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.