Polyhedral shape of log derivatives in one binding reaction

Chapter II: Polyhedral constraints enable holistic analysis of bioregulation

3.6 Polyhedral shape of log derivatives in one binding reaction

So we know the atomic chart with matrix𝐴=𝐴^𝑎=^[︁I_𝑑 0^]︁and the total chart with𝐴=𝐿 both resides in the same connected component of𝒜. Exactly which one depends on the choice of which direction is used for the stoichiometry vectors. Thus, without loss of generality, we assume𝐴^𝑎,𝐿∈ 𝒜⁺(𝑁).

To find other alternative chart, we note that given a𝐴∈ 𝒜⁺(𝑁), then for any𝑆 ∈GL⁺_𝑑(R) such that𝑆𝐴∈R^𝑑×𝑛_≥0 , then𝑆𝐴∈ 𝒜⁺(𝑁). HereGL⁺_𝑑(R)is the identity component of the general linear group of𝑑×𝑑matrices, i.e. 𝑑×𝑑matrices with positive determinant. This states that any invertible matrix with positive determinant can be left-multiplied to𝐴, and if the resulting matrix is non-negative, then it is in𝒜⁺(𝑁). There are several elementary matrices of GL⁺_𝑑(R)worth noting. It includes positive scaling Λ_𝛼, where 𝛼 ∈ R^𝑑_>0 is a positive vector inR^𝑑. It also includes permutations with positive sign, i.e. consisting of an even number of transpositions. It also includes row additionsI+𝑎𝐸_𝑖𝑗, where𝑖 ̸=𝑗, 𝑖, 𝑗 ∈ {1, . . . , 𝑑}, 𝐸_𝑖𝑗 has 1at (𝑖, 𝑗)entry and zero everywhere else, and 𝑎 ∈ R is a real number. I+𝑎𝐸_𝑖𝑗 takes the𝑗th row of𝐴, multiply by𝑎, and adds to the𝑖th row of𝐴. We caution that combinations of these elementary operations may go out ofGL⁺_𝑑(R), and the resulting matrix may no longer be non-negative.

As for right multiplication, given𝐴∈ 𝒜⁺(𝑁), then𝐴Λ_𝛼 ∈ 𝒜⁺(𝑁)for any positive vector 𝛼 ∈ R^𝑛_>0. This is because this multiplication is just a scaling of variables 𝑥, without changing its domainR^𝑛_>0.

These operations give a basic approach to explore other alternative charts ofℳ. It is an interesting question for further research to characterize the set𝒜⁺(𝑁).

3.6 Polyhedral shape of log derivatives in one binding reac-

particular log derivative above, ^𝜕^log^𝑥^𝑖

𝜕(log𝑡,log𝑘), if 𝑥_𝑖 is the active catalytic species, hence the name reaction order. Reaction order can also be equivalently interpreted as infinitesimal fold change, or the exponents of a local expression of𝑥_𝑖in terms of𝑡and𝑘.

This does not answer the question of why reaction orders are used to study bioregulation though. For example, why the linear direction ^𝜕𝑥^𝑖

𝜕(𝑡,𝑘) is not used? We demonstrate several useful properties of reaction orders in this section by examining the reaction orders of a single binding reaction in close detail. The key properties we demonstrate are summarized here. (1) Constrained by binding network stoichiometry, reaction orders are bounded within some polyhedral set, with vertices correspond to biologically meaningful regimes.

This is in contrast to linear derivatives, which are unbounded in most directions. (2) The polyhedral sets has hierarchical structures corresponding to robustness to variations, so that points concentrate to vertices, edges, and faces under asymptotic limits. This means reaction orders are robust to variations in rates and concentrations, so it can be controlled to precise values using noisy and uncertain actuations, and robust behaviors can be built on top of it.

Explicit reaction order calculation for one binding reaction

We do explicit calculation for one binding reaction to reveal the geometric shape of log derivatives. For transparency, we calculate the log derivatives directly without using any of the formula developed in previous section, although they produce the same result.

A binding network consisting of just one binding reaction can be written as follows:

𝐸+𝑆 ^𝑘

−⇀+

↽−

𝑘⁻

𝐶, (3.25)

where component species enzyme𝐸and substrate𝑆bind to form complex𝐶, and𝑘⁺and 𝑘⁻ are forward and backward reaction rate constants. Although we used enzyme and substrate to denote the two component species out of tradition, we do not assume they have any special properties, and by symmetry𝐸and𝑆 are equivalent by re-labeling.

