Modelling the Calvin cycle

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 . 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

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 . 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 , in the steady state solution is independent of the parameters . (The other concentrations will in general depend on the .) 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.

Talk on mathematical modelling in Karlstad

Yesterday I was in Karlstad in Sweden to give a talk on the uses of mathematical modelling in the natural sciences. I was invited to do this by Claes Uggla and I was very happy to have the opportunity to present some of my ideas on this subject. The talk was structured as a series of examples involving applications of different mathematical techniques. Many of these examples have been discussed in some form in this blog during the past few years and indeed a lot of my ideas on the subject were developed in conjunction with the blog posts. The subjects were William Harvey and the circulation of the blood, multidrug therapy for HIV-AIDS, the lizard Uta stansburiana, oscillations near the big bang, Liesegang rings, modelling oscillations in vole populations using a reaction-diffusion system, signal transduction in T cells.

As well as presenting a variety of applications of different types of mathematics I also wanted to explain some mathematical connections between these subjects. One central idea is that structural stability is an issue of key importance in modelling natural phenomena. Most phenomenological models involve parameters or other elements which are not known exactly. Thus to be of interest for applications features of the dynamics of the model should be invariant under arbitrary small perturbations of the system. More precisely, if a model does not possess an invariance of this type but is nevertheless useful this requires some explanation. One possible source of an explanation is the presence of what I call ‘absolute elements’ in the model. For instance, in population dynamics if a population is zero at some time then it will definitely remain zero. This fact is independent of the details of how the population grows when it is non-zero. Similarly a spacetime singularity can define an absolute element in cosmology. When the spacetime metric breaks down this ends the dynamics in a way which is independent of the details of the dynamics of the matter away from the singularity. Thus structural stability can be weakened to the condition of invariance under small perturbations which leave certain submanifolds fixed. This can lead to the appearance of relevant heteroclinic cycles although these are not structurally stable in the absolute sense. It explains the appearance of heteroclinic cycles in the models for lizards and for the big bang in a unified way. In a similar way, restricting the perturbations of a system of chemical reactions to those which leave a particular reaction irreversible can furnish the homoclinic orbit needed to model Liesegang rings.

I have now put a slightly extended version of this talk with references on my web page. On the same day there was a talk by Bernt Wennberg on models for the collective motion of birds and fish, concentrating mainly on kinetic models related to the Boltzmann equation. At the start of his talk he showed some of the well-known pictures of flocks of Starlings over Rome. In the evening I had my own pleasant experience with a flock of birds. A large number of Jackdaws (a couple of hundred) were flying around the central square in Karlstad and calling. For some reason I have become increasingly attached to the Jackdaw over the years. At this point, and without a good excuse, I want to tell a story about Jackdaws from the book ‘King Solomon’s Ring’ by Konrad Lorenz. It is a long time since I read the book and so I hope I do not distort the story too much. At one time Lorenz was living in a small village in Austria where he was regarded by the locals as a bit crazy. One of his interests was the social life of Jackdaws. There were Jackdaws living on the roofs of the houses and he climbed up to get close to them. In order to fit in better with his black subjects he decided to dress in black. The only ‘suitable’ black clothing he could find was a devil’s costume left over from a fancy dress party. No doubt the spectacle of him climbing over the roofs dressed as the devil perfected his reputation with the local inhabitants.

Conference on systems biology of T cells in Baeza, part 2

In the remaining one and a half days of the conference there were another fourteen talks and I will mention some aspects of their contents which attracted my attention. One recurring theme was that the encounter of a T cell receptor (TCR) with the peptide it recognizes bound to an MHC molelcule (pMHC) is often not just the encounter of one TCR with one pMHC but of multiple players. It can be shown by electron microscopy that the TCR tend to cluster on the surface of a T cell even before it has encountered antigen. This is done by attaching gold particles to the TCR so that they show up as black dots on the electron micrograph. It was shown in the talk of Hisse van Santen that a similar thing happens with the pMHC on the surface of antigen presenting cells. Judging from the discussion after the talk it seems that the explanation for this is that the pMHC, which are well known to be produced in the interior of the cell, are exported to the surface in groups. There also seems to be a widely held opinion that signalling through the T cell receptor is absolutely dependent on clustering of TCR. This makes life more complicated than it otherwise might have been. I learned at this conference that experiments on T cell signalling in vitro are often done by using tetramers, i.e. groups of four pMHC which are bound together covalently. In the talk of Wolfgang Schamel described experiments using tetramer binding. He said that this work was linked with some mathematical modelling, done by Thomas Höfer and others, but he did not want to take questions on that. My impression was that the model was an extension of the kinetic proofreading model. It has not yet been published and so I did not yet have an opportunity to look at it. Carmen Molina-París and Balbino Alarcón discussed cooperative effects in T cell receptor binding.

