝜕𝐺

𝜕(𝑁₁,· · · , 𝑁_𝑛), ^[︁𝜇^𝑡₁ · · · 𝜇^𝑡_𝑑 𝜅₁ · · ·𝜅_𝑟^]︁= 𝜕𝐺

𝜕(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡, 𝐾₁, . . . , 𝐾_𝑟). We relate them by a coordinate change:

[︁𝜇^𝑡₁ · · · 𝜇^𝑡_𝑑 𝜅₁ · · ·𝜅_𝑟^]︁= 𝜕𝐺

𝜕(𝑁₁,· · · , 𝑁_𝑛)

𝜕(𝑁₁,· · · , 𝑁_𝑛)

𝜕(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡, 𝐾₁, . . . , 𝐾_𝑟)

=^[︁𝜇₁ · · · 𝜇𝑛

]︁ 𝜕(𝑁₁,· · · , 𝑁𝑛)

𝜕(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡, 𝐾₁, . . . , 𝐾_𝑟).

This gives a map between the chemical potentials. However, chemical potentials are per- molecule changes of Gibbs free energy. Therefore we should multiply chemical potentials with molecule numbers to yield free energy changes. This exactly gives log derivatives, or reaction orders.

[︁𝜇^𝑡₁𝑁₁^𝑡 · · · 𝜇^𝑡_𝑑𝑁_𝑑^𝑡 𝜅₁𝐾₁ · · ·𝜅_𝑟𝐾_𝑟^]︁=^[︁𝜇₁𝑁₁ · · · 𝜇_𝑛𝑁_𝑛^]︁ 𝜕log(𝑁₁,· · · , 𝑁_𝑛)

𝜕log(𝑁₁^𝑡, . . . , 𝑁_𝑑^𝑡, 𝐾₁, . . . , 𝐾_𝑟). (2.17) Here ^{𝜕log𝑁𝑖}

𝜕log𝑁_𝑗^𝑡 is the log derivatives, or reaction orders, of𝑋_𝑖 to the𝑗th total, in per-molecule units. This is the same as ^{𝜕log𝑥𝑖}

𝜕log𝑡𝑗

in concentration units, where𝑥_𝑖and𝑡_𝑗are concentrations of 𝑋_𝑖and𝑗th total, since the units are ignored in log derivatives, which captures fold changes.

Eqn (2.17) shows that reaction orders, defined as log derivatives, have the natural physical meaning as conversion between external and internal free energy changes. In fact, since free energies are well defined when molecule counts are discrete while log derivatives are not, Eqn (2.17) can be used as a physical definition of reaction orders, and log derivatives emerge as bulk-scale approximation of this definition.

Now we do the calculation in detail for a simple example to illustrate this. We first begin with a system consisting of two types of particles,𝑋₁and𝑋₂ (see (a) of Figure2.6). The state of the system is therefore the number of these particles, (𝑁₁, 𝑁₂). The Gibbs free energy of the system is described in these coordinates,𝐺(𝑁₁, 𝑁2). To study how the Gibbs free energy of the system would change, we consider that both types of particles can be added or removed externally. Once particles are added or removed, the system equilibrates quickly, so equilibrium relations hold. So we have𝑑𝐺=𝜇₁𝑑𝑁₁+𝜇₂𝑑𝑁₂, where𝜇₁ and𝜇₂ are the chemical potentials of the two types of particles. In this setting, all changes to the system are caused by external exchanges. When no particles are added or removed, the state of the system does not change either, so𝑑𝑁₁ =𝑑𝑁₂ = 0resulting in𝑑𝐺= 0in a trivial way.

