Log derivative as transform between two coordinate charts

Chapter II: Polyhedral constraints enable holistic analysis of bioregulation

3.5 Log derivative as transform between two coordinate charts

Note that although the theorem states that this log derivative ^𝜕log𝑥

𝜕(log𝑡,log𝑘) can always be calculated at any point𝑥in this fashion through matrix inversion, we have not proved this, as all previous calculations assumed𝜕_𝑥𝐹 is invertible. To show this, we can prove something stronger, namely the map𝑓 : log𝑥↦→ (log𝑡,log𝑘)is a diffeomorphism. This map can be explicitly written as follows:

𝑓(𝑧) =

⎡

⎣

log𝐿exp𝑧 𝑁 𝑧

⎤

⎦, (3.20)

where the exponential map is applied component wise. We can show this map is a diffeomorphism fromR^𝑛toR^𝑛. By Hadamard-Caccioppoli Theorem,𝑓is a diffeomorphism if𝑓 is proper and𝑑𝑓 is bijective at all points. 𝑓 is proper because for every sequence of𝑧 escaping to infinity,𝑓(𝑧)also escapes to infinity, since𝐿is non-negative and every column is nonzero. So it is left to show that𝑑𝑓 is bijective at all points, i.e. 𝑑𝑓, the log derivative

𝜕(log𝑡,log𝑘)

𝜕log𝑥 is invertible for all𝑥. From (3.19), we see that this is equivalent to the matrix 𝑀(𝑥;𝐿) :=

⎡

⎣

𝐿Λ_𝑥 𝑁

⎤

⎦ (3.21)

is invertible for all𝑥. We prove this in the following proposition.

Proposition 3.5.2. 𝑀(𝑥;𝐿)is invertible for all𝑥∈R^𝑛_>0.

Proof. Proof by contradiction. If it is not invertible, then there exists a nonzero vector𝑣s.t.

𝑀(𝑥;𝐿)𝑣 = 0. This implies𝑁 𝑣 = 0, i.e. 𝑣∈rowspan𝐿, so there exists a nonzero vector𝑐 s.t. 𝑣 =𝐿^⊺𝑐. So𝐿Λ_𝑥𝑣 =𝐿Λ_𝑥𝐿^⊺𝑐= 0. But this is impossible, sinceΛ_𝑥is positive definite, so𝑐^⊺𝐿Λ_𝑥𝐿^⊺𝑐>0.

Hence,𝑓 : log𝑥 ↦→ (log𝑡,log𝑘)is a diffeomorphism, and the log derivative ^𝜕log𝑥

𝜕(log𝑡,log𝑘) is well defined on all points inℳ. The formula3.19is always applicable.

We can write the relationship of chartlog𝑥and chart(log𝑡,log𝑘)in the following diagram:

log𝑥−−−−−−−−−−−−−−−−−−−−−↽−−−−−−−−−−−−−−−−−−−−−^{𝑓(log𝑥)=(log𝐿𝑥,𝑁log𝑥)} ⇀

𝑑𝑓⁻¹=_𝜕(log^𝜕log_𝑡,log^𝑥_𝑘)=

⎡

⎣

Λ⁻¹_𝑡 𝐿Λ_𝑥 𝑁

⎤

⎦

−1 (log𝑡,log𝑘). (3.22)

Now we have established that for a generic equilibrium manifoldℳof a binding network, in addition to the natural chart log𝑥, we have an alternative chart in terms of totals and equilibrium constants (log𝑡,log𝑘), which is also a one-chart atlas. Recall that for isomer-atomic binding networks, we have another natural chart using atomic species (log𝑥^𝑎,log𝑘). We study how this chart relates to the total chart(log𝑡,log𝑘)below.

Log derivatives and chart transform for isomer-atomic binding networks

Recall that from the previous section, we established that if the binding network is isomer- atomic, then we have another chart using the atomic species, namely(log𝑥^𝑎,log𝑘), where 𝑥= (𝑥^𝑎,𝑥^𝑐)is split into the atomic species and the complex species. To relate this atomic chart to the total chart(log𝑡,log𝑘), since the equilibrium constants𝑘)is kept the same, we just need to study how 𝑥^𝑎 is mapped to 𝑡. The chart transform in Eq (3.16) tells us log𝑥^𝑐 =𝐿^⊺₂log𝑥^𝑎+𝑁₂⁻¹log𝑘, so we have

𝑡 =𝐿𝑥 =𝑥^𝑎+𝐿₂𝑥^𝑐=𝑥^𝑎+𝐿₂exp(𝐿^⊺₂log𝑥^𝑎+𝑁₂⁻¹log𝑘).

