Quantifying selection and diversity in viruses by entropy methods, with application to the haemagglutinin of H3N2 influenza

Abstract

Many viruses evolve rapidly. For example, haemagglutinin (HA) of the H3N2 influenza A virus evolves to escape antibody binding. This evolution of the H3N2 virus means that people who have previously been exposed to an influenza strain may be infected by a newly emerged virus. In this paper, we use Shannon entropy and relative entropy to measure the diversity and selection pressure by an antibody in each amino acid site of H3 HA between the 1992-1993 season and the 2009-2010 season. Shannon entropy and relative entropy are two independent state variables that we use to characterize H3N2 evolution. The entropy method estimates future H3N2 evolution and migration using currently available H3 HA sequences. First, we show that the rate of evolution increases with the virus diversity in the current season. The Shannon entropy of the sequence in the current season predicts relative entropy between sequences in the current season and those in the next season. Second, a global migration pattern of H3N2 is assembled by comparing the relative entropy flows of sequences sampled in China, Japan, the USA and Europe. We verify this entropy method by describing two aspects of historical H3N2 evolution. First, we identify 54 amino acid sites in HA that have evolved in the past to evade the immune system. Second, the entropy method shows that epitopes A and B on the top of HA evolve most vigorously to escape antibody binding. Our work provides a novel entropy-based method to predict and quantify future H3N2 evolution and to describe the evolutionary history of H3N2.

Figure 1.

The tertiary structure of the HA1 domain of H3 HA (PDB code: 1HGF). The surface of HA1 facing outward is the exposed surface when the HA trimer is formed. The other two HA1 domains (not shown) in the HA trimer are located at the back of the structure displayed here. The solid balls represent five epitopes. Colour code: blue, epitope A; red, epitope B; cyan, epitope C; yellow, epitope D; green, epitope E.

Figure 2.

Mean and standard error of relative entropy S_{i +1,j} in each bin of Shannon entropy. Shannon entropy and relative entropy in each of the 329 positions and in each of the 17 seasons between 1992–1993 (i = 0) and 2008–2009 (i = 16) fall into one of the eight bins. The first bin with Shannon entropy less than 0.1 is discarded. Bins with larger Shannon entropy D_i,j also have larger relative entropy S_{i +1,j} . Shannon entropy D_i,j and relative entropy S_{i +1,j} in iteration i = 51–100 of the neutral evolution model (crosses) are used to calculate mean and standard error of relative entropy in each bin of Shannon entropy distribution in the same way. No increasing trend is found. Error bar is one standard error. Open squares represent H3 data.

Figure 3.

Average Shannon entropy 〈D〉_i versus average relative entropy 〈S〉_{i +1} for each season between 1992–1993 (i = 0) and 2008–2009 (i = 16). For each season i, a set of amino acid positions j with Shannon entropy D_i,j greater than 0.1 are chosen. For all the j in this set of positions, 〈D〉_i is the average of the Shannon entropy D_i,j values and 〈S〉_{i +1} is the average of relative entropy S_{i +1,j} values. Horizontal and vertical error bars are the standard errors of Shannon entropy and relative entropy, respectively. The solid line, 〈S〉_{i +1} = 1.82〈D〉_i−0.23, is a least squares fit of 〈D〉_i to 〈S〉_{i +1} (i = 0,2, …,16). A strong correlation with R² = 0.50 exists between 〈D〉_i and 〈S〉_{i +1} excluding the point (0.22,1.38) with N_i = 1, which has a large standard error of the relative entropy S_{i +1,j}. Using the same method, 〈D〉_i and 〈S〉_{i +1} are calculated from a neutral evolution model, i = 51–100, and plotted. No visible correlation exists between 〈D〉_i and 〈S〉_{i +1} from the neutral evolution model (crosses). Open squares represent H3 data.

Figure 4.

(a) Average selection in each position quantified by relative entropy during the past 17 seasons from 1993–1994 to 2009–2010, calculated by . The colours represent positions in epitopes A to E and positions outside the epitopes, as in figure 1. (b) Number of seasons for each position when the relative entropy was greater than the threshold S_i^thres, i.e. the position was under selection. (c) Average diversity in each position quantified by Shannon entropy in the seasons from 1993–1994 to 2009–2010, calculated by . (d) Distribution of the average selection in each position displayed in (a). (e) Distribution of the numbers of seasons under selection displayed in (b). (f) Distribution of the average diversity in each position shown in (c).