This binding reaction has the following deterministic dynamics from the law of mass-action:

𝑑

𝑑𝑡𝐶(𝑡) = 𝑘⁺𝐸(𝑡)𝑆(𝑡)−𝑘⁻𝐶(𝑡), (3.26) where by slight abuse of notation, the symbol for a species is also used to denote the concentration of that species. This system has steady-state equation

𝐸𝑆 =𝐾𝐶, (3.27)

where𝐾 := ^𝑘_𝑘⁻+ is the equilibrium constant in the dissociation direction. As guaranteed, this corresponds to 𝑁log𝑥 = log𝑘, where 𝑁 = ^[︁−1 −1 1^]︁, 𝑥 = (𝐸, 𝑆, 𝐶), and 𝑘 = 𝐾. Note that in this particular case, the steady state necessarily satisfy detailed balance, so we do not need to begin with detailed balance condition to defineℳ, but instead deriveℳ from the specified rates.

The conserved quantities of this binding reaction are the total enzyme and total substrate:

𝑡_𝑆 :=𝑆+𝐶, 𝑡_𝐸 :=𝐸+𝐶.

(3.28) This can be written as 𝑡 = 𝐿𝑥 with the conservation matrix below. Here the species ordering is(𝐸, 𝑆, 𝐶), and the total ordering is(𝑡_𝐸, 𝑡_𝑆). We also attach the stoichiometry matrix here for clear comparison.

⎡

⎣

𝐿 𝑁

⎤

⎦=

⎡

⎢

⎣

1 0 1 0 1 1 1 1 −1

⎤

⎥

⎦

(3.29)

This in turn defines the equilibrium manifold:

ℳ=^{︁(𝑥,𝑡,𝑘)∈R⁶_>0 :𝑁 log𝑥= log𝑘,𝑡 =𝐿𝑥^}︁

=^{︁(𝐸, 𝑆, 𝐶, 𝑡_𝐸, 𝑡_𝑆, 𝐾)∈R⁶_>0 :𝐸𝑆 =𝐾𝐶, 𝑡_𝑆 =𝑆+𝐶, 𝑡_𝐸 =𝐸+𝐶^}︁.

(3.30) In this case, we can explicitly solve for 𝑥 = (𝐸, 𝑆, 𝐶) expressed in (𝑡,𝑘) = (𝑡_𝐸, 𝑡_𝑆, 𝐾). Namely,

2𝐶 =𝑡_𝐸+𝑡_𝑆+𝐾−^√︁(𝑡_𝐸 +𝑡_𝑆+𝐾)²−4𝑡_𝐸𝑡_𝑆, (3.31) which is then easily used to derive expressions for𝐸 and𝑆. This formula can be used as the exact solution for us to compare with in this case. But for the derivative to follow the workflow in the general case as described in Section3.4, we do not rely on this formula.

Instead, we use the implicit function theorem as we did in the general case to calculate the reaction order, i.e. the log derivative ^𝜕^{log(𝐸,𝑆,𝐶)}

𝜕log(𝑡𝐸,𝑡𝑆,𝐾). Define 𝐹 :R⁶_>0 →R³ whose zero set is the equilibrium manifoldℳ:

𝐹(𝐸, 𝑆, 𝐶, 𝑡_𝐸, 𝑡_𝑆, 𝐾) =

⎡

⎣

𝐹1(𝐸, 𝑆, 𝐶, 𝑡_𝐸, 𝑡𝑆) 𝐹₂(𝐸, 𝑆, 𝐶, 𝐾)

⎤

⎦=

⎡

⎢

⎣

𝐸+𝐶−𝑡_𝐸 𝑆+𝐶−𝑡_𝑆 𝐸𝑆−𝐶𝐾

⎤

⎥

⎦

, (3.32)

By implicit function theorem,

𝜕(𝐸, 𝑆, 𝐶)

𝜕(𝑡_𝐸, 𝑡_𝑆, 𝐾) =−

⎡

⎢

⎣

1 0 1 0 1 1 𝑆 𝐸 −𝐾

⎤

⎥

⎦

−1⎡

⎢

⎣

−1 0 0 0 −1 0 0 0 −𝐶

⎤

⎥

⎦

= 1

𝐸+𝑆+𝐾

⎡

⎢

⎣

𝐸+𝐾 −𝐸 𝐶

−𝑆 𝑆+𝐾 𝐶

𝑆 𝐸 −𝐶

⎤

⎥

⎦

Since𝐶 = ^𝐸𝑆_𝐾 , we can express the above in dimensionless quantities𝑒:= _𝐾^𝐸 and𝑠= _𝐾^𝑆.

