Archive for the ‘mathematical biology’ Category

Breaking waves in Madrid

July 19, 2014

Last week I was at the 10th AIMS Conference on Dynamical Systems, Differential Equations and Applications in Madrid. It was very large, with more than 2700 participants and countless parallel sessions. This kind of situation necessarily generates a somewhat hectic atmosphere and I do not really like going to that type of conference. I have heard the same thing from many other paople. There is nevertheless an advantage, namely the possibility of meeting many people. To do this effectively it is necessary to proceed systematically since it is easy to go for days without seeing a particular person of interest. This aspect was of particular importance for me since I am still at a relatively early stage in the process of entering the field of mathematical biology and I have few contacts there in comparison to my old field of mathematical relativity. In any case, the conference allowed me to meet a lot of interesting people and learn a lot of interesting things.

The first plenary talk, by Charles Fefferman, was on a subject related to a topic I was interested in many years ago. I learned that a lot has happened since I last thought about this. The attempt to model a body of fluid with a free surface leads to considerable mathematical difficulties. When I started working on dynamical models for this kind of situation few people seemed to be interested in proving theorems on the subject. The source of my interest in the subject was the influence of Jürgen Ehlers, who always had a clear vision of what were the important problems. In this way I found myself in the position of a pioneer in a certain research area. Being in that situation has the advantage of not being troubled by strong competition. On the other hand it can also mean that whatever you achieve can be largely ignored and it is not the best way to get wide recognition. Often finishing mathematical research directions gets more credit than starting them. This could no doubt be compensated by suitable advertizing but that was never my strong point. This is a configuration which I have often found myself in and in fact, comparing advantages and disadvantages, I do not feel I need to change it. Coming back to the fluids with free surface, this is now a hot topic and played a prominent role at the conference. When I was working on this the issue of local existence in the case of inviscid fluids was still open. A key step was the work of Sijue Wu on water waves. I learned from the talk of Fefferman that this has been extended in the meantime to global existence for small data. The question which is now the focus of interest is formation of singularities (i.e. breakdown of classical solutions) for large data. Instead of considering the breaking of one wave the idea is to consider two waves which are approaching each other while turning over until they meet. There are already analytical results on parts on this process by Fefferman and collaborators and they plan to extend this to a more global picture by using a computer-assisted proof. Another plenary was by Ingrid Daubechies, who talked about applications of image processing to art history. I must admit that beforehand the theme did not appear very attractive to me but in fact the talk was very entertaining and I am glad I went to hear it.

I gave a talk on my recent work with Juliette Hell on the MAPK cascade in a session organized by Bernold Fiedler and Atsushi Mochizuki. I found the session very interesting and the highlight for me was Mochizuki’s talk on his work with Fiedler. The subject is how much information can be obtained about a network of chemical reactions by observing a few nodes, i.e. by observing a few concentrations. What I find particularly interesting are the direct connections to biological problems. Applied to the gene regulatory network of an ascidian (sea squirt) this theoretical approach suggests that the network known from experimental observations is incomplete and motivates searching for the missing links experimentally. Among the many other talks I heard at the conference, one which I found particularly impressive concerned the analysis of successive MRT pictures of patients with metastases in the lung. The speaker was using numerical simulations with these pictures as input to provide the surgeon with indications which of the many lesions present was likely to develop in a dangerous way and should therefore be removed. One point raised in the talk is that it is not really clear what information about the tissue is really contained in an MRT picture and that this could be an interesting mathematical problem in itself. In fact there was an encouragingly (from my point of view) large number of sessions and other individual talks at the conference on subjects related to mathematical biology.

The conference took place on the campus of the Universidad Autonoma somewhat outside the city. A bonus for me was hearing and seeing my first bee-eater for many years. It was quite far away (flying high) but it gave me real pleasure. I was grateful that the temperatures during the week were very moderate, so that I could enjoy walking through the streets of Madrid in the evening without feeling disturbed by heat or excessive sun.