The inverse map from (log𝑥^𝑎,log𝑘)to(log𝑡,log𝑘)again requires solving an intractible polynomial system, so we resort to differentials.

Theorem 3.5.3(Isomer-atomic log derivative formula). Givenℳ ⊂ R^2𝑛_>0, the equilibrium manifold of an isomer-atomic binding network with transpose-reduced stoichiometry matrix𝑁 ∈R^𝑟×𝑛and conservation law matrix𝐿∈R^𝑑×𝑛, with an atom-first ordering so that the first𝑑species are atomic species, i.e. 𝐿 = ^[︁I_𝑑 𝐿2

]︁

,𝐿₂ ∈ R^𝑑×𝑟, and 𝑁 = ^[︁𝑁1 𝑁2

]︁

, with𝐿^⊺₂ = −𝑁₂⁻¹𝑁₁. Then at any point𝑝= (𝑥,𝑡,𝑘)∈ ℳ, we have

𝜕log𝑥^𝑎

𝜕(log𝑡,log𝑘) =^[︁(𝐿Λ_𝑥𝐿^⊺)⁻¹Λ_𝑡 −(𝐿Λ_𝑥𝐿^⊺)⁻¹𝐿₂Λ_𝑥^𝑐𝑁₂^−1]︁,

𝜕log𝑥^𝑐

𝜕(log𝑡,log𝑘) =𝐿^⊺₂ 𝜕log𝑥^𝑎

𝜕(log𝑡,log𝑘) +^[︁0 𝑁₂^−1]︁,

(3.23)

whereI_𝑟 is the identity matrix of dimension𝑟.

Proof. The second formula is immediately obtained by using chain rule and Eq (3.16):

𝜕log𝑥^𝑐

𝜕(log𝑡,log𝑘) = 𝜕𝐿^⊺₂log𝑥^𝑎+𝑁₂⁻¹log𝑘

𝜕(log𝑡,log𝑘) =𝐿^⊺₂ 𝜕log𝑥^𝑎

𝜕(log𝑡,log𝑘) +𝑁₂^−1[︁0 I_𝑟^]︁.

The first formula is obtained by block-matrix inversion of Eq (3.19). Block-matrix inversion satisfies

⎡

⎣

𝐴 𝐵 𝐶 𝐷

⎤

⎦=

⎡

⎣

(𝐴−𝐵𝐷⁻¹𝐶)⁻¹ 0

0 (𝐷−𝐶𝐴⁻¹𝐵)⁻¹

⎤

⎦

⎡

⎣

I −𝐵𝐷⁻¹

−𝐶𝐴⁻¹ I

⎤

⎦, if blocks𝐴and𝐷are both invertible. So

[︁𝐴 𝐵^]︁= (𝐴−𝐵𝐷⁻¹𝐶)^−1[︁I−𝐵𝐷^−1]︁.

Applying this to our case,

𝜕log𝑥^𝑎

𝜕(log𝑡,log𝑘) =^[︁I_𝑑 0^]︁𝜕(log𝑥^𝑎,log𝑥^𝑐)

𝜕(log𝑡,log𝑘) =^[︁I_𝑑 0^]︁

⎡

⎣

Λ_𝑥^𝑎 𝐿₂Λ_𝑥^𝑐 𝑁₁ 𝑁₂

⎤

⎦

−1⎡

⎣

Λ_𝑡 0 0 I_𝑟

⎤

⎦

=(Λ_𝑥^𝑎 −𝐿2Λ_𝑥^𝑐𝑁₂⁻¹𝑁1)^−1[︁I −𝐿₂Λ_𝑥^𝑐𝑁₂^−1]︁

⎡

⎣

Λ_𝑡 0 0 I_𝑟

⎤

⎦

=(Λ_𝑥^𝑎 +𝐿₂Λ_𝑥^𝑐𝐿^⊺₂)^−1[︁Λ_𝑡 −𝐿₂Λ_𝑥^𝑐𝑁₂^−1]︁. This is the desired formula by recognizingΛ_𝑥^𝑎+𝐿₂Λ_𝑥^𝑐𝐿^⊺₂ =𝐿Λ_𝑥𝐿^⊺.

We remark that the atomic log derivative formula in Eq (3.23) highlights an internal symmetric structure not obviously seen in the more general log derivative formula in Eq (3.19). In particular, the symmetric positive-semidefinite matrix(𝐿Λ_𝑥𝐿^⊺)⁻¹constitute the core structure from which all the log derivatives arise.