Michal Polonsky showed pictures of individual T cells trapped in small wells in a microfluidic device. When activated they wriggle very vigorously. These are the kind of pictures which could easily make you take a very anthropomorphic view of T cells. The aim of this work is to observe the differentiation, division and death of the cells over long periods (several days). If they were not trapped it would be extremely difficult to follow them under the microscope since they would be liable to run away. A break from the purely scientific talks was provided by a presentation of Dinah Singer about the systems biology programme at the National Cancer Institute in the US, a programme which she runs. Apart from concrete information about funding another aspect of this was the question of what might be learned about the potential for applying systems biology to immunology from existing applications of these ideas to cancer research. Dipankar Nandi talked about a phenomenon I had never heard of before and would never have expected – atrophy of the thymus as a consequence of certain diseases. Finally, I was on more familiar ground with the talk of Isabel Mérida about certain signalling pathways in T cell activation. The substance at the centre of her talk, diacylglycerol kinase, was not familiar to me but the context was. Right at the end of the conference there was a general discussion session planned. This session, which was led by Ed Palmer, ended up being very short. This was due to the (in itself positive) fact that the discussions after (and during) the individual talks had taken up more time than planned. The final discussion was interesting despite its brevity. The basic theme was: if mathematicians are collaborating with immunologists what can each side do to help the other in this process? Interesting points were brought up and we were all sent home with some things to think about.

Conference on systems biology of T cells in Baeza

At the moment I am attending a conference on systems biology of T cells in Baeza. Of the eleven talks today the first nine made no mention of mathematics – there was not a single equation. The tenth, by Zvi Grossmann, did show a couple. Thus the bias today was very much towards experimental immunology. It was interesting for me to be immersed in this atmosphere and I learned a lot of things. There are three things which stick in my mind particularly. The first is the fact, mentioned in the talk of Bruno Kyewski, that antigens mimicking all tissues of the body are presented by medullary epithelial cells in the thymus. This allows future T cells to learn about all self antigens. I asked him afterwards if this includes tissues which are in the immunologically privileged sites, usually poorly accessible to the immune system, like the central nervous system. He confirmed that this is the case. The second is the fact, which came up in the talk of Marisa Torio, that T cell precursors in the thymus have the potential to develop into almost any type of white blood cell. This means that the fate of a cell to become a T cell is in general not decided before it reaches the thymus, the answer to a question I had often asked myself. The third is the description in the talk of Alfred Singer of the way in which it is decided which of the surface molecules CD4 or CD8 a T cell carries. I had already watched a video by Singer on this subject on the NIH web page but one thing I was not aware of was the fact that by binding the protein Lck it is possible for CD4 and CD8 to interfere with T cell signalling. Lck is sequestered and hence is not available for use by the T cell receptor.