The Higgins-Selkov oscillator

May 14, 2014

In a previous post I wrote about glycolytic oscillations and mentioned a mathematical model for them, the Higgins-Selkov oscillator. Higgins introduced this as a chemical model while Selkov also wrote about some mathematical aspects of modelling this system. When I was preparing my course on dynamical systems I wanted to present an example where the existence of periodic solutions can be concluded by using the existence of a confined region in a two-dimensional system and Poincare-Bendixson theory. An example which is frequently treated in textbooks is the Brusselator and I wanted to do something different. I decided to try the Higgins-Selkov oscillator. Unfortunately I came up against difficulties since that model has unbounded solutions and it is hard to show that there are any bounded solutions except a stationary solution which can be calculated explicitly. For the purposes of the course I went over to considering the Schnakenberg model, a modification of the Higgins-Selkov oscillator where it is not hard to see that all solutions are bounded.

More recently I decided to try to finally find out what happens with the Higgins-Selkov oscillator itself. Reading Selkov’s paper I originally had the impression that he had proved the essential properties of the solutions. This turned out to be mistaken. One obstacle for me was that Selkov cited a theorem from a famous Russian textbook of Andronov et. al. and I did not know what the theorem was. An English translation of the book exists in the university library here but since Selkov only cites a page number I did not know how to find the theorem. I was able to get further when Jan Fuhrmann got hold of a copy of the page in question from the Russian original. This page has an easily identifiable picture on it and this allowed me to identify the corresponding page of the English translation and hence the theorem. I found that, as far as it is applicable to the oscillator problem this was something I could prove myself by a simple centre manifold argument. Thus I realized that the results quoted by Selkov only resolve some of the simpler issues in this problem.

At this stage I decided to follow the direction pointed out by Selkov on my own. The first stage, which can be used to obtain information about solutions which tend to infinity, is to do a Poincare compactification. This leads to a dynamical system on a compact subset of Euclidean space. In general it leads to the creation of new stationary points on the boundary which are not always hyperbolic. In this particular example two new stationary points are created. One of these has a one-dimensional centre manifold and it is relatively easy to determine its qualitative nature. This is what Selkov was quoting the result of Andronov et. al. for. The other new stationary solution is more problematic since the linearization of the system at that point is identically zero. More information can be obtained by transforming to polar coordinates about that point. This creates two new stationary points. One is hyperbolic and hence unproblematic. The linearization about the other is identically zero. Passing to polar coordinates about that point creates three new stationary points. One of them is hyperbolic while the other two have one-dimensional centre manifolds. The process comes to an end. When trying this kind of thing in the past I was haunted by the nightmare that the process would never stop. Is there a theorem which forbids that? In any case, in this example it is possible to proceed in this way and determine the qualitative behaviour near all points of the boundary. The problem is that this does not seem to help with the original issue. I see no way in which, even using all this information, it is possible to rule out that every solution except the stationary solution tends to infinity as t tends to infinity.

Given that this appeared to be a dead end I decided to try an alternative strategy in order to at least prove that there are some parameter values for which there exists a stable periodic solution. It is possible to do this by showing that a generic supercritical Hopf bifurcation occurs and I went to the trouble of computing the Lyapunov coefficient needed to prove this. I am not sure how much future there is for the Higgins-Selkov oscillator since there are more modern and more complicated models for glycolysis on the market which have been studied more intensively from a mathematical point of view. More information about this can be found in a paper of Kosiuk and Szmolyan, SIAM J. Appl. Dyn. Sys. 10, 1307.

Finally I want to say something about the concept of feedback, something I find very confusing. Often it is said in the literature that oscillations are related to negative feedback. On the other hand the oscillations in glycolysis are often said to result from positive feedback. How can this be consistent? The most transparent definition of feedback I have seen is the one from a paper of Sontag which I discussed in the context of monotone systems. In that sense the feedback in the Higgins-Selkov oscillator is definitely negative. An increase in the concentration of the substrate leads to an increase in the rate of production of the product. An increase in the concentration of the product leads to an increase of the rate of consumption of the substrate. The combination of a positive and a negative sign gives a negative loop. The other way of talking about this seems to be related to the fact that an increase in the concentration of the product leads to an increase in the reaction rate between substrate and product. This is consistent with what was said before. The difference is what aspects of the system are being regarded as cause and effect, which can lead to a different assignment of the labels positive and negative. The problem as I see it is that feedback is frequently invoked but rarely defined, with the implicit suggestion that the definition should be obvious to anyone with an ounce of understanding. I seem to be lacking that ounce.

