## Archive for November, 2015

### Feinberg’s proof of the deficiency zero theorem

November 23, 2015

I discussed the deficiency zero theorem of chemical reaction network theory (CRNT) in a previous post. (Some further comments on this can be found here and here.) This semester I am giving a lecture course on chemical reaction network theory. Lecture notes are growing with the course and can be found in German and English versions on the web page of the course. The English version can also be found here. Apart from introductory material the first main part of the course was a proof of the Deficiency Zero Theorem. There are different related proofs in the literature and I have followed the approach in the classic lecture notes of Feinberg on the subject closely. The proof in those notes is essentially self-contained apart from one major input from a paper of Feinberg and Horn (Arch. Rat. Mech. Anal. 66, 83). In this post I want to give a high-level overview of the proof.

The starting point of CRNT is a reaction network. It can be represented by a directed graph where the nodes are the complexes (left or right hand sides of reactions) and the directed edges correspond to the reactions themselves. The connected components of this graph are called the linkage classes of the network and their number is usually denoted by $l$. If two nodes can be connected by oriented paths in both directions they are said to be strongly equivalent. The corresponding equivalence classes are called strong linkage classes. A strong linkage class is called terminal if there is no directed edge leaving it. The number of terminal strong linkage classes is usually denoted by $t$. From the starting point of the network making the assumption of mass action kinetics allows a system of ODE $\dot c=f(c)$ to be obtained in an algorithmic way. The quantity $c(t)$ is a vector of concentrations as a function of time. Basic mathematical objects involved in the definition of the network are the set $\cal S$ of chemical species, the set $\cal C$ of complexes and the set $\cal R$ of reactions. An important role is also played by the vector spaces of real-valued functions on these finite sets which I will denote by $F({\cal S})$, $F({\cal C})$ and $F({\cal R})$, respectively. Using natural bases they can be identified with $R^m$, $R^n$ and $R^r$. The vector $c(t)$ is an element of $F({\cal S})$. The mapping $f$ from $F({\cal S})$ to itself can be written as a composition of three mappings, two of them linear, $f=YA_k\Psi$. Here $Y$, the complex matrix, is a linear mapping from $F({\cal C})$ to $F({\cal S})$. $A_k$ is a linear mapping from $F({\cal C})$ to itself. The subscript $k$ is there because this matrix is dependent on the reaction constants, which are typically denoted by $k$. It is also possible to write $f$ in the form $Nv$ where $v$ describes the reaction rates and $N$ is the stoichiometric matrix. The image of $N$ is called the stoichiometric subspace and its dimension, the rank of the network, is usually denoted by $s$. The additive cosets of the stoichiometric subspace are called stoichiometric compatibility classes and are clearly invariant under the time evolution. Finally, $\Psi$ is a nonlinear mapping from $F({\cal S})$ to $F({\cal C})$. The mapping $\Psi$ is a generalized polynomial mapping in the sense that its components are products of powers of the components of $c$. This means that $\log\Psi$ depends linearly on the logarithms of the components of $c$. The condition for a stationary solution can be written as $\Psi(c)\in {\rm ker} (YA_k)$. The image of $\Psi$ is got by exponentiating the image of a linear mapping. The matrix of this linear mapping in natural bases is $Y^T$. Thus in looking for stationary solutions we are interested in finding the intersection of the manifold which is the image of $\Psi$ with the kernel of $YA_k$. The simplest way to define the deficiency of the network is to declare it to be $\delta=n-l-s$. A fact which is not evident from this definition is that $\delta$ is always non-negative. In fact $\delta$ is the dimension of the vector space ${\rm ker} Y\cap {\rm span}(\Delta)$ where $\Delta$ is the set of complexes of the network. An alternative concept of deficiency, which can be found in lecture notes of Gunawardena, is the dimension $\delta'$ of the space ${\rm ker} Y\cap{\rm im} A_k$. Since this vector space is a subspace of the other we have the inequality $\delta'\le\delta$. The two spaces are equal precisely when each linkage class contains exactly one terminal strong linkage class. This is, in particular, true for weakly reversible networks. The distinction between the two definitions is often not mentioned since they are equal for most networks usually considered.

If $c^*$ is a stationary solution then $A_k\Psi (c^*)$ belongs to ${\rm ker} Y\cap{\rm im} A_k$. If $\delta'=0$ (and in particular if $\delta=0$) then this means that $A_k\Psi (c^*)=0$. In other words $\Psi (c^*)$ belongs to the kernel of $A_k$. Stationary solutions of this type are called complex balanced. It turns out that if $c^*$ is a complex balanced stationary solution the stationary solutions are precisely those points $c$ for which $\log c-\log c_*$ lies in the orthogonal complement of the stoichiometric subspace. It follows that whenever we have one solution we get a whole manifold of them of dimension $n-s$. It can be shown that each manifold of this type meets each stoichiometric class in precisely one point. This is proved using a variational argument and a little convex analysis.

