Distribution of the second virial coefficients of globular proteins

George and Wilson [Acta. Cryst. D 50, 361 (1994)] looked at the distribution of values of the second virial coefficient of globular proteins, under the conditions at which they crystallise. They found the values to lie within a fairly narrow range. We have defined a simple model of a generic globular protein. We then generate a set of proteins by picking values for the parameters of the model from a probability distribution. At fixed solubility, this set of proteins is found to have values of the second virial coefficient that fall within a fairly narrow range. The shape of the probability distribution of the second virial coefficient is Gaussian because the second virial coefficient is a sum of contributions from different patches on the protein surface. PACS: 87.14.Ee, 87.15Nn. Protein crystallisation is an important problem yet our grasp of the details of how it occurs is very poor. Proteins need to be crystallised from solution in order to determine their structure via X-ray crystallography [1, 2]. The crystallisation presumably starts with heterogeneous nucleation of the crystalline phase in the protein solution, but there has been no systematic experimental study of this, as far as the author is aware. Without an understanding of how proteins crystallise, protein crystallisation is almost totally ad hoc: essentially the only way to know if a protein will crystallise under a certain set of conditions is to try it. It would be enormously useful if we could predict the conditions under which a protein was most likely to crystallise. Here by protein we mean globular protein, which are proteins that are soluble in solution, as opposed to membrane proteins which exist embedded in a surfactant bilayer. The hope that it is possible to predict the conditions that promote crystallisation motivated George and Wilson [3] to look at the values of the (osmotic) second virial coefficient of a number of proteins under the conditions where they were crystallised. They found that the second virial coefficient was always negative and lay within what they called ‘a fairly narrow range’. If we ignore outliers then second virial coefficients gathered together by Haas and Drenth [4], and converted to reduced units by Vliegenthart and Lekkerkerker [5], lie in the range −8 to −40, in units of the volume of the protein; see Table V of Ref. [5]. The simplest explanation of this range is that the upper limit is set by the requirement that the attractive interactions be strong enough to pull the molecules into a crystal from a dilute solution. The lower limit is set by the dynamics of the solution, if the attractive interactions are too strong the protein molecules tend to aggregate irreversibly and this aggregation preempts and prevents crystallisation. Testing these explanations is all but impossible due to our poor understanding of crystallisation so we turn to a welldefined, and easily calculated, property of a protein solution: its solubility. We consider the solubility of the protein, i.e., the concentration of protein in the solution which coexists with the crystal, in preference to the process of crystallisation. We ask the question: For a given solubility, say 5% by volume, what is the distribution of values that we expect for B2? If we have 1000 proteins, say, all with the same solubility, then is their distribution of values of B2 very broad, or is it narrow? What is the shape of the distribution, i.e., what is its functional form? The distribution of values of B2 of a large number of proteins defines a probability distribution function P (B2). We will consider a constraint, that of fixed solubility, and so will obtain a probability distribution function that also depends on this constraint. We are inspired to study this function by a range of work on protein solutions and crystals [6–12] that has shown that protein-protein interactions are well described by a sum over contacts between the proteins. Where by a contact between two proteins, we mean that a specific patch on the surface of one protein approaches closely, a couple of Å, to a specific patch on the surface of the other protein. Now, if these contacts are more-or-less independent then we expect them to contribute essentially independently to B2. But if B2 is the sum of many independent contributions for each protein, then we know the form of the distribution function P (B2): it is a Gaussian. The central limit theorem states that the probability distribution function of some property Y , which is a sum over a large number of independent random variables, is a Gaussian [13]. Also of course the more independent patches there are on the surface the narrower will be the distribution of values of B2. The physical picture is that the surface of a protein has a number of patches on its surface. Under the conditions where the protein’s solubility is low, these patches attract each other. The strength of each patch attraction is then a random variable selected from some distribution. It is a random variable if the strength of the attraction of one patch on the surface is independent of the attraction of