Proofs of dynamical properties of the MAPK cascade

April 3, 2014

The MAP kinase cascade, which I mentioned in a previous post, is a biochemical network which has been subject to a lot of theoretical and experimental study. Although a number of results about mathematical models for this network have been proved, many widely accepted results are based on numerical and/or heuristic approaches. Together with Juliette Hell we set out to extend the coverage of rigorous results in this area. Our first results on this can be found in a paper we just posted on q-bio.

The system of equations which is fundamental for this work is that of Huang and Ferrell discussed in my previous post on the subject. I call it the MM-MA system (for Michaelis-Menten via mass action). When this system can be reduced to a smaller system by means of a quasistationary approximation the result will be called the MM system (for Michaelis-Menten) (cf. this post). With a suitable formulation the MM system is a singular limit of the MM-MA system. The MAPK cascade consists of three coupled layers. The first main result of our paper concerns the dual futile cycle, which can be thought of as the second layer of the cascade in isolation (cf. this post). We proved that the MM system for the dual futile cycle exhibits a generic cusp bifurcation and hence that for suitable values of the parameters there exist two different stable stationary solutions (bistability). Using the fact that this system is a singular limit of the system arising from the MM-MA description of the same biological system we then used geometric singular perturbation theory (cf. this post) to conclude that the MM-MA system also shows bistability.

The second main result concerns the system obtained by truncating that of Huang-Ferrell by keeping only the first two layers. It is subtle to find a useful quasistationary approximation for this system and we were put on the right track by a paper of Ventura et. al. (PLoS Comp. Biol. 4(3):e1000041). This allowed us to obtained an MM system which is a limit of the MM-MA system in a way which allows geometric singular perturbation theory to be applied. This leads to the following relative statement: if the MM system for the truncated MAPK cascade has a hyperbolic periodic solution then the same is true for the MM-MA system. To get an absolute statement it remains to prove the existence of periodic solutions of the MM system, which in this case is of dimension three. That there are solutions of this kind is indicated by numerical work of Ventura et. al.

The Hopf-Hopf bifurcation and chaos in ecological systems

February 11, 2014

This post arises from the fact that there seems to be some constructive interference between various directions I am pursuing at the moment. The first has to do with the course on dynamical systems I just finished giving. This course was intended not only to provide students in Mainz with an extended introduction to the subject but also to broaden my own knowledge. I wrote lecture notes for this in German and having gone to the effort of producing this resource I thought I should translate the notes into English so as to make them more widely available. The English version can be found here. Both versions are on the course web page. The second thing is that I will be organizing a seminar on bifurcation theory next semester and I want it to achieve a wide coverage, even at the risk that its waters, being broad, may be shallow (this is paraphrase of a quote I vaguely remember from Nietzsche). The connections between these two things are that I treated simple bifurcation theory and a little chaos in the course and that going further into the landscape of bifurcations necessarily means that at some point chaos rears its ugly head. The third thing is the fact that it has been suggested that the MAPK cascade, a dynamical system I am very interested in from the point of view of my own research, may exhibit chaotic behaviour, as described in a paper of Zumsande and Gross (J. Theor. Biol. 265, 481). This paper attracted my attention when it appeared on arXiv but it is only now that I understood some of the underlying ideas and, in particular, that the Hopf-Hopf bifurcation plays a central role. This in turn led me to a paper by Stiefs et. al. on chaos in ecological systems (Math. Biosci. Eng. 6, 855). They consider models with predator-prey interaction and a disease of the predators.