Next, we allow some internal dynamics to happen through a chemical reaction (see (b) of Figure2.6). Namely, we have reaction2𝑋₁ ⇌𝑋₂, that two𝑋₁form a𝑋₂. This gives us the physical notion that𝑋₂ has two units of𝑋₁. To keep this notion separate from that 𝑋₁and𝑋₂ are two distinct types of particles, we denote𝑋as the unit particle, so𝑋₁is a monomer of𝑋, and𝑋₂ is a dimer of𝑋. Since the reaction can only happen within the system and quickly reaches equilibrium, the external distinction of𝑋₁ and𝑋₂becomes trivial, since they inter-convert inside the system. Addition and removal of𝑋₁ and 𝑋₂ molecules is then the same as just adding or removing 𝑋. Each addition of𝑋₂ can be considered as adding two𝑋, for example. In other words, externally𝑋₁and𝑋₂ are simply one and two𝑋 molecules respectively, and the difference in their chemical properties is no

Figure 2.6Illustration of the biophysical setting considered in connecting chemical potential with reaction orders. (a) A system consisting of two types of particles,𝑋1and𝑋2, in equilibrium within the system. The number of these two particles,𝑁1and𝑁2, can be added or removed via exchange with external environments.

Once the particle number changes, the system quickly re-equilibrates. So the Gibbs free energy of the system is described by𝐺(𝑁1, 𝑁2)and satisfy the equilibrium relation𝑑𝐺=𝜇1𝑑𝑁1+𝜇2𝑑𝑁2, where𝜇𝑖is the chemical potential of particle𝑋𝑖. (b) A system consisting of two types of particles𝑋1and𝑋2with internal chemical reaction2𝑋1⇌𝑋2that two𝑋1dimerize to form𝑋2. To distinguish free monomers𝑋1and the monomer molecules bound in𝑋2, we denote𝑋 as the generic monomer, so𝑋1is monomer of𝑋, and𝑋2is dimer of𝑋. The dimerization reaction only happens inside the system, not outside, so we color particles orange as reaction-capable, while grey is not reaction-capable. The exchange with external environment can add or remove monomers. But once a monomer is added or removed externally, the internal reaction quickly equilibrates, therefore causing a net increase or decrease of total monomers, or total𝑋, denoted𝑁_𝑡. The reaction equilibrium internal of the system is therefore𝑑𝐺= 0for fixed𝑁_𝑡, captured by equilibrium constant 𝐾for dimer dissociation. From an external point of view, where we can only add or remove𝑋to change𝑁_𝑡, but not modify the equilibrium internal to the system, we want𝐺(𝑁𝑡, 𝐾)expressed in terms of𝑁𝑡that can be externally modified, and𝐾that cannot be modified. So the equilibrium relation is𝑑𝐺=𝜇𝑡𝑑𝑁𝑡+𝜅𝑑𝐾, where𝜇𝑡is chemical potential for a generic𝑋, regardless of monomer or dimer form, and𝜅is how𝐺changes with𝐾.

longer distinguishable, due to the chemical reaction inside the system. Thus the addition or removal of𝑋externally correspond to changing the total number of𝑋inside the system, which we denote𝑁_𝑡=𝑁₁+ 2𝑁₂.

While𝑁𝑡can be adjusted externally, the reaction equilibrium inside the system can not.

Since the system equilibrates quickly, 𝑁_𝑡 is fixed when reaching towards equilibrium.

The condition is therefore 𝑑𝐺 =𝜇₁𝑑𝑁₁+𝜇₂𝑑𝑁₂ = 0subject to the constraint that𝑑𝑁_𝑡 = 𝑑𝑁₁+ 2𝑑𝑁₂ = 0. This results in the equilibrium condition2𝜇₁ =𝜇₂. So the system’s state variables(𝑁₁, 𝑁₂)are no longer completely free to vary, even if𝑁_𝑡can be adjusted. For each value of𝑁_𝑡,𝑁₁and𝑁₂ are uniquely determined by the equilibrium condition2𝜇₁ =𝜇₂. To explicitly represent this restriction, we need to relate 𝜇_𝑖 with 𝑁_𝑖. A simple relation for ideal or dilute solution is that𝜇𝑖 = 𝜇⁰_𝑖 +𝑘𝐵𝑇 log𝑥𝑖, where 𝑥𝑖 is the concentration of 𝑋𝑖, proportional to𝑁𝑖,𝜇⁰_𝑖 is a standard chemical potential for𝑋𝑖, capturing the internal energies of an 𝑋_𝑖 molecule, and 𝑘_𝐵 is Boltzmann’s constant. Non-ideal solutions will