𝜕(𝐸, 𝑆, 𝐶)

𝜕(𝑡_𝐸, 𝑡_𝑆, 𝐾) = 1 1 +𝑒+𝑠

⎡

⎢

⎣

1 +𝑒 −𝑒 𝑒𝑠

−𝑠 1 +𝑠 𝑒𝑠

𝑠 𝑒 −𝑒𝑠

⎤

⎥

⎦

. (3.33)

To calculate log derivative, we note that ^𝜕^log^𝑔(𝑥)

𝜕log𝑥 = Λ⁻¹_𝑔 ^{𝜕𝑔(𝑥)}_𝜕𝑥 Λ_𝑥, whereΛ_𝑣 = diag(𝑣), since

𝜕log𝑔𝑖(𝑥)

𝜕log𝑥𝑗 = ^𝜕𝑔_𝜕𝑥^𝑖^(𝑥)

𝑗

𝑥𝑗

𝑔𝑖(𝑥). Therefore,

𝜕log(𝐸, 𝑆, 𝐶)

𝜕log(𝑡_𝐸, 𝑡_𝑆, 𝐾) = 1 𝐸+𝑆+𝐾

⎡

⎢

⎣

𝐸 0 0 0 𝑆 0 0 0 𝐶

⎤

⎥

⎦

−1⎡

⎢

⎣

𝐸+𝐾 −𝐸 𝐶

−𝑆 𝑆+𝐾 𝐶

𝑆 𝐸 −𝐶

⎤

⎥

⎦

⎡

⎢

⎣

𝑡_𝐸 0 0 0 𝑡_𝑆 0 0 0 𝐾

⎤

⎥

⎦

= 1

𝐸+𝑆+𝐾

⎡

⎢

⎣

(1 + _𝐾^𝑆)(𝐸+𝐾) −𝑡_𝑆 𝑆

−𝑡_𝐸 (1 + ^𝐸_𝐾)(𝑆+𝐾) 𝐸

𝑆+𝐾 𝐸+𝐾 −𝐾

⎤

⎥

⎦

Expressing this in terms of𝑒and𝑠yields the following result.

Theorem 3.6.1(Reaction orders of a simple binding reaction). The reaction orders of a simple binding reaction𝐸+𝑆 ^𝑘

−⇀+

↽−

𝑘⁻

𝐶 can be expressed as

𝜕log(𝐸, 𝑆, 𝐶)

𝜕log(𝑡_𝐸, 𝑡_𝑆, 𝐾) = 1 1 +𝑒+𝑠

⎡

⎢

⎣

(1 +𝑒)(1 +𝑠) −𝑠(1 +𝑒) 𝑠

−𝑒(1 +𝑠) (1 +𝑠)(1 +𝑒) 𝑒

1 +𝑠 1 +𝑒 −1

⎤

⎥

⎦

, (3.34)

where 𝑒 = _𝐾^𝐸, 𝑠 = _𝐾^𝑆, and 𝐾 := ^𝑘_𝑘⁻+ the equilibrium constant of this binding reaction in the dissociation direction.

Below we re-do the calculation for reaction order using the formula in Eq (3.19) to show that indeed they yield the same result.

Example 5(Simple binding reaction.). 𝐸+𝑆 ⇌𝐶with binding constant𝐾. Let variables be 𝑥= (𝐸, 𝑆, 𝐶),𝑡 = (𝑡_𝐸, 𝑡_𝑆)and𝑘=𝐾. The corresponding stoichiometry matrix (note we use the forward or dissociation direction), conservation laws, and part of the matrix to be inverted are

𝑁 =^[︁1 1 −1^]︁, 𝐿=

⎡

⎣

1 0 1 0 1 1

⎤

⎦,Λ⁻¹_𝑡 𝐿Λ_𝑥=

⎡

⎣

𝐸

𝑡𝐸 0 ^𝐶_𝑡^𝐸𝑆

𝐸

0 _𝑡^𝑆

𝑆

𝐶𝐸𝑆

𝑡𝑆

⎤

⎦

𝜕log(𝐸, 𝑆, 𝐶)

𝜕log(𝑡_𝐸, 𝑡_𝑆, 𝐾) =

⎡

⎢

⎣

𝐸 𝑡𝐸 0 _𝑡^𝐶

𝐸

0 _𝑡^𝑆

𝑆

𝐶 𝑡𝑆

1 1 −1

⎤

⎥

⎦

−1

(︂𝐶𝐸+𝐶𝑆+𝐸𝑆 𝑡_𝐸𝑡_𝑆

)︂⁻¹

⎡

⎢

⎣

1 −_𝑡^𝐶

𝐸

𝐶𝑆 𝑡𝐸𝑡𝑆

−_𝑡^𝐶

𝑆 1 _𝑡^𝐶𝐸

𝐸𝑡𝑆

𝑆 𝑡𝑆

𝐸

𝑡𝐸 −_𝑡^𝐸𝑆

𝐸𝑡𝑆

⎤

⎥

⎦

Now we express the above in terms of(𝑒, 𝑠) := (_𝐾^𝐸,_𝐾^𝑆), we get the same result as Eq (3.34).