Grossmann’s talk was mainly concerned with rather abstract ideas about cell signalling and it was hard for me to get to grips with them. I had the impression that the right mathematical context for these things should be control theory. The last and only really mathematical talk of the day, by Rob de Boer, was a highlight for me and not only for me. At dinner the air was buzzing with conversations on the subject. The talk was on monitoring the dynamics of immune cells by labelling with deuterium and drawing conclusions about their lifetimes. I had heard a talk on a similar subject by de Boer before at a conference in Dresden and I wrote about it briefly in a previous post. I liked that earlier talk but I liked the talk today much more. This was probably less due to the difference in content as to the fact that for whatever reason I now appreciated the significance of this work much better. This is an example where a mathematical model can be used to obtain information about processes in immunology which it is difficult or impossible to obtain in any other way. It is not that the mathematics is complicated, just some explicitly solvable linear ODE. The impressive thing is the direct contact this work makes with real biological questions like ‘how long does a memory T cell live’. Analysing different experiments both using deuterium in human subjects and other more poisonous substances which can only be used in mice originally gave inconsistent answers for lifetimes. With hindsight this arose from the assumption in the models of just one population of cells with a definite death rate. Passing to a model with two classes of cells largely removed the discrepancy. There was another interesting aspect of this lecture and its reception which explains its prevalence at dinner. It has to do with communication between different fields, in this case mathematics and biology. There was a lot of confusion among the audience which was due not to the factual content of the work but to the way the results were described and to the choice of language in describing the results. I should remember for the future that it is not enough to get an interesting result in mathematical biology. It is also necessary to be very careful about formulating it in the right way so as to make its meaning transparent for biologists.

The multiple futile cycle

The multiple futile cycle is a simple type of network of chemical reactions which is often found in biological systems. In a previous post I mentioned it as a component of a slightly more complicated network found in many cells, the MAP kinase cascade. One concrete realization of the multiple futile cycle is a protein which can be phosphorylated at up to sites. All the phosphorylation steps are carried out by one kinase while all dephosphorylation steps are carried out by one phosphatase. Each step is modelled in the Michaelis-Menten way, including an enzyme-substrate complex as one of the species and using mass action kinetics. There results a system of ordinary differential equations with three conservation laws. These represent the conservation of the total amount of the two enzymes and of the substrate protein. In the case , which might be called the simple futile cycle, using the conservation laws to eliminate some of the variables leads to a three-dimensional dynamical system. A basic question is what can be said about the dynamics of solutions of this system.

It has been shown by Angeli and Sontag (Nonlinear Analysis RWA, 9, 128) that in the case every solution converges to a stationary solution and that this stationary solution is unique for given values of the conserved quantities. The proof uses the theory of monotone dynamical systems. The original dynamical system is not monotone and so the first step in their proof is to replace it by another system which is monotone and show that convergence properties of solutions of the second imply convergence properties of solutions of the first. The second step is to prove the convergence of solutions of monotone systems under the additional condition of the existence of a translational symmetry. The paper mentions that this result is dual to a previously known result due to Mierczyński about monotone systems with a conserved quantity. Up to now I thought that the only benefit of knowing that a dynamical system is monotone is the possibility of reducing it to an effective system of one dimension less. This is only interesting if the initial system is of dimension no more than three. What this work has shown me is that knowing that a system is monotone can sometimes be the key to concluding much more. One aspect of the paper of Angeli and Sontag which was a source of confusion for me was a difference in conventions to what I am familiar with from chemical reaction network theory. This seems to be essential for the monotonicity argument and not just a matter of taste. The stoichiometric matrix (or stoichiometry matrix) is defined differently because a reversible reaction is treated as a single reaction rather than as a pair.. I feel a spontaneous preference for the CRNT convention but here it seems that a different one can be a real advantage. In the case of the simple futile cycle an important effect is that the dimension of the kernel of the stoichiometric matrix is three with the CRNT convention and one with the Angeli-Sontag convention.

In another paper (J. Math. Biol. 61, 581) Angeli, De Leenheer and Sontag present a more general theory related to this. Here the hypotheses needed to obtain a suitable monotone system involve the properties of certain graphs constructed from the reaction network. In this theory the notion of persistence of the dynamical system plays an important role. This is the property that a positive solution can never have any limit points on the boundary of the positive region. The case (dual futile cycle) has been considered in a paper of Wang and Sontag (J. Nonlin. Sci. 18, 527). There they are able to show that for certain ranges of the parameters generic solutions converge to stationary solutions. To emphasize the power of the techniques developed in these papers it should be pointed out that they can be applied to systems with arbitarily large numbers of unkowns and parameters and that when they apply they give strong conclusions.