have a more sophisticated relationship between chemical potentials and concentrations, which can be captured via linearization of𝜇_𝑖 inlog𝑥_𝑖at a certain concentration to obtain 𝜇_𝑖 = 𝜇⁰_𝑖 +𝑎(𝑥_𝑖)𝑘_𝐵𝑇log𝑥_𝑖. So the ideal behavior can be considered as the case when 𝑎(𝑥_𝑖)≡1. With this ideal solution formula, we find that the equilibrium condition becomes 2𝜇⁰₁+ 2𝑘_𝐵𝑇 log𝑥₁ =𝜇⁰₂+ 2𝑘_𝐵𝑇 log𝑥₂. This allows us to define the binding constant𝐾so that log𝐾 = log^𝑥_𝑥²¹

2 = _𝑘¹

𝐵𝑇(2𝜇⁰₁−𝜇⁰₂). So the relation𝐾𝑥₂ =𝑥²₁ is an explicit parameterization of the restriction on the numbers or concentrations of 𝑋₁ and 𝑋₂ from the reaction equilibrium. Note that𝐾𝑥₂ = 𝑥²₁ is the same as what we obtain from the steady state equation in mass-action kinetics of the binding reaction.

Our discussion above yields an alternative description for the state of the system. Given fixed𝑁_𝑡 and𝐾, the internal state variables(𝑁₁, 𝑁₂)are uniquely determined. In other words,(𝑁𝑡, 𝐾)is an alternative set of state variables. For example, Gibbs free energy can also be considered a state function in these state variables𝐺(𝑁_𝑡, 𝐾). This is the natural set of variables for the external view, since𝑁_𝑡is exactly what is adjustable externally, while 𝐾 describes the internal equilibrium not externally accessible. When we externally add or remove𝑋 molecules, therefore, we are changing the system by𝑑𝑁_𝑡. To relate this to Gibbs free energy changes, we have equilibrium relation𝑑𝐺=𝜇_𝑡𝑑𝑁_𝑡+𝜅𝑑𝐾, where𝜇_𝑡is the chemical potential of an𝑋molecule, and𝜅is the partial derivative ^𝜕𝐺

𝜕𝐾 while keeping 𝑁_𝑡constant. When𝐾 is not adjustable, as is the case when we are restricted to external exchanges, we have𝑑𝐺=𝜇_𝑡𝑑𝑁_𝑡.

In order to see how the external adjustment in terms of𝑑𝑁_𝑡causes changes in internal 𝜇𝑡= 𝜕𝐺

𝜕𝑁_𝑡 = 𝜕𝐺

𝜕(𝑁₁, 𝑁₂)

𝜕(𝑁₁, 𝑁₂)

𝜕𝑁_𝑡 =^[︁𝜇₁ 𝜇₂^]︁𝜕(𝑁₁, 𝑁₂)

𝜕𝑁_𝑡 ,

where the partial derivatives with respect to 𝑁_𝑡 keeps 𝐾 constant, since (𝑁_𝑡, 𝐾)is the alternative coordinate. To relate to free energy changes, we need to multiply chemical potentials by molecule numbers, so we have

𝜇_𝑡𝑁_𝑡 =^[︁𝜇₁𝑁₁ 𝜇₂𝑁₂^]︁𝜕log(𝑁₁, 𝑁₂)