We can also utilize the stoichiometry-atomic property of this network, with atomic species 𝑥^𝑎= (𝐸, 𝑆). This splits the stoichiometry and the conservation law matrix into atomic and non-atomic parts:

𝑁 =^[︁𝑁₁ 𝑁₂^]︁=^[︁1 1 −1^]︁, 𝐿=^[︁I₂ 𝐿₂^]︁=

⎡

⎣

1 0 1 0 1 1

⎤

⎦, and the core symmetric structure is

(𝐿Λ_𝑥𝐿^⊺)⁻¹ =

⎡

⎣

𝐸+𝐶 𝐶 𝐶 𝑆+𝐶

⎤

⎦

−1

= 1

𝐸𝑆+𝐶𝑆+𝑆𝐶

⎡

⎣

𝑆+𝐶 −𝐶

−𝐶 𝐸+𝐶

⎤

⎦= 1

1 +𝑒+𝑠

⎡

⎣

1 + ¹_𝑒 −1

−1 1 + ¹_𝑠

⎤

⎦.

So𝑁₂⁻¹ =−1,−𝐿₂Λ_𝑥^𝑐𝑁₂⁻¹ =

⎡

⎣

𝐶 𝐶

⎤

⎦. From this we can calculate

𝜕log(𝐸, 𝑆)

𝜕log(𝑡_𝐸, 𝑡_𝑆) = (𝐿Λ_𝑥𝐿^⊺)⁻¹Λ_𝑡= 1 1 +𝑒+𝑠

⎡

⎣

(1 +𝑒)(1 +𝑠) −𝑠(1 +𝑒)

−𝑒(1 +𝑠) (1 +𝑒)(1 +𝑠)

⎤

⎦,

𝜕log(𝐸, 𝑆)

𝜕log𝐾 =−(𝐿Λ_𝑥𝐿^⊺)⁻¹𝐿₂Λ_𝑥^𝑐𝑁₂⁻¹ = 1 1 +𝑒+𝑠

⎡

⎣

𝑠 𝑒

⎤

⎦,

𝜕log𝐶

𝜕log(𝑡_𝐸, 𝑡𝑆, 𝐾) =𝐿^⊺₂ 𝜕log(𝐸, 𝑆)

𝜕log(𝑡_𝐸, 𝑡𝑆, 𝐾)+^[︁0 𝑁₂⁻¹^]︁= 1 1 +𝑒+𝑠

[︁1 +𝑠 1 +𝑒 −1^]︁.

△

These calculations have yielded an explicit formula for the reaction orders of one binding reaction (Eq (3.34)). We derived this formula in both ways, one by explicitly step-by-step calculate through the definition of manifold and implicit function theorem, as an illustration of the abstract derivations in Section3.4, and another by directly applying the resulting log

derivative formula in Eq (3.19) from our general derivations. Below, we use these formula to investigate what properties the reaction orders have, and how are these properties related to biological behaviors.

Polyhedral shape of reaction order and its biological implications

We use the formula Eq (3.34) to study the properties of reaction orders and their biological implications.

We first look at the overall shape of all possible values that the reaction orders can take.

In order to visualize in 2D, we can consider the equilibrium constant 𝐾 as fixed and normalize all the variables so that 𝐾 is the unit of concentration. This encodes our biological assumption that we focus on changes to reaction orders caused by varying total concentrations𝑡_𝐸, 𝑡_𝑆, rather than varying the equilibrium constant𝐾. This makes sense when the biomolecules making up the binding network are already fixed, and we are studying how to regulate it. If instead we are studying what biomolecules to use to achieve a certain behavior in a binding network, then we should allow𝐾 to vary and be an explicit variable in the reaction orders. For this particular example, the value that𝐾 is fixed at does not matter because Eq (3.34) shows that the value of the reaction orders can take only depends on(𝑒, 𝑠), the ratios of𝐸 and𝑆 to𝐾, so the range of values that the reaction orders can take is independent of what value𝐾 is fixed at. Now with𝐾 held fixed and normalized away, the reaction orders of interest are reaction order of the complex𝐶to total enzyme and total substrate ^𝜕^log^𝐶

𝜕log(𝑡𝐸,𝑡𝑆), and reaction order of free enzyme to total enzyme and total substrate ^𝜕^log^𝐸

𝜕log(𝑡𝐸,𝑡𝑆). Note that 𝑆 and 𝐸 are symmetric since we have just one binding reaction. Which species’ reaction order is of interest depends on which species is catalytically active. For example, if𝐶 is the active complex for downstream catalysis, such as a gene bound with an activating transcription factor or the activated form of an enzyme or receptor, then reaction orders of𝐶corresponds to response of biological activity. On the other hand, if𝑆 is a repressive ligand or repressing transcription factor, then the free form 𝐸 is catalytically active, so the reaction order of𝐸corresponds to biological activity.

We computationally sample the values of reaction orders in Eq (3.34) to obtain Figure 3.5. Several features arise from visual inspection. First, it appears that these sets take a polyhedral shape. Second, the polyhedral shape suggests it can be formed as combinations of simpler shapes, e.g. vertices and edges. Third, the points concentrate at the edges and vertices, suggesting robustness at those locations. Lastly, the reaction order of𝐸 is unbounded, going towards infinity in the(−1,1)direction, suggesting hypersensitivity.

