Immunotherapy for cancer

A promising innovative approach to cancer therapy is to try to persuade the immune system to attack cancer cells effectively. The immune system does kill cancer cells and presumably removes many tumours which we never suspect we had. At the same time established tumours are able to successfully resist this type of attack in many cases. The idea of taking advantage of the immune system in this way is an old one but it took a long time before it became successful enough to reach the stage of an approved drug. This goal was achieved with the approval of ipilimumab for the treatment of melanoma by the FDA in 2011. This drug is a monoclonal antibody which binds the molecule CTLA4 occurring on the surface of T cells.

To explain the background to this treatment I first recall some facts about T cells. T cells are white blood cells which recognize foreign substances (antigens) in the body. The antigen binds to a molecule called the T cell receptor on the surface of the cell and this gives the T cell an activation signal. Since an inappropriate activation of the immune system could be very harmful there are built-in safety mechanisms. In order to be effective the primary activation signal has to be delivered together with a kind of certificate that action is really necessary. This is a second signal which is given via another surface molecule on the T cell, CD28. The T cell receptor only binds to an antigen when the latter is presented on the surface of another cell (an antigen-presenting cell, APC) in a groove within another molecule, an MHC molecule (major histocompatibility complex). On the surface of the APC there are under appropriate circumstances other molecules called B7.1 and B7.2 which can bind to CD28 and give the second signal. Once this has happened the activated T cell takes appropriate action. What this is depends on the type of T cell involved but for a cytotoxic T cell (one which carries the surface molecule CD8) it means that the T cell kills cells presenting the antigen. If the cell was a virus-infected cell and the antigen is derived from the virus then this is exactly what is desired. Coming back to the safety mechanisms, it is not only important that the T cell is not erroneously switched on. It is also important that when it is switched on in a justified case it should also be switched off after a certain time. Having it switched on for an unlimited time would never be justified. This is where CTLA4 comes in. This protein can bind to B7.1 and B7.2 and in fact does so more strongly than CD28. Thus it can crowd out CD28 and switch off the second signal. By binding to CTLA4 the antibody in ipilimumab stops it from binding to B7.1 and B7.2, thus leaving the activated T cell switched on. In some cases cancer cells present unusual antigens and become a target for T cells. The killing of these cells can be increased by CTLA4 via the mechanism just explained. At this point I should say that it may not be quite clear whether this is really the mechanism of action of CTLA4 in causing tumours to shrink. Alternative possibilities are mentioned in the Wikipedia article on CTLA4.

There are various things which have contributed to my interest in this subject. One is lectures I heard in the series ‘Universität im Rathaus’ [University in the Town Hall] in Mainz last February. The speakers were Matthias Theobald and Ugur Sahin and the theme was personalized cancer medicine. The central theme of what they were talking about is one step beyond what I have just sketched. A weakness of the therapy using antibodies to CTLA4 or the related approach using antibodies to another molecule PD-1 is that they are unspecific. In other words they lead to an increase not only in the activity of the T cells specific to cancer cells but of all T cells which have been activated by some antigen. This means that serious side effects are very likely. An approach which is theoretically better but as yet in a relatively early stage of development is to produce T cells which are specific for antigens belonging to the tumour of a specific patient and for an MHC molecule of that patient capable of presenting that antigen. From the talk I had the impression that doing this requires a lot of input from bioinformatics but I was not able to understand what kind of input it is. I would like to know more about that. Coming back to CTLA4, I have been interested for some time in modelling the activation of T cells and in that context it would be natural to think about also modelling the deactivating effects of CTLA4 or PD-1. I do not know whether this has been tried.

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Oscillations in the MAP kinase cascade

In a recent post I mentioned my work with Juliette Hell on the existence of oscillations in the Huang-Ferrell model for the MAP kinase cascade. We recently put our paper on the subject on ArXiv. The starting point of this project was the numerical and neuristic work of Qiao et. al., PLoS Comp. Biol. 3, 1819. Within their framework these authors did an extensive search of parameter space and found Hopf bifurcations and periodic solutions for many parameters. The size of the system is sufficiently large that it represents a significant obstacle to analytical investigations. One way of improving this situation is to pass to a limiting system (MM system) by a Michaelis-Menten reduction. In addition it turns out that the periodic solutions already occur in a truncated system consisting of the first two layers of the cascade. This leaves one layer with a single phosphorylation and one with a double phosphorylation. In a previous paper we had shown how to do Michaelis-Menten reduction for the truncated system. Now we have generalized this to the full cascade. In the truncated system the MM system is of dimension three, which is still quite convenient for doing bifurcation theory. Without truncation the MM system is of dimension five, which is already much more difficult. It is however possible to represent the system for the truncated cascade as a (singular) limit of that for the full cascade and thus transport information from the truncated to the full cascade.

