Archive for January, 2016

The advanced deficiency algorithm

January 23, 2016

Here I discuss another tool for analysing chemical reaction networks of deficiency greater than one. This is the Advanced Deficiency Algorithm developed by Feinberg and Ellison. It seems that the only direct reference for the mathematical proofs is Ellison’s PhD thesis. There is a later PhD thesis by Haixia Ji in which she introduces an extension of this called the Higher Deficiency Algorithm and where some of the ideas of Ellison are also recapitulated. In my lecture course, which ends next week, I will only have time to discuss the structure of the algorithm and give an extended example without proving much.

The Advanced Deficiency Algorithm has a general structure which is similar to that of the Deficiency One Algorithm. In some cases it can rule out multistationarity. Otherwise it gives rise to several sets of inequalities. If one of these has a solution then there is multistationarity and if none of them does there is no multistationarity. It is not clear to me if this is really an algorithm which is guaranteed to give a diagostic test in all cases. I think that this is probably not the case and that one of the themes of Ji’s thesis is trying to improve on this. An important feature of this algorithm is that the inequalities it produces are in general nonlinear and thus may be much more difficult to analyse than the linear inequalities obtained in the case of the Deficiency One Algorithm.

Now I have come to the end of my survey of deficiency theory for chemical reaction networks. I feel I have learned a lot and now is the time to profit from that by applying these techniques. The obvious next step is to try out the techniques on some of my favourite biological examples. Even if the result is only that I see why the techniques do not give anything interesting in this cases it will be useful to understand why. Of course I hope that I will also find some positive results.

Elementary flux modes

January 16, 2016

Elementary flux modes are a tool which can be used to take a reaction network of deficiency greater than one and produce subnetworks of deficiency one. If it can be shown that one of these subnetworks admits multistationarity then it can sometimes be concluded that the original network also does so. There are two conditions that need to be checked in order to allow this conclusion. The first is that the network under consideration must satisfy the condition t=l. The second is that genericity conditions must be satisfied which consist of requiring the non-vanishing of the determinants of certain matrices. If none of the subnetworks admit multistationarity then no conclusion is obtained for the full network. The core element of the proof is an application of the implicit function theorem. These conclusions are contained in a paper of Conradi et. al. (Proc. Nat. Acad. Sci. USA 104, 19175).

One of the ways of writing the condition for stationary solutions is Nv(c)=0, where as usual N is the stoichiometric matrix and v(c) is the vector of reaction rates. Since v(c) is positive this means that we are looking for a positive element of the kernel of N. This suggests that it is interesting to look at the cone K which is the intersection of the kernel of N with the non-negative orthant. According to a general theorem of convex analysis K consists of the linear combinations with non-negative coefficients of a finite set of vectors which have a mininum (non-zero) number of non-zero components. In the case of reaction networks these are the elementary flux modes. Recalling that N=YI_a we see that positive vectors in the kernel of the incidence matrix I_a are a special type of elementary flux modes. Those which are not in the kernel of I_a are called stoichiometric generators. Each stoichiometric generator defines a subnetwork where those reaction constants of the full network are set to zero where the corresponding reactions are not in the support of the generator. It is these subnetworks which are the ones mentioned above in the context of multistationarity. The application of the implicit function theorem involves using a linear transformation to introduce adapted coordinates. Roughly speaking the new coordinates are of three types. The first are conserved quantities for the full network. The second are additional conserved quantities for the subnetwork, complementing those of the full network. Finally the third type represents quantities which are dynamical even for the subnetwork.

Here are some simple examples. In the extended Michaelis-Menten description of a single reaction there is just one elementary flux mode (up to multiplication by a positive constant) and it is not a stoichiometric generator. In the case of the simple futile cycle there are three elementary flux modes. Two of these, which are not stoichoimetric generators correspond to the binding and unbinding of one of the enzymes with its substrate. The third is a stoichoimetric generator and the associated subnetwork is obtained by removing the reactions where a substrate-enzyme complex dissociates into its original components. The dual futile cycle has four elementary flux modes of which two are stoichiometric generators. In the latter case we get the (stoichiometric generators of the) two simple futile cycles contained in the network. Of course these are not helpful for proving multistationarity. Another type of example is given by the simple models for the Calvin cycle which I have discussed elsewhere. The MA system considered there has two stoichiometric generators with components (3,6,6,1,3,0,1) and (3,5,5,1,3,1,0). I got these by working backwards from corresponding modes for the MM-MA system found by Grimbs et. al. This is purely on the level of a vague analogy. I wish I had a better understanding of how to get this type of answer more directly. Those authors used those modes to prove bistability for the MM-MA system so that this is an example where this machinery produces real results.