A Hopf-Hopf (or double Hopf) bifurcation arises at a stationary point where the linearization has two pairs of non-vanishing purely imaginary eigenvalues. Of course it is necessary to have a system of at least dimension four in order for this to occur. The subset of parameter space where it occurs has codimension two and lies at the intersection of two hypersurfaces on which there are Hopf bifurcations. For this system there is an approximate normal form. In other words the system is topologically equivalent to a system given by simple explicit formulae plus higher order error terms. The dynamics of the model system ignoring error terms can be analysed in detail. For simple bifurcations a system in approximate normal form is topologically equivalent to the model system. For the Hopf-Hopf bifurcation (and for the simpler fold-Hopf bifurcation with one zero and one pair of non-zero purely imaginary eigenvalues) this is no longer the case and the perturbation leads to more complicated dynamics. For instance, a heteroclinic orbit in the model system can break as a result of the perturbation. A lot of information on these things can be found in the book of Kuznetsov. In the paper on ecological systems mentioned above a Hopf-Hopf bifurcation is found using computer calculations and this is described as ‘clear evidence for the existence of chaotic parameter regions’. My understanding of chaos is still too weak to be able to appreciate the precise meaning of this statement.

Using computer calculations Zumsande and Gross find fold-Hopf bifurcations in the MAPK cascade (without explicit feedback) indicating the presence of complex dynamics. If chaos occurs in the ecological system and the MAPK cascade what biological meaning could this have? Ecosystems can often be thought of as spatially localized communities with their own dynamics which are coupled to each other. If the dynamics of the individual communities is of a simple oscillatory type then they may become synchronized and this could lead to global extinctions. If the local dynamics are chaotic this cannot happen so easily and even if a fluctuation which is too big leads to extinctions in one local community, these can be avoided in neighbouring communities, giving the ecosystem a greater global stability. One point of view of chaos in the MAPK cascade is that it is an undesirable effect which might interfere with the signalling function. It might be an undesirable side effect of other desirable features of the system. In reality MAPK cascades are usually embedded in various feedback loops and these might suppress the complex  behaviour in the free cascade. Zumsande and Gross investigated this possibility with the conclusion that the feedback loops tend to make things worse rather than better.

Monotone dynamical systems

December 29, 2013

In previous posts I have written a little about monotone dynamical systems, a class of systems which in some sense have simpler dynamical properties than general dynamical systems. Unfortunately this subject was always accompanied by some confusion in my mind. This results from the necessity of a certain type of bookkeeping which I was never really able to get straight. Now I think my understanding of the topic has improved and I want to fix this knowledge here. There are two things which have led to this improvement. One is that I read an expository article by Eduardo Sontag which discusses monotone systems in the context of biochemical networks (Systems and Synthetic Biology 1, 59). The other is that I had the chance to talk to David Angeli who patiently answered some of my elementary questions as well as providing other insights. In what follows I only discuss continuous dynamical systems. Information on the corresponding theory in the case of discrete dynamical systems can be found in the paper of Sontag.

Consider a dynamical system \dot x=f(x) defined on an open subset of R^n. The system is called monotone if \frac{\partial f_i}{\partial x_j}\ge 0 for all i\ne j. This is a rather restrictive definition – we will see alternative possibilities later – but I want to start in a simple context. There is a theorem of Müller and Kamke which says that if two solutions x and y of a monotone system satisfy x_i(0)\le y_i(0) for all i then they satisfy x_i(t)\le y_i(t) for all i and all t\ge 0. This can be equivalently expressed as the fact that for each t\ge 0 the time t flow of the dynamical system preserves the partial order defined by the condition that x_i\le y_i for all i. This can be further reexpressed as the condition that y-x belongs to the positive convex cone in R^n defined by the conditions that the values of all Cartesian coordinates are non-negative. This shows the way to more general definitions of monotone flows on vector spaces, possibly infinite dimensional. These definitions may be useful for the study of certain PDE such as reaction-diffusion equations. The starting point is the choice of a suitable cone. This direction will not be followed further here except to consider some other simple cones in R^n.