Consider the MM system for the truncated cascade. The aim is then to find a Hopf bifurcation in a three-dimensional dynamical system with a lot of free parameters. Because of the many parameters is it not difficult to find a large class of stationary solutions. The strategy is then to linearize the right hand side of the equations about these stationary solutions and try show that there are parameter values where a suitable bifurcation takes place. To do this we would like to control the eigenvalues of the linearization, showing that it can happen that at some point one pair of complex conjugate eigenvalues passes through the imaginary axis with non-zero velocity as a parameter is varied, while the remaining eigenvalue has non-zero real part. The behaviour of the eigenvalues can largely be controlled by the trace, the determinant and an additional Hurwitz determinant. It suffices to arrange that there is a point where the trace is negative, the determinant is zero and the Hurwitz quantity passes through zero with non-zero velocity. This we did. A superficially similar situation is obtained by modelling an in vitro model for the MAPK cascade due to Prabakaran, Gunawardena and Sontag mentioned in a previous post in a way strictly analogous to that done in the Huang-Ferrell model. In that case the layers are in the opposite order and a crucial sign is changed. Up to now we have not been able to show the existence of a Hopf bifurcation in that system and our attempts up to now suggest that there may be a real obstruction to doing so. It should be mentioned that the known necessary condition for a stable hyperbolic periodic solution, the existence of a negative feedback loop, is satisfied by this system.

Now I will say some more about the model of Prabakaran et. al. Its purpose is to obtain insights on the issue of network reconstruction. Here is a summary of some things I understood. The in vitro biological system considered in the paper is a kind of simplification of the Raf-MEK-ERK MAPK cascade. By the use of certain mutations a situation is obtained where Raf is constitutively active and where ERK can only be phosphorylated once, instead of twice as in vivo. This comes down to a system containing only the second and third layers of the MAPK cascade with the length of the third layer reduced from three to two phosphorylation states. The second layer is modelled using simple mass action (MA) kinetics with the two phosphorylation steps being treated as one while in the third layer the enzyme concentrations are included explicitly in the dynamics in a standard Michaelis-Menten way (MM-MA). The resulting mathematical model is a system of seven ODE with three conservation laws. In the paper it is shown that for given values of the conserved quantities the system has a unique steady state. This is an application of a theorem of Angeli and Sontag. Note that this is not the same system of equations as the system analogous to that of Huang-Ferrell mentioned above.

The idea now is to vary one of the conserved quantities and monitor the behaviour of two functions and of the unknowns of the system at steady state. It is shown that for one choice of the conserved quantity and change in the same direction while for a different choice of the conserved quantity they change in opposite directions when the conserved quantity is varied. From a mathematical point of view this is not very surprising since there is no obvious reason forbidding behaviour of this kind. The significance of the result is that apparently biologists often use this type of variation in experiments to reach conclusions about causal relationships between the concentrations of different substances (activation and inhibition), which can be represented by certain signed oriented graphs. In this context ‘network reconstruction’ is the process of determining a graph of this type. The main conclusion of the paper, as I understand it, is that doing different experiments can lead to inconsistent results for this graph. Note that there is perfect agreement between the experimental results in the paper and the results obtained from the mathematical model. In a biological system if two experiments give conflicting results it is always possible to offer the explanation that some additional substance which was not included in the model is responsible for the difference. The advantage of the in vitro model is that there are no other substances which could play that role.

Models for photosynthesis, part 3

Here I continue the discussion of models for photosynthesis in two previous posts. There I described the Pettersson and Poolman models and indicated the possibility of introducing variants of these which use exclusively mass action kinetics. I call these the Pettersson-MA and Poolman-MA models. I was interested in obtaining information about the qualitative behaviour of solutions of these ODE systems. This gave rise to the MSc project of Dorothea Möhring which she recently completed successfully. Now we have extended this work a little further and have written up the results in a paper which has just been uploaded to ArXiv. The central issue is that of overload breakdown which is related to the mathematical notion of persistence. We would like to know under what circumstances a positive solution can have -limit points where some concentrations vanish and, if so, which concentrations vanish in that case. It seems that there was almost no information on the latter point in the literature so that the question of what exactly overload breakdown is remained a bit nebulous. The general idea is that the Pettersson model should have a stronger tendency to undergo overload breakdown while the Poolman model should have a stronger tendency to avoid it. The Pettersson-MA and Poolman-MA models represent a simpler context to work in to start with.

For the Pettersson-MA model we were able to identify a regime in which overload breakdown takes place. This is where the initial concentrations of all sugar phosphates and inorganic phosphate in the chloroplast are sufficiently small. In that case the concentrations of all sugar phosphates tend to zero at late times with two exceptions. The concentrations of xylulose-4-phosphate and sedoheptulose-7-phosphate do not tend to zero. These results are obtained by linearizing the system around a simple stationary solution on the boundary and applying the centre manifold theorem. Another result is that if the reaction constants satisfy a certain inequality a positive solution can have no positive -limit points. In particular, there are no positive stationary solutions in that case. This is proved using a Lyapunov function related to the total number of carbon atoms. In the case of the Poolman-MA model it was shown that the stationary point which was stable in the Pettersson case becomes unstable. Moreover, a quantitative lower bound for concentration of sugar phosphates at late times in obtained.These results fit well with the intuitive picture of what should happen. Some of the results on the Poolman-MA model can be extended to analogous ones for the original Poolman model. On the other hand the task of giving a full rigorous definition of the Pettersson model was postponed for later work. The direction in which this could go has been sketched in a previous post.