We discuss these features in detail below.

(a) (b)

Figure 3.5Sampling of the log derivative of𝐶(subfigure(a)) and𝐸(subfigure(b)) with respect to𝑡_𝐸and𝑡_𝑆 for the binding network with just one binding reaction𝐸+𝑆⇌𝐶. Sampling is via the variables𝑒= _𝐾^𝐸 and 𝑠= _𝐾^𝑆 taking values between10⁻⁶to10⁶, uniformly sampled on the log scale.

Polyhedral shape. The first thing we notice is that the overall shape for both𝐶 and𝐸’s reaction orders arepolyhedral. This implies that the full range of reaction orders can be captured as the convex combination of several extreme cases. This can be seen in an explicit and analytical way as follows. Take𝐶’s reaction orders for example. We see from the sampled shape that it is the convex combination of vertices (1,0), (0,1), and(1,1). This means any point in the set can be expressed as 𝜆₁(1,0) +𝜆₂(0,1) +𝜆₃(1,1)with 𝜆_𝑖 non-negative and sum to one^∑︀³

𝑖=1𝜆_𝑖 = 1. We can then compare this expression with the formula in Eq (3.34) to have the following equation. Note that to be consistent with the figure, we have taken the order for the totals to be(𝑡_𝑆, 𝑡_𝐸)instead of(𝑡_𝐸, 𝑡_𝑆).

𝜕log𝐶

𝜕log(𝑡_𝑆, 𝑡𝐸) =^[︁_1+𝑒+𝑠^1+𝑒 _1+𝑒+𝑠^1+𝑠 ^]︁=^[︁𝜆₁ +𝜆₃ 𝜆₂+𝜆₃^]︁. (3.35) This together with that𝜆_𝑖’s sum to1can be used to formulate the following linear system of equations for𝜆_𝑖’s:

⎡

⎢

⎣

1 1 1 1 0 1 0 1 1

⎤

⎥

⎦

⎡

⎢

⎣

𝜆₁ 𝜆₂ 𝜆₃

⎤

⎥

⎦

⎡

⎢

⎣

𝜕log𝐶

𝜕log𝑡𝑆

𝜕log𝐶

𝜕log𝑡𝐸

⎤

⎥

⎦

⎡

⎢

⎣

1+𝑒 1+𝑒+𝑠

1+𝑠 1+𝑒+𝑠

⎤

⎥

⎦

. (3.36)

Solving this linear system of equations yields

⎡

⎢

⎣

𝜆₁ 𝜆₂ 𝜆3

⎤

⎥

⎦

⎡

⎢

⎣

1 0 −1 1 −1 0

−1 1 1

⎤

⎥

⎦

⎡

⎢

⎣

1+𝑒 1+𝑒+𝑠

1+𝑠 1+𝑒+𝑠

⎤

⎥

⎦

= 1

1 +𝑒+𝑠

⎡

⎢

⎣

𝑒 𝑠 1

⎤

⎥

⎦

. (3.37)

This tells us that we can write the reaction orders of 𝐶 in an explicit way as convex combinations of a set of vertices:

𝜕log𝐶

𝜕log(𝑡_𝑆, 𝑡_𝐸) = 𝑒

1 +𝑒+𝑠(1,0) + 𝑠

1 +𝑒+𝑠(0,1) + 1

1 +𝑒+𝑠(1,1). (3.38) As a quick note, although log derivatives are often used to compute “weighted average of exponents” for polynomials in statistical mechanics, and the log derivatives form a convex polytope as well, the polytope for𝐶 above is not obtainable from a polynomial.

For a polynomial𝑓(𝑥)in variables taking positive values𝑥∈R^𝑛_>0, the log derivatives are contained in its Newton polytope, which is the polytope formed as a convex combination of the exponents for each monomial term. For example, if𝑓(𝑥) = 1 +𝑥₁+𝑥₂, then the log derivative ^𝜕^log^𝑓

𝜕log(𝑥1,𝑥2) is contained in polytope with vertices(0,0),(1,0), and(0,1), and the convex coefficients are ¹

1+𝑥1+𝑥2

, ^𝑥¹

1+𝑥1+𝑥2

, and ^𝑥²

1+𝑥1+𝑥2

respectively. We see that although these coefficients are the same as the coefficients in Eq (3.38), one of the vertices are different.

If we want a polynomial with the(1,1)vertex and similar coefficients, then the polynomial can be 𝑓(𝑥) = 𝑥₁ +𝑥₂ +𝑥₁𝑥₂. But then the coefficient for the (1,1) vertex is ^𝑥¹^𝑥²

𝑥1+𝑥2+𝑥1𝑥2

, a different form compared to that in Eq (3.38). So we see polynomials cannot yield the convex combination in Eq (3.38). Either the vertices are different, or the convex coefficients (i.e. when the vertices are achieved) are different. Indeed, we know Eq (3.38) comes from taking the log derivative of the explicit expression in Eq (3.31) which involves square roots.