The German Dr. House

January 14, 2016

The central figure in the American TV series Dr. House is a doctor who is brilliant in the diagnosis of unusual medical conditions but personally very difficult. When I first saw this series I found the character so unpleasant that I did not want to watch the programme. However in the course of time I got drawn in to watching it by the interest of the medical content. While some aspects of this series are quite exaggerated and far from reality the medical parts are very accurate and well researched. As I learned yesterday even details seen there like the numbers on heart monitors accurately reflect the situation being portrayed. I have this information from a lecture I attended yesterday at the Medizinische Gesellschaft Mainz [Mainz Medical Society]. The speaker was Professor Jürgen Schäfer, a man who has become known in the media as the German Dr. House. I am pleased to report that I detected no trace of the social incompetence of Dr. House in Dr. Schäfer.

Jürgen Schäfer is trained as a cardiologist. He and his wife, who is a gastroenterologist, got so interested by the series Dr. House that they would spend time discussing the details of the diagnoses and researching the background after they has seen each programme. Then Schäfer had the idea that he could use Dr. House in his lectures at the University of Marburg. The first obstacle was to know if he could legally make use of this material. After a casual conversation with one of his patients who is a lawyer he contacted the necessary people and signed a suitable contract. At this time his project attracted considerable attention in the media even before it had started. In the lectures he analyses the cases occurring in the series. The students are encouraged to develop their own diagnoses in dialogue with the professor. These lectures are held in the evenings and are very popular with the students. In the evaluations the highest score was obtained for the statement that ‘the lectures are a lot of fun’.

This is only the start of the story. During a consultation in one of the episodes of Dr. House he suddenly makes a deep cut with a scalpel in the body of the patient (one of the melodramatic elements), opens the wound and shows that the flesh inside is black. The diagnosis is cobalt poisoning. After seeing this it occurred to Dr. Schäfer that this diagnosis might also apply to one of his own patients and this turned out to be true. In addition to serious heart problems this patient was becoming blind and deaf. He had had a hip joint replacement with an implant made of a ceramic material. At some point this became damaged and was replaced. In order to try to avoid the implant breaking again the new one was made of metal. The old implant fragmented and left splitters in the body. These had acted like sandpaper on the new joint and at the time of removal it had been reduced to 70% of its original size by this process. As a result large quantities of cobalt was released, resulting in the poisoning. The speaker showed a picture of the operation of another of his patients with a similar problem where the wound could be seen to be filled with a black oily liquid. Together with colleagues Schäfer published an account of this case in The Lancet with the title ‘Cobalt intoxication diagnosed with the help of Dr. House’. Not all his coauthors were happy with this title but Schäfer wanted to acknowledge his debt to the series. At the same time it was a great piece of advertizing for him which lead to a lot of attention in the international media.

Due to his growing fame Schäfer started to get a lot of letters from patients with mysterious illnesses. This was more than he could handle. He informed the administration of the university clinic where he worked that he was going to start sending back letters of this type unopened, since he just did not have the time to cope with them. To his surprise they wanted him to continue with this work and arranged from him to be relieved from other duties. They set up a new institute for him called Zentrum für unerkannte Krankheiten [centre for unrecognized diseases]. This was perhaps particularly surprising since this is a privately funded clinic and the work of this institute costs money rather than making money. The techniques used there include toxicological and genomic analyses.

Here is another example from the lecture. Schäfer’s institute uses large scale DNA analysis to screen for a broad range of parasites in patients with unclear symptoms. In one patient they found DNA of the parasite causing schistosomiasis. This disease is usually got by bathing in infected water in tropical or subtropical areas. The patient tested negatively for the parasite and had never been to a place where this disease occurs. The mystery was cleared up due to the help of a vet of Egyptian origin. He was familiar with schistosomiasis and due to his experience with large animals he was not afraid of analysing very large stool samples. He succeeded in finding eggs of the parasite in the patient’s stool. The diffculty was that the numbers of eggs were very low and that for certain reasons they were difficult to recognise in this case, except by an expert. The patient was treated for schistosomiasis as soon as the genetic results were available but it was very satisfying to have a confirmation by more classical techniques. The mystery of how the patient got infected was solved as follows. As a hobby he kept lots of fish and he imported these from tropical regions. The infection presumably came from the water in his aquarium. We see that in the modern world it is easy to import tropical diseases by express delivery after placing an order in the internet

I do not want to end before mentioning that Schäfer said something nice about how mathematicians can help medical doctors. He had a patient who is a mathematics professor and had the following problem. From time to time he would collapse and was temporarily paralysed although fully conscious. A possible explanation for this would have been an excessively high level of sodium in the body. On the other hand measurements showed that the concentration of sodium in his blood was normal, even after an attack. The patient then did a calculation (just simple arithmetic). On the basis of known data he worked out the amount of sodium and potassium in different types of food and noted a correlation between negative effects of a food on his health and the ratio of the sodium to potassium concentrations. This supported the hypothesis of sodium as a cause and encouraged the doctors to look more deeply into the matter. It turned out that in this type of disease the sodium is concentrated near the cell membrane and cannot be seen in the blood. A genetic analysis revealed that the patient had a mutation in a little-known sodium channel.