A monotone system in the sense defined above is also sometimes called cooperative. The name comes from population models where the species are beneficial to each other. Changing the sign in the defining conditions leads to the class of competitive systems. These can be transformed into cooperative systems by changing the direction of time. However for a given choice of time direction the competitive systems need not have the pleasant properties of cooperative systems. Another simple type of coordinate transformation is to reverse the signs of some of the coordinates x_i. When can this be used to transform a given system into a monotone one?. Two necessary conditions are that each partial derivative of a component of f must have a (non-strict) sign which is independent of x and that the derivatives are symmetric under interchange of their indices. What remains is a condition which can be expressed in terms of the so-called species graph. This has one node for each variable x_i and an arrow from node i to node j if \frac{\partial f_j}{\partial x_i} is not identically zero. If the derivative is positive the arrow bears a positive sign and if it is negative a negative sign. Alternatively, the arrows with positive sign have a normal arrowhead while those with negative sign have a blunt end. In this way the system gives rise to a labelled oriented graph. To each (not necessarily oriented) path in the graph we associate a sign which is the product of the signs of the individual edges composing the path. The graph is said to be consistent if signs can be associated to the vertices in such a way that the sign of an edge is always the product of the signs of its endpoints. This is equivalent to the condition that every closed loop in the graph has a positive sign. In other words, every feedback in the system is positive. Given that the other two necessary conditions are satisfied the condition of consistency characterizes those networks which can be transformed by changes of sign of the x_i to a monotone system. A transformation of this type can also be thought of as replacing the positive orthant by another orthant as the cone defining the partial order.

Next I consider some examples. Every one-dimensional system is monotone. In a two-dimensional system we can have the sign patterns (+,+), (-,-) and (+,-). In the first case the system is monotone. In the second case it is not but can be made so by reversing the sign of one of the coordinates. This is the case of a two-dimensional competitive system. In the third case the system cannot be made monotone. A three-dimensional competitive system cannot be made monotone. The species graph contains a negative loop. Higher dimensional competitive systems are no better since their graphs all contain copies of that negative loop.

A general message in Sontag’s paper is that consistent systems tend to be particularly robust to various types of disturbances. Large biochemical networks are in general not consistent in this sense but they are close to being consistent in the sense that removing a few edges from the network make them consistent. This also means that they can be thought of as a few consistent subsystems joined together. Since biological systems need robustness this suggests a topological property which biochemical networks should have compared to random networks. Sontag presents an example where this has been observed in the transcription network of yeast.

A more sophisticated method which can often be used to obtain monotone systems from systems of chemical reactions by a change of variables has been discussed in a previous post. The advantage of this is that together with other conditions it can be use to show that generic solutions, or sometimes even all solutions, of the original system converge to stationary solutions.

Geometric singular perturbation theory

November 29, 2013

I have already written two posts about the Michaelis-Menten limit, one not very long ago. I found some old results on this subject and I was on the look-out for some more modern accounts. Now it seems to me that what I need is something called geometric singular perturbation theory which goes back to a paper of Fenichel (J. Diff. Eq. 31, 53). An interesting aspect of this is that it involves using purely geometric statements to help solve analytical problems. If we take the system of two equations given in my last post on this subject, we can reformulate them by introducing a new time coordinate \tau=t/\epsilon, called the fast time, and adding the parameter as a new variable with zero time derivative. This gives the equations x'=\epsilon f(x,y), y'=g(x,y) and \epsilon'=0, where the prime denotes the derivative with respect to \tau. We are interested in the situation where the equation g(x,y)=0, which follows from the equations written in terms of the original time coordinate t, is equivalent to y=h_0(x) for a smooth function h_0. The linearization of the system in \tau along the zero set of g automatically has at least two zero eigenvalues. For Fenichel’s theorem it should be assumed that it does not have any more zero (or purely imaginary) eigenvalues. Then each point on that manifold has a two-dimensional centre manifold. Fenichel proves that there exists one manifold which is a centre manifold for all of these points. This is sometimes called a slow manifold. (Sometimes the part of it for a fixed value of \epsilon is given that name.) Its intersection with the plane \epsilon=0  coincides with the zero set of g. The original equations have a singular limit as \epsilon tends to zero, because \epsilon multiplies the time derivative in one of the equations. The remarkable thing is that the restriction of the system to the slow manifold is regular. This allows statements to be made that qualitative properties of the dynamics of solutions of the system with \epsilon=0 are inherited by the system with \epsilon small but non-zero.