In addition to characterizing how the atomic chart(log𝑥^𝑎,log𝑘)is mapped from the total chart(log𝑡,log𝑘), Eq (3.23) also implies a simpler way to calculate the log derive ^𝜕log𝑥

𝜕(log𝑡,log𝑘)

for isomer-atomic binding entworks. Namely, we can first calculate just the log derivative of 𝑥^𝑎, then use this to obtain the log derivative of𝑥^𝑐. For large scale problems, the bottleneck in log dervative computation would be matrix inversion, which often has𝑂(𝑛³)complexity to invert a square matrix with dimension𝑛(could be lower by more advanced algorithms, but still larger than𝑂(𝑛^2.3)). We see that in this formula, we can compute all log derivatives by inverting only one matrix with dimension 𝑑 and one with dimension 𝑟, instead of inverting a matrix with dimension𝑛in Eq (3.19), so roughly reducing complexity from𝑛³ to𝑑³+𝑟³. If we are only interested in the log derivative of𝑥with respect to𝑡, we only need to compute ^𝜕log𝑥

𝑎

𝜕log𝑡 = (𝐿Λ_𝑥𝐿^⊺)⁻¹Λ_𝑡, which only requires to invert a𝑑×𝑑matrix, and then compute ^𝜕log𝑥

𝑐

𝜕log𝑡 =𝐿^⊺₂^𝜕_𝜕^log_log^𝑥_𝑡^𝑎 =𝐿^⊺₂(𝐿Λ_𝑥𝐿^⊺)⁻¹Λ_𝑡.

We summarize the three charts studied so far and the transform between them in Figure 3.4.

Other alternative charts

The two alternative charts we have studied so far have equilibrium constants fixed, and the remaining𝑑degrees of freedom are represented as𝑥^𝑎 =𝐴^𝑎𝑥where𝐴^𝑎 :=^[︁I_𝑑 0^]︁in atomic chart, and𝑡=𝐿𝑥in total chart. This brings the question about what are the other charts of the form(log𝐴𝑥,log𝑘), for some non-negative matrix𝐴∈R^𝑑×𝑛?

We know(log𝐴𝑥,log𝑘)is a chart if and only if the maplog𝑥↦→(log𝐴𝑥,log𝑘)is invertible for all𝑥. From log derivative formula (3.19), we know this is equivalent to matrix𝑀(𝑥;𝐴)

Figure 3.4The three charts of equilibrium manifoldℳof a binding network, and the transform between them.

is invertible for all𝑥. Hence we define𝒜 =^{︁𝐴∈R^𝑑×𝑛_≥0 : det𝑀(𝑥;𝐴)̸= 0,∀𝑥^}︁, the set of 𝐴such that(log𝐴𝑥,log𝑘)are charts.

Topologically, since determinant is a continuous function, a matrix𝐴∈ 𝒜has determinant of 𝑀(𝑥;𝐴) either positive for all 𝑥, or negative for all 𝑥. This separates 𝒜 into two connected components,

𝒜⁺(𝑁) :=^{︁𝐴∈R^𝑑×𝑛_≥0 : det𝑀(𝑥;𝐴)>0,∀𝑥^}︁, (3.24) and correspondingly𝒜⁻(𝑁).

For the atomic chart,det𝑀(𝑥;𝐴^𝑎) = det𝑁₂. So

[︁

I_𝑑 0^]︁ ∈ 𝒜⁺(𝑁)ifdet𝑁₂ >0. Similarly, we show below that the total chart’s sign is determined by the same condition for isomer- atomic CRNs.

Proposition 3.5.4. Assume isomer-atomic, then𝐿∈ 𝒜⁺(𝑁)if and only ifdet𝑁₂ >0. Proof. Calculate

⎡

⎣

𝐿 𝑁

⎤

⎦

⎡

⎣

I_𝑑 0 0 𝑁₂^⊺

⎤

⎦=

⎡

⎣

I_𝑑 −𝑁₁^⊺𝑁₂^−⊺ 𝑁₁ 𝑁₂

⎤

⎦

⎡

⎣

I_𝑑 0 0 𝑁₂^⊺

⎤

⎦=

⎡

⎣

I −𝑁₁^⊺ 𝑁₁ 𝑁₂𝑁₂^⊺

⎤

⎦. Take determinant of both sides,

det

⎛

⎝

⎡

⎣

𝐿 𝑁

⎤

⎦

⎞

⎠det(𝑁₂^⊺) = det(𝑁₂𝑁₂^⊺+𝑁₁𝑁₁^⊺)>0.

The right hand side is a positive definite matrix, so determinant is positive. Since det𝑀(1;𝐿)>0if and only ifdet𝑀(𝑥;𝐿)>0for all𝑥∈R^𝑛_>0from previous Lemma, we have the desired result.

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-