More generally, reaction order polyhedra come from log derivative of possibly non-analytic expressions that are roots of systems of polynomial equations.

Employing a similar approach we can parameterize a point in the reaction order polyhedron for𝐸as𝜆₁(0,1) +𝜆₂(−1,1) +𝜏(−1,1), with𝜆₁, 𝜆₂, 𝜏 ≥0and𝜆₁ +𝜆₂ = 1. This expression uses our observation from Figure3.5bthat the polyhedron is defined by two vertices at (0,1)and(−1,1), and a ray in direction(−1,1). The equation relating these coefficients to

the reaction order expression in Eq (3.34) is

⎡

⎢

⎣

1 1 0 0 −1 −1 1 1 1

⎤

⎥

⎦

⎡

⎢

⎣

𝜆₁ 𝜆₂ 𝜏

⎤

⎥

⎦

⎡

⎢

⎣

𝜕log𝐸

𝜕log𝑡𝑆

𝜕log𝐸

𝜕log𝑡𝐸

⎤

⎥

⎦

⎡

⎢

⎣

−𝑠(1+𝑒) 1+𝑒+𝑠 (1+𝑒)(1+𝑠)

1+𝑒+𝑠

⎤

⎥

⎦

(3.39)

Solve this system of linear equations yield

⎡

⎢

⎣

𝜆₁ 𝜆₂ 𝜏

⎤

⎥

⎦

⎡

⎢

⎣

0 1 1 1 −1 −1

−1 0 1

⎤

⎥

⎦

⎡

⎢

⎣

−𝑠(1+𝑒) 1+𝑒+𝑠 (1+𝑒)(1+𝑠)

1+𝑒+𝑠

⎤

⎥

⎦

= 1

1 +𝑒+𝑠

⎡

⎢

⎣

1 +𝑒 𝑠 𝑒𝑠

⎤

⎥

⎦

. (3.40)

So we can explicitly write

𝜕log𝐸

𝜕log(𝑡𝑆, 𝑡𝐸) = 1 +𝑒

1 +𝑒+𝑠(0,1) + 𝑠

1 +𝑒+𝑠(−1,1) + 𝑒𝑠

1 +𝑒+𝑠(−1,1). (3.41) We can also use the above results to write the reaction orders of all species in the form a polyhedron. Define𝜆_𝑒 = _1+𝑒+𝑠^𝑒 ,𝜆_𝑠 = _1+𝑒+𝑠^𝑠 ,𝜆₁ = _1+𝑒+𝑠¹ , and𝜏 = _1+𝑒+𝑠^𝑒𝑠 . Then

𝜕log(𝑆, 𝐸, 𝐶)

𝜕log(𝑡_𝑆, 𝑡_𝐸, 𝐾) =𝜆_𝑒

⎡

⎢

⎣

1 −1 1 0 1 0 1 0 0

⎤

⎥

⎦

+𝜆_𝑠

⎡

⎢

⎣

1 0 0

−1 1 1 0 1 0

⎤

⎥

⎦

+𝜆₁

⎡

⎢

⎣

1 0 0 0 1 0 1 1 −1

⎤

⎥

⎦

+𝜏

⎡

⎢

⎣

1 −1 0

−1 1 0 0 0 0

⎤

⎥

⎦

. (3.42)

From the above, we see both through computational sampling and analytical derivations that indeed the range of values that reaction orders can take form a polyhedral set. The mathematical reason for this polyhedral shape is studied in Section3.8. We investigate the biological implications of the polyhedral shape below.

Vertices and edges as asymptotic approximations. Roughly speaking, a polyhedral set is the set formed by the convex combination of vertices (with vertices generalized to include rays as well). This suggests we can consider the general behavior of a catalysis reaction, i.e.

the reaction order of the active species in a binding network, as the “convex combination”

of the behaviors at the vertices of reaction order polyhedron. If the behavior is simple at the vertices, then this gives us a way to describe the complicated general behavior through simple extreme-case behavior at the vertices.

We take the𝐶 polyhedron to illustrate this. From Eq (3.38), we see each vertex is achieved at a certain extreme of the(𝑒, 𝑠)variables. Namely, (1,0)is achieved when𝑒 ≫1, 𝑠(𝑒is much larger than1and 𝑠),(0,1)is achieved when𝑠 ≫ 1, 𝑒 and(1,1)is achieved when 1≫𝑒, 𝑠. We can relate these vertices’ reaction order to the approximate expressions of𝐶 in(𝑡_𝑆, 𝑡_𝐸, 𝐾)at these vertices by applying these asymptotic conditions to the equations defining the equilibrium manifold in Eq (3.30), or to the explicit solution in Eq (3.31).