D. Angeli and E. D. Sontag (2008). Translation-invariant monotone systems, and a global convergence result for enzymatic futile cycles Nonlinear Analysis: Real World Applications, 9 DOI: 10.1016/j.nonrwa.2006.09.006

D. Angeli, P. De Leenheer and E. D. Sontag (2010). Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates. Journal of mathematical biology, 61 (4) PMID: 19949950
 

The Goodwin oscillator

During a talk by Jae Kyoung Kim at last week’s SMB meeting the speaker showed a system of equations and called it ‘a system you all know’. This revealed to me a gap in my knowledge of mathematical biology. The system is the Goodwin oscillator. It is described in Murray’s book on mathematical biology and I am sure I have read the relevant section on some level. This just shows that there a big difference between reading something and understanding its significance and being able to situate in a wider context. Now I have done my homework on this and I will write something about it here. The system, in the form it is given in Murray’s book, is an example of systems of the form . Here the labels on the are supposed to be interpreted modulo . In other words there are equations and is interpreted as . In the Goodwin model itself and the functions are linear for . The function is equal to for constants , and . This function is positive and its derivative is negative. Thus it can be interpreted as representing a negative feedback on the production of . In the context in which it was introduced by Goodwin the quantities , and represent concentrations of mRNA, the enzyme it codes for and the product of a reaction it catalyzes. The substrate of the enzyme is assumed present at a constant level and is not modelled explicitly.

It is known that the system admits a periodic solution if the Hill coefficient is greater than eight and not otherwise. Since this number is considered unrealistically large for the application which inspired the model modifications of this have been considered where periodic solutions can be obtained for lower values of . It is proved by Hastings, Tyson and Webster (J. Diff. Eq. 25, 39) that for the Goodwin system and a larger class of similar models the following is true. The system has a unique steady state and if the linearization of the system at that point has no repeated eigenvalues and at least one eigenvalue with positive real part there exist periodic solutions. This reduces the existence question to the analysis of the linearization. The existence proof relies on the Brouwer fixed point theorem and is similar to a proof I described in a previous post. Although the Goodwin system is three dimensional the method is not restricted to that case. The proof does not give information about the stability of the periodic solutions. In the paper of Hastings et. al. they indicate that an alternative analysis using a Hopf bifurcation can give stability in some cases. However no details of the stability argument are given in that paper.

The Goodwin model was inspired by the fundamental work of Monod and Jacob on gene regulation.  Various things have given me an appetite for learning more about gene regulatory networks and this was increased by some of the talks I heard last week.

SMB annual meeting in Knoxville, part 2

The music did seem to have a positive effect on the synchronization of lectures. Unfortunately it was not always there – for instance it was not there before my talk – and it seems to have been getting less and less. One good thing is that the name tags, as well as showing the usual information have the first name (or nickname) printed in large letters at the top. I find that this can be very useful for recognizing people after only having met them fleetingly.

The plenary talk of Claire Tomlin yesterday was about the HER2 receptor which plays an important role in breast cancer. It is connected to transcription factors in the nucleus by a signalling network containing two main pathways. One of these includes the MAP kinase cascade while another passes through the substance Akt. Excessive activity of this type of signalling can be reduced by a drug called lapatinib, which is a tyrosine kinase inhibitor. There is, however, a problem that this beneficial effect can be neutralized after some time. The speaker described ideas for overcoming this effect based on a study of the signalling network. A result of this analysis is that, counterintuitively, combining the administration of lapatinib with another treatment which increases the concentration of Akt at a different time could lead to a more effective therapy. I did not get the details but this seems like a case where mathematical modelling could actually contribute effectively to cancer treatment by suggesting new strategies. Relations were mentioned to the pattern of hairs on the wings of Drosophila. In her research on biomedical themes she benefits from her background in control engineering and aerodynamics.

