Archive for August, 2011

Kinetic proofreading

August 25, 2011

In biological systems information is propagated from one form to another by chemical reactions. An example is the translation of mRNA into protein by the ribosome. Under certain circumstances there are limits to the accuracy of this kind of process. In a one-step process with two possible outcomes the accuracy is bounded above in terms of the difference of the free energies of the two alternative reactions. In other words, it is bounded in terms of the ratio of the reaction constants. Putting in the numbers for some important biological processes shows that this bound is exceeded by a large factor. This led to a proposal by Hopfield (PNAS 71, 4135) of a way in which this accuracy can be achieved by using more complicated reactions with several steps. He called it kinetic proofreading. (There was other related work by Ninio (Biochimie 57, 587) at about the same time.) Later McKeithan (PNAS 92, 5042) applied this idea to the question of how the T cell receptor can discriminate so accurately between different antigens. This model was studied mathematically by Eduardo Sontag (IEEE Transactions on Automatic Control, 46, 1028), who related it to chemical reaction network theory (CRNT). Here I will take Sontag’s work as starting point for my description.

Let T be the concentration of T cell receptors not bound to a ligand and M the concentration of peptide-MHC complexes not bound to a receptor. When a peptide-MHC complex binds to a receptor this gives the basic form of the occupied receptor and the concentration of these is denoted by C_0. The rate constant for this process is denoted by k_1 This basic form can be modified by phosphorylation at up to N sites, giving rise to quantities C_i. There are successive phosphorylation reactions leading from C_i to C_{i+1} and the corresponding rate constants are denoted by k_{p,i}. There are dissociation reactions where the peptide-MHC complex detaches from the receptor and the receptor is simultaneously completely dephosphorylated. The rate constants are denoted by k_{-1,i}. The total concentrations of T cell receptors and peptide-MHC complexes (both bound and free) are denoted by T^* and M^* respectively. They are conserved quantities and can be used to eliminate the variables T and M from the system if desired. Doing so gives the system for the variables C_i, i=0,1,\dots,N at the beginning of Sontag’s paper. In the terminology of CRNT this corresponds to restricting to a stoichiometric compatibility class. It is elementary to calculate the stationary solutions of the original system and there is exactly one in each stoichiometric compatibility class. In terms of CRNT the system is weakly reversible and of deficiency zero. General theory then implies that there is exactly one stationary solution in each stoichiometric compatibility class and that it is asymptotically stable. Sontag strengthens this result, proving that all solutions converge to the corresponding stationary solutions at late times.

Now I come back to the original motivation. For simplicity let us suppose that k_{p,i} and k_{-1,i} are independent of i. Let \alpha=\frac{k_p}{k_p+k_{-1}}. Then it turns out, as computed by McKeithan, that the ratio of the fully phosphorylated complex C_N to the total complex is \alpha^N. This means that if N is not too small this ratio depends very sensitively on the value of the dissociation constant k_{-1}. If it is C_N which gives rise to further signalling within the cell this gives a way of magnifying differences between the binding properties of ligands.


The life of François Jacob

August 19, 2011

I have just read the autobiography ‘La statue intérieure’ by François Jacob. I find that this is a book of high literary quality. An indication of this going beyond my personal judgement is that after the book came out Jacob was invited to talk about it by Bernard Pivot on his TV programme ‘Apostrophes’. I understand that at the time when Pivot was active an invitation from him was a kind of certificate of quality for any new book which appeared in French. Jacob is best known as a biologist but this book convinces me that he is also a very gifted writer. The book is not a monumental work, but rather a collection of anecdotes which illuminate many aspects of Jacob’s life and, more generally, many aspects of the human condition. I find it difficult to say what makes up the charm of his writing – I can only suggest reading the book in order to experience it at first hand.

The first three quarters of the book describe the part of his life before he began his career as a scientific researcher in his late twenties. He had started out studying medicine with the aim of becoming a surgeon. After less than two years this was interrupted by the outbreak of the second world war. Jacob, who is Jewish, fled France and joined the French army in exile led by de Gaulle. He spent a large part of the war in Africa. Shortly after his return to France he was very seriously wounded. This made it impossible for him to become a surgeon and left him somewhat at a loss what to do. An interesting point, which he does not emphasize in the book, is that in a sense his being wounded in this case was a result of a decision of his own. He was tending to an officer who had just been wounded when the group was bombed again. The officer could not be moved and begged Jacob not to leave him alone. Jacob could not do anything to protect the man but he nevertheless stayed with him instead of taking cover. As a result of this he was almost killed himself. His decision was very honourable but maybe not very reasonable. In any case, he ended up spending many difficult months in hospital.

Jacob’s way into research was quite indirect and dependent on a lot of chance factors. For some time he worked in an institute which was supposed to produce penicillin in France but never came close to doing so. He became involved in developing and marketing an antibiotic called tyrothricin. Somewhat later he was able to enter the research group of André Lwoff at the Institut Pasteur. This paved the way for the work for which he was awarded the Nobel prize for medicine with Lwoff and Jacques Monod. He describes how each year to commemorate the anniversary of Pasteur’s death everyone working at the institute (not just the scientists) would make a formal visit to the tomb of Pasteur in the basement of one of the buildings. This was made more vivid for me by the fact that I had visited this tomb myself a couple of years ago. I was at the Pasteur museum quite late in the afternoon when not many visitors were there. I was shown the tomb by a very friendly employee of the museum. It is an impressive structure with lots of marble. I remarked to her that Robert Koch did not have such an impressive mausoleum. She replied that the Germans do not honour their great scientists in the way the French do. I am not sure how true this is but it is at least food for thought.

A large part of the last quarter of the book is a description of the work with Monod. There are also a lot of general reflections on the way in which science is done and how the process by which scientific ideas are developed contrasts with the final product as found in research papers and textbooks. It also gives a good picture of who did what in this collaboration. A key mechanism was the interaction between what Jacob was doing on prophages and what Monod was doing on the lac operon. From a certain point on they were always looking for analogies between the two. This part of the book gives a vivid portrait of the early days of the discipline of molecular biology. It includes a description of Jacob working feverishly with Sydney Brenner at Caltech to establish the existence of messenger RNA, in an atmosphere of general scepticism. The narrative ends after the completion of the project with Monod. What happened afterwards in Jacob’s (scientific) life? According to the book ‘In the beginning was the worm’ by Andrew Brown, Jacob tried to work on Caenorhabditis elegans but without success and he failed to get funding to set up an ‘Institut de la Souris’. Jacob later wrote a book called ‘La souris, la mouche et l’homme’ and perhaps I will read that sometime. But since my summer holiday is at an end it will not be very soon.