𝜕log𝑁_𝑡 =𝜇₁𝑁₁𝜕log𝑁₁

𝜕log𝑁_𝑡 +𝜇₂𝑁₂𝜕log𝑁₂

𝜕log𝑁_𝑡. (2.18) So we see that log derivatives, or reaction orders, transforms changes in free energy𝜇_𝑡𝑁_𝑡 from external adjustments to corresponding changes internally, namely𝜇₁𝑁₁and𝜇₂𝑁₂. Since this example is simple, we can calculate this conversion explicitly to see how the transformation is done in detail. Write this in terms of concentration variables,𝑥_𝑖for𝑁_𝑖 and𝑡for𝑁_𝑡, we have

𝜇_𝑡𝑡 =𝜇₁𝑥₁𝜕log𝑥1

𝜕log𝑡 +𝜇₂𝑥₂𝜕log𝑥2

𝜕log𝑡 . (2.19)

We can calculate the reaction orders by brute force using the equilibrium relation𝐾𝑥₂ =𝑥²₁ and the definition of totals𝑡=𝑥₁+ 2𝑥₂.

𝜕log𝑥₁

𝜕log𝑡 = 𝜕log𝑥₁

𝜕log(𝑥₁ + 2𝑥₂) =

(︃𝜕log(𝑥₁+ 2𝑥₂)

𝜕log𝑥₁

)︃−1

=

(︃ 𝑥₁

𝑥₁+ 2𝑥₂ · 𝜕log𝑥₁

𝜕log𝑥₁ + 2𝑥₂

𝑥₁+ 2𝑥₂ · 𝜕log 2𝑥₂

𝜕log𝑥₁

)︃−1

=

(︂ 𝑥₁

𝑥₁+ 2𝑥₂ ·1 + 2𝑥₂ 𝑥₁+ 2𝑥₂ ·2

)︂−1

=𝑥₁+ 2𝑥₂ 𝑥₁+ 4𝑥₂. We can define𝜆₁ = _𝑥 ^𝑥1

1+4𝑥2

, and𝜆₂ = _𝑥^4𝑥2

1+4𝑥2

so that𝜆₁+𝜆₂ = 1. Then

𝜕log𝑥₁

𝜕log𝑡 =𝜆₁·1 +𝜆₂· 1

2, 𝜕log𝑥₂

𝜕log𝑡 = 2𝜕log𝑥₁

𝜕log𝑡 =𝜆₁·2 +𝜆₂·1.

So we have

𝜇_𝑡𝑡=

(︂

𝜆₁·1 +𝜆₂· 1 2

)︂

(𝜇₁𝑥₁+ 2𝜇₂𝑥₂).

The change in free energy from external adjustments is propagated to internal ones via a transformation factor𝜆₁·1 +𝜆₂·¹₂ between1and ¹

2. When𝑥₁ is dominant, it is close to1; when𝑥2 is dominant, it is close to ¹

2.

In summary, through our generic arguments and this simple example, we see that reaction orders describe how changes in Gibbs free energy from external addition and removal of molecules are mapped to free energy changes of internal components. Importantly, we consider scenarios where a system has internal reactions reaching equilibria that is not adjustable externally, therefore requiring an internal-external split. Hence, reaction orders is a fundamental tool when studying regulations of biomolecular systems where only parts of the species’ concentrations are adjustable, and there exists reactions internal to the system.

Part II

Part 2. Mathematical underpinnings

66

Chapter 3

Polyhedral Representation of Binding Net- work Steady States

In earlier chapters, we see that binding reaction networks regulate catalysis reactions. In this chapter, we show that the regulatory profiles of a binding network can be characterized as constrained in polyhedral sets in terms of reaction orders (log derivatives). We investigate the mathematical properties of log derivatives from binding networks. In particular, we make the following contributions: (1) we define what the set of binding networks is that makes biological sense; (2) we characterize the manifold of all possible detailed balanced steady states of a binding network; (3) we derive a formula for log derivatives, which can be used for computational sampling; and (4) we show that the polyhedral shape of log derivatives fundamentally comes from decomposition rules of log derivative operators.

This further yields a calculus method to analytically obtain log derivative polyhedra, either top-down via dominance-decomposition tree (DDT) or bottom-up via summation of matrix representations.