There remains a lot to be done. It is possible to define a kind of hybrid model by setting in the Poolman model. It would be desirable to completely clarify the definition of the Pettersson model and then, perhaps, to show that it can be obtained as a well-behaved limiting system of the hybrid system in the sense of geometric singular perturbation theory. This might allow the dynamical properties of solutions of the different systems to be related to each other. The only result on stationary solutions obtained so far is a non-existence theorem. It would be of great interest to have positive results on the existence, multiplicity and stability of stationary solutions. A related question is that of classifying possible -limit points of positive solutions where some of the concentrations are zero. This was done in part in the paper but what was not settled is whether potential -limit points with positive concentrations of the hexose phosphates can actually occur. Finally, there are a lot of other models for the Calvin cycle on the market and it would be interesting to see to what extent they are susceptible to methods similar to those used in our paper.

Phosphorylation systems

In order to react to their environment living cells use signalling networks to propagate information from receptors, which are often on the surface of the cell, to the nucleus where transcription factors can change the behaviour of the cell by changing the rate of production of different proteins. Signalling networks often make use of phosphorylation systems. These are networks of proteins whose enzymatic activity is switched on or off by phosphorylation or dephosphorylation. When switched on they catalyse the (de-)phophorylation of other proteins. The information passing through the network is encoded in the phosphate groups attached to specific amino acids in the proteins concerned. A frequently occurring example of this type of system is the MAPK cascade discussed in a previous post. There the phosphate groups are attached to the amino acids serine, threonine and tyrosine. Another type of system, which is common in bacteria, are the two-component systems where the phosphate groups are attached to histidine and aspartic acid.

There is a standard mathematical model for the MAPK cascade due to Huang and Ferrell. It consists of three layers, each of which is a simple or dual futile cycle. Numerical and heuristic investigations indicate that the Huang-Ferrell model admits periodic solutions for certain values of the parameters. Together with Juliette Hell we set out to find a rigorous proof of this fact. In the beginning we pursued the strategy of showing that there are relaxation oscillations. An important element of this is to prove that the dual futile cycle exhibits bistability, a fact which is interesting in its own right, and we were able to prove this, as has been discussed here. In the end we shifted to a different strategy in order to prove the existence of periodic solutions. The bistability proof used a quasistationary (Michaelis-Menten) reduction of the Huang-Ferrell system. It applied bifurcation theory to the Michaelis-Menten system and geometric singular perturbation theory to lift this result to the original system. To prove the existence of periodic solutions we used a similar strategy. This time we showed the presence of Hopf bifurcations in a Michaelis-Menten system and lifted those. The details are contained in a paper which is close to being finished. In the meantime we wrote a review article on phosphorylation systems. Here I want to mention some of the topics covered there.

The MAPK cascade, which is the central subject of the paper is not isolated in its natural biological context. It is connected with other biochemical reactions which can be thought of as feedback loops, positive and negative. As already mentioned, the cascade itself consists of layers which are futile cycles. The paper first reviews what is known about the dynamics of futile cycles and the stand-alone MAPK cascasde. The focus is on phenomena such as multistability, sustained oscillations and (marginally) chaos and what can be proved about these things rigorously. The techniques which can be used in proofs of this kind are also reviewed. Given the theoretical results on oscillations it is interesting to ask whether these can be observed experimentally. This has been done for the Raf-MEK-ERK cascade by Shankaran et. al. (Mol. Syst. Biol. 5, 332). In that paper it is found that the experimental results do not fit well to the oscillations in the isolated cascade but they can be modelled better when the cascade is embedded in a negative feedback loop. Two other aspects are also built into the models used – the translocation of ERK from the cytosol to the nucleus (which is what is actually measured) and the fact that when ERK and MEK are not fully phosphorylated they can bind to each other. It is also briefly mentioned in our paper that a negative feedback can arise through the interaction of ERK with its substrates, as explained in Liu et. al. (Biophys. J. 101, 2572). For the cascade as treated in the Huang-Ferrell model with feedback added no rigorous results are known yet. (For a somewhat different system there is result on oscillations due to Gedeon and Sontag, J. Diff. Eq. 239, 273, which uses the strategy based on relaxation oscillations.)

In our paper there is also an introduction to two-component systems. A general conclusion of the paper is that phosphorylation systems give rise to a variety of interesting mathematical problems which are waiting to be investigated. It may also be hoped that a better mathematical understanding of this subject can lead to new insights concerning the biological systems being modelled. Biological questions of interest in this context include the following. Are dynamical features of the MAPK cascade such as oscillations desirable for the encoding of information or are they undesirable side effects? To what extent do feedback loops tend to encourage the occurrence of features of this type and to what extent do they tend to suppress them? What are their practical uses, if any? If the function of the system is damaged by mutations how can it be repaired? The last question is of special interest due to the fact that many cancer cells have mutations in the Raf-MEK-ERK cascade and there have already been many attempts to overcome their negative effects using kinase inhibitors, some of them successful. A prominent example is the Raf inhibitor Vemurafenib which has been used to treat metastatic melanoma.