Mathematics enters the canon of molecular biology

On the shelf in my office I have a heavy red tome. It is the fifth edition of ‘Molecular Biology of the Cell’ by Alberts et al., a standard work. While reading an editorial by Arup Chakraborty I learned that the sixth edition of the book contains a section on mathematical models. Its title is ‘Mathematical Analysis of Cell Functions’ and it is almost 20 pages long. It is perhaps not very deep mathematically but I am nevertheless delighted that it is there. I find it very important that a textbook of central importance in cell biology takes time to discuss mathematics in that context and presents arguments why mathematics is valuable for biology. I wonder how much of what is written there would be understood by a mathematician with no background in molecular biology. Of course that is not the intended audience of the book and it is just my idle curiosity that makes me ask the question.

Let me list some of the main themes treated in the section. I give them as informal statements. (1) Negative feedback can stabilize a steady state. (2) Negative feedback plus delay can give rise to sustained oscillations. (3) Positive feedback plus cooperativity can give rise to multistability. (4) An incoherent feed-forward loop can give rise to a transient response to a signal. (5) A coherent feed-forward loop can give rise to a delay. I should emphasize that in all cases mentioned here the models involved use ODE. The delay in (2) does not have to do with a delay equation but just with a sufficiently large number of steps which take place one after another.

I would like to make some connections between the points (1)-(5) above and mathematical theorems. I will start with (2) since that seems the easiest case. I will replace (2) with another statement which is even easier: (2a) negative feedback is a necessary condition for the existence of an attracting periodic solution. This is related to ideas of Rene Thomas which I disussed in a previous post. Suppose we have a system of ODE \dot x=f(x) where the signs of the elements of the linearization Df(x) are independent of x. Here a ‘sign’ may be positive, negative or zero. As explained elsewhere a system of this kind defines a graph called the species graph or influence graph. A feedback loop is an oriented cycle in this graph and its sign is the product of the signs of the edges making up the cycle. The system exhibits negative feedback if its influence graph contains a negative feedback loop. The claim is then that there exist no attracting periodic solutions. Before I go further I should acknowledge my debt to Frederic Beck, who wrote his MSc under my supervision on the subject of feedback. The discussion which follows has benefitted a lot from his exposition of this subject. A system without negative feedback loops as defined above is called coherent. It was proved by Angeli, Hirsch and Sontag (J. Diff. Eq. 246, 3058) that by reversing the signs of some of the variables the system can be made quasicooperative. This means that along any feedback loop all signs are equal. Note that this does not mean that signs need be equal along an unoriented cycle. This can be illustrated by the case of an incoherent feedforward loop. It follows from the results of Angeli et al. that in a coherent system there exist no attracting periodic solutions. This result can be strengthened by replacing ‘attracting’ by ‘stable’. This has been proved by Richard and Comet (J. Math. Biol 63, 593). They also claim that if the condition that the signs of the entries in the linearization are spatially constant is dropped then the analogous statement is false. Their proof of this latter statement contained an error, as noticed by Frederic Beck. They repaired it in an erratum (J. Math. Biol. 70, 957).

This discussion shows that a lot is understood about negative feedback as a necessary condition for periodic solutions although there may still be more to be discovered on that subject. I think that much less is understood about the possibility of negative feedback being a sufficient condition for periodic solutions. As a concrete example let us consider the Selkov oscillator. It does contain a negative feedback loop consisting of one positive and one negative edge. For some parameter values it exhibits periodic solutions but for others there are only damped oscillations. Can the assumption of a negative feedback loop be supplemented in some way by an assumption of delay to give a sufficient condition? What should that assumption of delay be?

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