The talk of Becca Asquith which I mentioned in the last post was cancelled. Instead there was a lecture by Sandy Anderson who seems to like to cultivate the image of the hard-drinking Scotsman. He started his career in mathematical modelling and then moved a long way towards medical research, now heading a lab at the Moffit Cancer Center in Florida. The subject of his talk was the role of heterogeneity in cancer. He started by giving a view of the importance of cancer (in terms of the number of people it kills) and the trends in the numbers for the different forms. They have mostly been decreasing for many years with the notable exception of lung cancer (for well-known reasons) but the rate of decrease is not very large despite the huge amount of effort, and money, put into cancer research. He said that death in cancer usually does not result from a tumour which stays in its original site but as a result of metastasis. Thus that is the key phenomenon to be understood. This requires an understanding of many different scales and for the talk he concentrated on the cellular scale. He claimed that an important fact that cancer researchers had not taken into account sufficiently until very recently is how heterogeneous tumours are. There is a large variation in the phenotype of the individual cancer cells and the phenotypes are evolving. This evolution is strongly influenced by the environment of the tumour, for instance the structure of the surrounding extracellular matrix. Experiments done on cell cultures may give misleading results since the ‘happy’ cells in the Petri dish with all modern comforts are not under the same pressure as corresponding cells in the body. The more the external pressures are the more the dangerous cells which are going to metastasize dominate over the others. In some cases treatment can accelerate the growth of a tumour. This danger exists if the treatment is given too late. These ideas have arisen by the use of mathematical modelling. These are ‘hybrid models’ which combine discrete and continuous dynamical systems and this is a terms which I have met in several other talks at this conference. One of the conclusions of this research is that it may be a good idea to control cancer cells rather to destroy them. For the attempt to destroy cells may destroy the relatively harmless ones and unleash the dangerous one on their surroundings. Anderson’s talk conveyed the excitement of the application of mathematical modelling in cancer research at this moment and I wonder if some of the young people in the audience might have been recruited.

This afternoon I went to a session on wound healing. There was an introductory lecture by Rebecca Segal and this was helpful for me since I knew very little about the subject. Two of the things I found interesting – I was already primed for this by talking to Angela Reynolds at her poster yesterday – is that immunology (dynamics of neutrophils and macrophages) plays a big role and that ODE models can be useful. Useful means that they can help doctors make decisions how to treat wounds they are confronted with in practise.

SMB annual meeting in Knoxville

On Tuesday I will travel to Knoxville for the annual meeting of the Society for Mathematical Biology. On Wednesday I will give a talk there about my work on the NFAT signalling pathway. The programme of the conference is very dense: apart from the times when there are plenary talks there are seven sessions in parallel. My usual tactics at conferences of this type is to choose whole sessions to attend rather than individual talks. Anything else is usually frustrating due to the poor synchronization of the talks in different sessions. Maybe it will be better in this case. It is planned to have music to mark the breaks between talks which will be heard in all the rooms. This could overcome any lack of discipline imposed by the chairs of the individual sessions. Since all the rooms are in one building and, to judge by their numbering, close together it may really be practicable to attend individual talks.

What is the advantage of going to a big conference like this? The primary one is the opportunity of networking with people working in the field. Given that so much of the time is filled up with lectures this will require serious effort. It is good that the list of lectures was available well in advance of the conference. This allows a certain overview of who is taking part. It would have been even better if a full list of participants had also been available in advance. The second most important aspect of the conference is learning new things by actually listening to the talks. Since this is not a subject that I know so well that I can almost predict what the talks will be like just by seeing the titles and authors, there is plenty of opportunity for me here. Making the best of this opportunity will nevertheless require careful planning.