Alternatively, we can include the𝐾 variable in the reaction orders to see the vertices are

𝜕log𝐶

𝜕log(𝑡𝑆,𝑡𝐸,𝐾) taking values(1,0,0),(0,1,0), and(1,1,−1), which we can integrate to get𝐶

expressed in(𝑡_𝑆, 𝑡_𝐸, 𝐾)with a multiplicative constant which we can set to1. The result is summarized as follows,

𝐶 ≈

⎧

⎪⎪

⎨

⎪⎪

⎩

𝑡_𝐸, reaction order(0,1), 𝑡_𝑆 ≫𝑡_𝐸, 𝐾;

𝑡𝑆𝑡𝐸

𝐾 , reaction order(1,1), 𝐾 ≫𝑡𝐸, 𝑡𝑆; 𝑡_𝑆, reaction order(1,0), 𝑡_𝐸 ≫𝑡_𝑆, 𝐾.

(3.43)

Each vertex corresponds to a biologically meaningfulregimethat the reaction can operate in. When the total substrate𝑡_𝑆 is very large, the enzymes are saturated so that the speed of catalysis is only limited by the enzyme, therefore𝐶 ≈𝑡_𝐸, corresponding to vertex(0,1). When it is the other way around and the total enzyme is very large and total substrate is limiting,𝐶 ≈ 𝑡_𝑆, corresponding to vertex(1,0). When enzyme and substrate are not abundant relative to the binding affinity𝐾, the speed of catalysis is limited by the formation of complex𝐶. In this regime, the complex𝐶is low in number compared to total enzyme and total substrate, and increasing either enzyme and substrate creates more complexes.

Therefore 𝐶 ≈ ^𝑡^𝑆_𝐾^𝑡^𝐸, corresponding to vertex(1,1). Together, we see that the polyhedral shape highlights the vertices as theregimesthat a catalysis reaction can operate in, which corresponds to extreme cases of concentrations and equilibrium constants. The asymptotic conditions of these extreme cases in turn yield asymptotic approximations of the active species (𝐶 in this case) that are simple monomials and biologically interpretable. The general complicated behavior can then be considered as varying in between these simple extreme-case regimes.

The above extreme-case analysis may remind a reader of the Michaelis-Menten formula (or the Langmuir form more generally), where one asymptotic condition that substrate is overabundant compared to enzyme𝑡_𝑆 ≫𝑡_𝐸 is used to derive the formula𝐶≈𝑡_𝐸_𝑡 ^𝑡^𝑆

𝑆+𝐾

that spans two regimes. A common description of this formula is that it is responsive when substrate concentration is low 𝑡𝑆 ≪ 𝐾, and becomes saturated when substrate concentration is high𝑡𝑆 ≫ 𝐾. This can be viewed as theedgeconnecting vertices (0,1) and(1,1). This is also clear from the asymptotic conditions, as 𝑡_𝑆 ≫ 𝑡_𝐸 has non-empty intersection with the conditions for these two vertices. We caution that although the Michaelis-Menten formula indeed forms an edge connecting the(0,1)and(1,1)vertices, it does not capture all points on the edge. In other words, the Michaelis-Menten assumption is a sufficient but not necessary condition for the edge. Although the condition𝑡_𝑆 ≫𝑡_𝐸, 𝐾 for the(0,1)vertex is contained in the Michaelis-Menten condition 𝑡_𝑆 ≫ 𝑡_𝐸, the same does not hold for the(1,1)vertex. The condition(𝐾 ≫𝑡_𝐸, 𝑡_𝑆of the(1,1)vertex does not require𝑡_𝑆 ≫𝑡_𝐸, e.g. both𝐾 ≫ 𝑡_𝐸 ≫𝑡_𝑆 and𝐾 ≫𝑡_𝑆 ≫𝑡_𝐸 can achieve vertex(1,1). The necessary and sufficient condition for the(0,1)to(1,1)edge is𝐸 ≪𝐾 or𝑡𝐸 ≪𝑡𝑆. This is

more general than𝑡_𝐸 ≪𝑡_𝑆. For example, when𝑡_𝑆 ≪𝐾, the(1,1)vertex is still achieved by 𝑡_𝐸 ≪𝐾, without assuming𝑡_𝐸 ≪𝑡_𝑆.

In summary, the polyhedral set for the full range of reaction orders highlights that the Michaelis-Menten formula is a sufficient edge-approximation of the overall behavior.

Through one asymptotic condition 𝑡_𝑆 ≫ 𝑡_𝐸, it captures a behavior spanning the edge connecting two regimes corresponding to vertices(0,1)and(1,1), and misses the regime corresponding to vertex(1,0)and expression𝐶 ≈𝑡_𝑆.