Due to my growing interest in this subject I invited Peter Szmolyan from Vienna,who is a leading expert in this field, to come and give a colloquium here in Mainz, which he did yesterday. One of his main themes was that in many models arising in applications the splitting into the variables x and y cannot be done globally. Instead it may be necessary to use several splittings to describe different parts of the dynamics of one solution. He discussed two examples in which these ideas are helpful for understanding the dynamics better and establishing the existence of relaxation oscillations. The first is a model of Goldbeter and Lefever (Biophys J. 12, 1302) for glycolysis. It is different from the model I mentioned in a previous post but is also an important part of the chapter of Goldbeter’s book which I discussed there. The model of Goldbeter and Lefever was further studied theoretically by Segel and Goldbeter (J. Math. Biol. 32, 147). On this basis a rigorous analysis of the dynamics including a proof of the existence of relaxation oscillations was given in a recent paper by Szmolyan and Ilona Kosiuk (SIAM J. Appl. Dyn. Sys. 10, 1307). The other main example in the talk was a system of equations due to Goldbeter which is a kind of minimal model for the cell cycle. It is discussed in chapter 9 of Goldbeter’s book.

I have the feeling that GSPT is a body of theory which could be very useful for my future work and so I will do my best to continue to educate myself on the subject.

The Michaelis-Menten limit

July 2, 2013

In a previous post I wrote about the Michaelis-Menten reduction of reactions catalysed by enzymes in which a single equation (effective equation) is the limit of a system of two equations (extended equations) as a parameter \epsilon tends to zero. What I did not talk about is the sense in which solutions of the effective equation approximate those of the extended ones. I was sure that this must be well-known but I did not know a source for it. Now I discovered that what I had been seeking is to be found in a very nice form in a book which had been standing on a shelf in my office for many years. This is the book ‘Asymptotic Expansions for Ordinary Differential Equations’ by Wolfgang Wasow and the part of relevance to Michaelis-Menten reduction starts on p. 249. Michaelis-Menten is not mentioned there but the key mathematical result is exactly what is needed for that application. The theorem is due to Tikhonov but the original paper is in Russian and so not accessible to me. For convenience I repeat the equations from the previous post on this subject. \dot u=f(u,v),\epsilon\dot v= g(u,v). This is the type of system treated in Tikhonov’s theorem, including the possibility that u and v are vector-valued.

The statement of the theorem is as follows. On any finite time interval [0,T] the function u in the extended system converges uniformly to the solution of the reduced system as \epsilon\to 0. Given a solution of the reduced system it is possible to compute a corresponding function v. On the time interval (0,T] the function v in the extended system converges to the function v coming from the reduced system uniformly on compact subsets. Of course this conclusion  requires some hypotheses on the functions f and g. The key thing is that for a fixed value of u we have an asymptotically stable stationary solution of the equation for v (with \epsilon\ne 0).

With this result in hand it is possible to compute higher order corrections in \epsilon. This was first done by Vasileva and is also explained in the book of Wasow. The result was extended to a statement global in t by Hoppensteadt, Trans. Amer. Math. Soc. 123, 521. I expect that there are more modern treatments of these things in the literature but I find the sources quoted here very helpful for the beginner like myself. There remains the question of the relation to the usual Michaelis-Menten procedure. This is nicely discussed in a paper by Heineken et. al., Math. Biosci. 1, 95.

Population dynamics and chemical reactions

June 21, 2013

The seminar which I mentioned in a recent post has caused me to go back and look carefully at a number of different models in biology and chemistry. It has happened repeatedly that I felt I could glimpse some mathematical relations between the models. Now I have spent some time pursuing these ideas. One aspect is that many of the systems of ODE coming from biological models can be thought of as arising from chemical reaction networks with mass action kinetics, even when the unknowns are not chemical concentrations. In this context it should be mentioned that if an ODE system arises in this way the chemical network which leads to it need not be unique.