It is clear from what has been said up to now that it is important to understand the positive elements of the kernel of $A_k$. This kernel has dimension $t$ and a basis each of whose elements is positive on a terminal strong linkage class and zero otherwise. Weak reversibility is equivalent to the condition that the union of the terminal strong linkage classes is the set of all complexes. It can be concluded that when the network is not weakly reversible there exists no positive element of the kernel of $A_k$. Thus for a network which is not weakly reversible and has deficiency zero there exist no positive stationary solutions. This is part of the Deficiency Zero Theorem. Now consider the weakly reversible case. There a key statement of the Deficiency Zero Theorem is that there exists a complex balanced stationary solution $c^*$. Where does this $c^*$ come from? We sum the vectors in the basis of ${\rm ker} A_k$ and due to weak reversibility this gives something which is positive. Then we take the logarithm of the result. When $\delta=0$ this can be represented as a sum of two contributions where one is of the form $Y^T z$. Then $c^*=e^z$. A further part of the deficiency zero theorem is that the stationary solution $c^*$ in the weakly reversible case is asymptotically stable. This is proved using the fact that for a complex balanced stationary solution the function $h(c)=\sum_{s\in\cal S}[c(s)(\log c(s)-\log c^*(s)-1)+c(s^*)]$ is a Lyapunov function which vanishes for $c=c^*$

### Conference on biological oscillators at EMBL in Heidelberg

November 17, 2015

EMBL, the European Molecular Biology Laboratory, is an international institution consisting of laboratories at five sites, two in Germany, one in the UK, one in France and one in Italy. I recently attended a meeting on the theme ‘Biological Oscillators’ at the site in Heidelberg. The impressive building is in the form of a double helix. There are two spiral ramps over several stories which are linked by bridges (‘hydrogen bonds’, in German Wasserstoffbrücken). This helix provides an original setting for the poster sessions. The building is reached by ascending steep hills in the area behind the castle. I took the comfortable option of using the bus provided by the institute. This meeting had about 130 participants but I think that the capacity is much greater.

One of the most interesting talks on the first day from my point of view was by Petra Schwille from the Max Planck Institute for Biochemistry. She talked about the Min system which is used by bacteria to determine their plane of division. The idea is that certain proteins (whose identity is explicitly known) oscillate between the ends of the cell and that the plane of division is the nodal surface of the concentration of one of these. The speaker and her collaborators have been able to reconstitute this system in a cell-free context. A key role is played by the binding of the proteins to the cell membrane. Diffusion of bound proteins is much slower than that of proteins in solution and this situation of having two different diffusion constants in a coupled system is similar to the classical scenario known from the Turing instability. It sounds like modelling this system mathematically can be a lot of fun and that there is no lack of people interested in doing so.

There was also a ‘Keynote Lecture’ by Jordi Garcia-Ojalvo which lived up to the promise of its special title. The topic was the growth of a colony of Bacillus subtilis. (The published reference is Nature 523, 550.) In fact, to allow better control, the colony is constrained to be very thin and is contained in a microfluidic system which allows its environment to be manipulated precisely. A key observation is that the colony does not grow at a constant rate. Instead its growth rate is oscillatory. The speaker explained that this can be understood in terms of the competition between the cells near the edge of the colony and those in the centre. The colony is only provided with limited resources (glycerol, glutamate and salts). It may be asked which resource limits the growth rate. It is not the glycerol, which is the primary carbon source. Instead it is the glutamate, which is the primary source of nitrogen. An important intermediate compound in the use of glutamate is ammonium. If cells near the boundary of the colony produced ammonium it would be lost to the surroundings. Instead they use ammonium produced by the interior cells. It is the exterior cells which grow and they can deprive the inner cells of glutamate. This prevents the inner cells producing ammonium which is then lacking for the growth of the outer cells. This establishes a negative feedback loop which can be seen as the source of the oscillations in growth rate. The feasibility of this mechanism was checked using a mathematical model. The advantage of the set-up for the bacteria is that if the colony is exposed to damage from outside it can happen that only the exterior cells die and the interior cells generate a new colony. The talk also included a report on further work (Nature 527, 59) concerning the role of ion channels in biofilms. There are close analogies to the propagation of nerve signals and the processes taking place can be modelled by equations closely related to the Hodgkin-Huxley system.

I will now mention a collection of other topics at the conference which I found particularly interesting. One recurring theme was NF$\kappa$B. This transcription factor is known to exhibit oscillations. A key question is what their function is, if any. One of the pioneers in this area, Mike White, gave a talk at the conference. There were also a number of other people attending working on related topics. I do not want to go any deeper here since I think that this is a theme to which I might later devote a post of its own, if not more than one. I just note two points from White’s talk. One is that this substance is a kind of hub or bow-tie with a huge number of inputs and outputs. Another is that the textbook picture of the basic interactions of NF$\kappa$B is a serious oversimplification. Another transcription factor which came up to a comparable extent during the conference is Hes1, which I had never previously heard of. Jim Ferrell gave a talk about the coordination of mitosis in Xenopus eggs. These are huge cells where communication by means of diffusion would simply not be fast enough. The alternative proposed by Ferrell are trigger waves, which can travel much faster. Carl Johnson talked about mechanisms ensuring the stability of the KaiABC oscillator. He presented videos showing the binding of individual KaiA molecules to KaiC. I was was amazed that these things can be seen directly and are not limited to the cartoons to be found in biology textbooks. Other videos I found similarly impressive were those of Alexander Aulehla showing the development of early mouse embryos (segmentation clock) where it could be seen how waves of known chemical events propagating throught the tissues orchestrate the production of structures in the embryo. These pictures brought the usual type of explanations used in molecular biology to a new level of concreteness in my perception.