In the schedule there are eleven talks under the heading immunology and in addition a minisymposium on cancer immunology. These are things for me to focus on. There is also one plenary talk (by Becca Asquith) containing the phrase ‘immune response’. There is a session with the title ‘systems biology’ and four talks. My feelings towards this subject are ambivalent. On the one hand the idea – a concentrated theoretical approach to understanding biological systems – seems to me a very good thing. On the other hand I am not convinced by the way this idea has been realized up to now. One problem I see is that the definition of systems biology is rather vague and hence it is difficult to see what the content is. Another is that I have the impression that there is too much dominance of the quantitative over the qualitative (and high throughput over low throughput). My negative impression may just be due to lack of knowledge. In any case, I feel that I want to be enthusiastic about systems biology but I have not yet found the right point of access. A few weeks ago there was a conference on systems biology in Leipzig. I would have liked to attend but was prevented by other commitments. A highlight was a debate between Sydney Brenner and Denis Noble. I was not able to be there and so I was happy when I recently found that a video of it is available on the web. In fact the debate was not marked by strongly conflicting ideas. Both participants stressed that their views were not very far apart. I did not feel that I had a much clearer picture of the subject of the debate afterwards than I did before. Brenner dominated the proceedings. As usual he had a lot of interesting things to say. For instance he talked about a bacterium which adapted to live in I always find it inspiring listening to him and I recently had the opportunity to experience him live in a talk he gave in Berlin with the title ‘Reading the genome’. Through this I came upon a resource where short general articles by Brenner can be found. These one-page texts appeared under the names ‘Loose Ends’ and ‘False Starts’ and were published each month in the journal Current Biology between 1994 and 2000.

Low throughput biology

In modern biology there is a strong tendency to collect huge quantities of data with high throughput techniques. This data is only useful if we have good techniques of analysing it to obtain a better understanding of the biological systems being studied. One approach to doing this is to build mathematical models. An idea which is widespread is that the best models are those which are the closest to reality in the sense that they take account of as many effects as possible and use as many measured quantities as possible. Suppose for definiteness that the model is given by a system of ordinary differential equations. Then this idea translates into using systems with many variables and many parameters. There are several problems which may come up. The first is that some parameters have not been measured at all. The second is that those which have been measured are only known with poor accuracy and different parameters have been measured in different biological systems. A third problem is that even if the equations and parameters were known perfectly we are still faced with the difficult problem of analysing at least some aspects of the qualitative behaviour of solutions of a dynamical system of high dimension. The typical way of getting around this is to put the equations on the computer and calculate the solutions numerically for some initial data. Then we have the problem that we can only do the calculations for a finite number of initial data sets and it is difficult to know how typical the solutions obtained really are. To have a short name for the kind of model just described I will refer to it as a ‘complex model’.

In view of all these difficulties with complex models it makes sense to complement the above strategy by one which goes in a very different direction. The idea for this alternative approach is to build models which are as simple as possible subject to the condition that they include a biological effect of interest. The hope is then that a detailed analysis of the simple model will generate new and useful ideas for explaining biological phenomena or will give a picture of what is going on which may be crude but is nevertheless helpful in practise, perhaps even more helpful than a complex model.

It often happens that in analysing a complex model many of the parameters have to be guessed (perhaps just in an order of magnitude way) or estimated by some numerical technique. It is then justified to ask whether adding more variables and corresponding parameters really means adding information. How can we hope to understand complex models at all? If these were generic dynamical systems with the given number of unknowns and parameters this would be hopeless. Fortunately the dynamical systems arising in biology are far from generic. They have arisen by the action of evolution optimizing certain properties under strong constraints. Given that this is the case it makes sense to try and understand in what ways these systems are special. If key mechanisms can be identified then we can try to isolate them and study them intensively in relatively simple situations. My intention is not to deny the value of high throughput techniques. What I want to promote is the idea that it is bad if the pursuit of those approaches leads to the neglect of others which may be equally valuable. On a theoretical level this means the use of ‘simple models’ in contrast to ‘complex models’. There is a corresponding idea on the experimental side which may be even more necessary. This is to focus on the study of certain simple biological systems as a complement to high throughput techniques. This alternative might be called ‘low throughput biology’. It occurred to me that if I had this idea under this name then it might also have been introduced by others. Searching for the phrase with Google I only found a few references and as far as I could see the phrase was generally associated with a negative connotation. Rather than making an opposition between low throughput and high throughput techniques like David and Goliath it would be better to promote cooperation between the two. I have come across one good example in this in the work of Uri Alon and his collaborators on network motifs. This work is well explained in the lectures of Alon on systems biology which are available on Youtube. The idea is to take a large quantity of data (such as the network of all transcription factors of E. coli) and to use statistical analysis to identify qualitative features of the network which make it different from a random network. These features can then be isolated, analysed and, most importantly, understood in an intuitive way.