The first example I want to mention is the Lotka-Volterra system. Today it is usually presented as a model of population dynamics. Often the example of lynx and hares is used and this is natural due to the intrinsic attractiveness of furry animals. The story of Volterra and his son in law also has a certain human interest. The fact that Lotka found the equations earlier is usually just a side comment. In any case, the population model is equivalent to an ODE system coming from a reaction network which was described by Lotka in a paper in 1920 (J. Amer. Chem. Soc. 42, 1595). The network is defined by the reactions A_1\to 2A_1, A_1+A_2\to 2A_2, A_2\to 0 and A_1+A_2\to 0. The last entry in the list can be thought of as an alternative reaction producing another substance which is not included explicitly in the model. A simpler version, also considered in Lotka’s paper, omits this last reaction. In his book ‘Mathematical aspects of reacting and diffusing systems’ Paul Fife looks at the second system from the point of view of chemical reaction network theory. He computes its deficiency \delta in the sense of CRNT to be one. It has three linkage classes. The second model also has deficiency one. All the linkage classes have deficiency zero and so the deficiency one theorem does not apply. The chemical system introduced by Lotka was not supposed to correspond to a system of real reactions. He was just looking for a hypothetical reaction which would exhibit sustained oscillations.

Next I consider the fundamental model of virus dynamics as given in the book of Nowak and May which has previously been mentioned in this blog. Something which I only noticed now is that in a sense there is a term missing from the model. This represents the fact that when a virion enters a cell to infect it that virion is removed from the virus population. This fact is apparently not mentioned in the book. In an alternative model discussed in a paper of Perelson and Nelson (SIAM Rev. 41, 3) they also omit this term and discuss possible justifications for doing so. The fundamental model as found in the book of Nowak and May can be interpreted as the equations coming from a network of chemical reactions. This is also true of the modified version where the missing term is replaced. Both systems (at least the ones I found) have deficiency two.

Several well-known models in epidemiology can also be obtained from chemical networks. For instance the SIR model can be obtained from the reactions S+I\to 2I and I\to 0. This network has deficiency zero and is not weakly reversible. The deficiency zero theorem applies and tells us that there is no equilibrium. Of course this fact is nothing new for this example. The SIS model is similar but in that case the system has deficiency one and a positive stationary solution exists for certain parameter values. You might complain that the games I am playing do not lead to useful insights and you may be right. Nevertheless, seeing analogies between apparently unrelated things is a notorious strength of mathematics. There is also one success story related to the things I have been talking about here, namely the work of Korobeinikov on the standard model of virus dynamics mentioned in a previous post. He imported a Lyapunov function of a type known for epidemiological models in order to prove the global asymptotic stability of stationary solutions of the fundamental model of virus dynamics.

Modelling the Calvin cycle

March 18, 2013

Some years ago the Max Planck Institute for Molecular Plant Physiology organized a conference on metabolic networks. I decided to see what was going on in the institute next to the one where I work and I went to some of the talks. The one which I found most interesting was by Zoran Nikoloski. His subject was certain models for the Calvin cycle, which is part of photosynthesis. A motivating question was whether photosynthesis can work in two different stable steady states. If that were the case it might be possible to influence the plant to move from one state to another and, in the best case, to increase its production of biomass. This is of interest for biotechnology. Mathematically the question is that of multistationarity, i.e. whether a system of evolution equations admits more than one stationary solution. Beyond this it is of interest whether there can be more than one stable stationary solution. In fact in this context the issue is not that of absolute uniqueness of stationary solutions but of uniqueness within a given stoichiometric compatibility class. This means that the solution is unique when certain conserved quantities are fixed. One thing I found attractive about the presentation was that the speaker was talking about rigorous mathematical results on the dynamics and not just about numerically calculating a few solutions.

If the system is modelled deterministically and diffusion is neglected there results a system of ordinary differential equations for the concentrations of the substances involved as functions of time. It is necessary to choose which substances should be included in the description. In a basic model of the Calvin cycle there are five substances. In the work discussed in the talk of Nikoloski and in a paper he wrote with Sergio Grimbs and others (Biosystems 303, 212) various ODE systems based on this starting point are considered. They differ by the type of kinetics used. They consider mass action kinetics (MA), extended Michaelis-Menten kinetics where the enzymes catalysing the reactions are included explicitly (MM-MA) and effective Michaelis-Menten (MM) obtained from the system MM-MA by a singular limit. The systems MA and MM consist of five equations while the system MM-MA consists of nineteen equations. In the paper of Grimbs et. al. they show among other things that the system MM never admits a stable stationary solution, whatever the reaction constants, while the system MM-MA can exhibit two different stationary solutions.