I think that this lecture was very entertaining for the audience, including my wife and myself. However this is not just entertainment. With his institute Schäfer is providing essential help for many people in very difficult situations. He has files of over 4000 patients. This kind of work requires a high investment in time and money which is not possible for a usual university clinic, not to mention an ordinary GP. It is nevertheless the case that Schäfer is developing resources which could be used more widely, such as standard protocols for assessing patients of this type. As he emphasized, while by definition a rare disease only effects a small number of patients the collection of all rare diseases together affects a large number of people. If more money was invested in this kind of research it could result in  a net saving for the health system since it would reduce the number of people running from one doctor to another since they do not have a diagnosis.

The deficiency one algorithm

January 7, 2016

Here I continue the discussion of Chemical Reaction Network Theory with the Deficiency One Algorithm. As its name suggests this is an algorithm for investigating whether a reaction network allows multistationarity or not. It is also associated to a theorem. This theorem says that the satisfaction of certain inequalities associated to a reaction network is equivalent to the existence of a choice of reaction constants for which the corresponding system of equations with mass action kinetics has two distinct positive stationary solutions in a stoichiometric compatibility class. This is true in the context of networks of deficiency one which satisfy some extra conditions. The property of satisfying these, called regularity, seems to be relatively weak. In fact, as in the case of the Deficiency One Theorem, there is a second related result where a pair of zeroes of the right hand side f in a stoichiometric compatibility class is replaced by a vector in the kernel of Df which is tangent to that class.

An example where this theory can be applied is double phosphorylation in the processive case. The paper of Conradi et. al. cited in the last post contains the statement that in this system the Deficiency One Algorithm implies that multistationarity is not possible. For this it refers to the Chemical Reaction Network Toolbox, a software package which implements the calculations of the theorem. In my course I decided for illustrative purposes to carry out these calculations by hand. It turned out not to be very hard. The conclusion is that multistationarity is not possible for this system but the general machinery does not address the question of whether there are any positive stationary solutions. I showed that this is the case by checking by hand that \omega-limit points on the boundary of the positive orthant are impossible. The conclusion then follows by a well-known argument based on the Brouwer fixed point theorem. This little bit of practical experience with the Deficiency One Algorithm gave me the impression that it is really a powerful tool. At this point it is interesting to note a cross connection to another subject which I have discussed in this blog. It is a model for the Calvin cycle introduced by Grimbs et. al. These authors noted that the Deficiency One Algorithm can be applied to this system to show that it does not allow multistationarity. They do not present the explicit calculations but I found that they are not difficult to do. In this case the equations for stationary solutions can be solved explicitly so that using this tool is a bit of an overkill. It neverless shows the technique at work in another example.

Regularity consists of three conditions. The first (R1) is a necessary condition that there be any positive solutions at all. If it fails we get a strong conclusion. The second (R2) is the condition l=t familiar from the Deficiency One Theorem. (R3) is a purely graph theoretic condition on the network. A weakly reversible network which satisfies (R3) is reversible. Reading this backwards, if a network is weakly reversible but not reversible then the theorem does not apply. The inequalities in the theorem depend on certain partitions of a certain set of complexes (the reactive complexes) into three subsets. What is important is whether the inequalities hold for all partitions of a network or whether there is at least one partition for which they do not hold. The proof of the theorem uses a special basis of the kernel of YA_\kappa where \kappa is a function on \cal C constructed from the reaction constants. In the context of the theorem this space has dimension t+1 and t of the basis vectors come from a special basis of A_\kappa of the type which already comes up in the proof of the Deficiency Zero Theorem.

An obvious restriction on the applicability of the Deficiency One Algorithm is that it is only applicable to networks of deficiency one. What can be done with networks of higher deficiency? One alternative is the Advanced Deficiency Algorithm, which is implemented in the Chemical Reaction Network Toolbox. A complaint about this method which I have seen several times in the literature is that it is not able to treat large systems – apparently the algorithm becomes unmanageable. Another alternative uses the notion of elementary flux modes which is the next topic I will cover in my course. It is a way of producing certain subnetworks of deficiency one from a given network of higher deficiency. The subnetworks satisfy all the conditions needed to apply the Deficiency One Algorithm except perhaps t=l.