We can then investigate edge-approximation in general, with Michaelis-Menten approximation (or single molecule states approximations and external-bath approximations, see Section3.1) as a special case. In the example of𝐶’s reaction order, while two asymptotic conditions yield vertices, one asymptotic condition yield edges. In terms of convex coefficients in Eq (3.38), we can yield an edge by eliminating one vertex. For example, to obtain the Michaelis-Menten edge, we can eliminate the(1,0)vertex by letting coefficient ^𝑒

1+𝑒+𝑠

goes to zero, which corresponds to asymptotic condition𝑒≪ 𝑠or𝑒≪1. For simplicity, we follow Michaelis-Menten to use just one of the two conditions to represent an edge approximation. This yields the following summary for edge approximations of𝐶, and the graphical summarize of both edge and vertex approximations in Figure3.6.

𝐶 ≈

⎧

⎪⎪

⎨

⎪⎪

⎩

𝑡_𝐸_𝑡 ^𝑡^𝑆

𝑆+𝐾, edge from(0,1)to(1,1), 𝑡_𝑆 ≫𝑡_𝐸;

min{𝑡_𝑆, 𝑡_𝐸}, edge from(0,1)to(1,0), 𝐾 ≪𝑡_𝐸 or𝐾 ≪𝑡_𝑆; 𝑡_𝑆_𝑡 ^𝑡^𝐸

𝐸+𝐾, edge from(1,0)to(1,1), 𝑡_𝐸 ≫𝑡_𝑆.

(3.44)

In addition to the edge that is symmetric reflection of Michaelis-Menten, we also obtain an edge approximation connecting (0,1) to (1,0). Importantly, this is another class of approximations just as valid as the Michaelis-Menten or single molecule states or external bath approximations. To derive the formula 𝐶 ≈ min{𝑡_𝑆, 𝑡_𝐸}, we can apply the asymptotic condition𝐾 ≪ 𝑡𝐸 or 𝐾 ≪ 𝑡𝑆, which implies𝑡𝐸 +𝑡𝑆 +𝐾 ≈ 𝑡𝐸 +𝑡𝑆 and

√︁(𝑡_𝐸 +𝑡_𝑆+𝐾)²−4𝑡_𝐸𝑡_𝑆 ≈ |𝑡_𝐸 −𝑡_𝑆|, to the exact solution of𝐶 in Eq (3.31):

𝐶(𝑡_𝐸, 𝑡_𝑆)≈ (𝑡_𝐸 +𝑡_𝑆)− |𝑡_𝐸 −𝑡_𝑆|

2 = min{𝑡_𝐸, 𝑡_𝑆}.

The condition 𝐾 ≪ 𝑡_𝐸 or 𝐾 ≪ 𝑡_𝑆 corresponds to tight binding between enzyme and substrate. One natural scenario where this is true is in strong sequestrations between molecules, e.g. between sigma and anti-sigma factors, and in nucleic acid circuits. This edge approximation also plays a central role for antithetic integral control motifs for robust perfect adaptation proposed in [22]. Importantly, this edge approximation is an alternative

Figure 3.6Visualization of how the reaction order polyhedron captures the holistic regulatory profile, and how the vertex and edge approximations for𝐶in one binding reaction𝐸+𝑆 ⇌𝐶compare to the exact solution. Upper left is the exact solution of𝐶in terms of𝑡_𝐸and𝑡_𝑆in Eq (3.31). In the large𝑡_𝑆limit (close to 𝑡_𝑆 axis in the 3D plot), the Michaelis Menten formula (upper middle) is a good approximation of the exact solution. In the small𝐾limit (when𝑡_𝐸and/or𝑡_𝑆are large), the minimum formula corresponding to the diagonal edge is a good approximation of the exact solution.

to the Michaelis-Menten (or single molecule states or external bath) edge approximations of enzymatic reactions. This edge approximation is valid whenever tight binding is present.

Robustness of vertices and edges. Another prominent feature of the sampling of reaction orders in Figure 3.5is that the randomly sampled points concentrate at the edges and vertices. Since the points are uniformly sampled in log scale on enzyme and substrate concentrations, if we consider perturbations to the system as multiplicative variations in these concentrations, then the vertices and edges should be robust to such variations.

We can see why the points concentrate at vertices and edges by looking at the convex coefficients in Eq (3.38) and Eq (3.41). Because the coefficients of the vertices all take rational-function form such as ^𝑒

1+𝑒+𝑠, they approach extreme values of 0or 1when the variables(𝑒, 𝑠,1)are far apart in values. This means the condition for the reaction order to be in the interior of the polyhedra away from edges and vertices is quite fragile: the concentrations need to be “finely adjusted” so that they are close to each other. Once these concentrations drift apart from each other, we are at the vertices and edges.

Another way to describe this is that we can achieve precise values of reaction orders by very crude control of concentrations. For example, to push the reaction order of𝐶toward

Dalam dokumen Biocontrol of biomolecular systems (Halaman 123-138)