After the talk I started reading about this subject and I also talked to Nikoloski about it. Later I began doing some research on these systems myself. Some technical difficulties which arose (which I wrote about in a previous post) led me to consult Juan Velázquez and he joined me in this project. Now we have written a paper on models for the Calvin cycle. In a case where there is only one stationary solution and it is unstable it is of interest to consider the final fate of general solutions of the system. For some initial conditions the concentrations of all substances tend to zero at late times. For other data (a whole open set) we were able to show that all concentrations tend to infinity as t\to\infty. We called the latter class runaway solutions. These do not seem to be of direct biological relevance but they might be helpful in choosing between alternative models which are more or less appropriate. The proof of the existence of runaway solutions for the MA system is somewhat complicated since this turns out to be a system with two different timescales. The system MM-MA also admits runaway solutions. Although the system is larger than MA the existence proof is simpler and in fact can be carried out in the context of a larger class of systems. Runaway solutions are also found for the system MM.

In the paper of Grimbs et. al. one system is considered which includes the effect of diffusion. Restricting to homogeneous solutions of this system gives a system of ODE called MAdh which is different from the system MA. The difference is that while the concentration of ATP is a dynamical variable in MAdh it is taken to be constant in MA. We showed that the system MAdh has zero, one or two solutions depending on the values of the parameters and that all solutions are bounded. Thus runaway solutions are ruled out. Intuitively this is due to the fact that the supply of energy is bounded but this heuristic argument is far from providing a proof. There are many other models of the Calvin cycle in the literature. In general they consider the reactions between a larger class of substances. It is an interesting task for the future to extend the results obtained up to now to these more general models. This post has been very much concerned with the mathematics of the problem and has not said much about the biology. The reactions making up the Calvin cycle were determined experimentally by Melvin Calvin and I can highly recommend his Nobel lecture as a description of how this was achieved

Absolute concentration robustness

February 20, 2013

In the past years I have been on the committees for many PhD examinations. A few days ago, for the first time, I was was on the committee for a thesis on a subject belonging to the area of mathematical biology. This was the thesis of Jost Neigenfind and it was concerned with a concept called absolute concentration robustness (ACR).

The concentration of a given substance in cells of a given type varies widely between the individual cells. (Cf. also this previous post). It is of interest to identify mechanisms which can ensure that the steady state concentration of a particular substance is independent of initial data. (This is a way in which the output of a system can be independent of background variation.) In saying this I am assuming implicitly that more general solutions converge to steady states. A more satisfactory formulation can be obtained as follows. In a chemical reaction network there are usually a number of conserved quanitities, say C_\alpha. These define affine subspaces of the state space, the stoichiometric compatibility classes. For many systems there is exactly one stationary solution in each stoichiometric compatibility class. The condition of interest here is that the value of one of the concentrations, call it x_1, in the steady state solution is independent of the parameters C_\alpha. (The other concentrations x_i,i>1 will in general depend on the C_\alpha.) This property is ACR. I first heard of this in a talk by Uri Alon at the SMB conference in Krakow in the summer of 2011. The basic idea is explained clearly in a paper of Shinar and Feinberg (Science 327, 1389). They present a general theoretical approach but also describe some biological systems where ACR (in a suitable approximate sense) has been observed experimentally. In the terminology of Chemical Reaction Network Theory (CRNT) the examples they discuss have deficiency one. They mention that ACR is impossible in systems of deficiency zero. There is no reason why it should not occur in systems of deficiency greater than one but in those more complicated dynamics make it more difficult to decide whether the property holds or not.

The result of Shinar and Feinberg only covers a class of reaction networks which is probably very restricted. What Neigenfind does in his thesis is to develop more general criteria for ACR and computer algorithms which can check these criteria for given systems. The phenomenon of ACR is interesting since it is a feature which may be more common in reaction systems coming from biology than in generic systems. At least there is a good potential reason why this might be the case.


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