Analytic markovian rates for generalized protein structure evolution.
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A general understanding of the complex phenomenon of protein evolution requires the accurate description of the constraints that define the sub-space of proteins with mutations that do not appreciably reduce the fitness of the organism.By using a coarse-grained representation for the protein structures, we derive an analytic formulation of the transition rates between each of the intermediate structures.Knowledge of the transition rates allows for the study of complex evolutionary pathways represented by trajectories through a set of intermediate structures.
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Affiliation: Department of Physics, University of Vienna, Vienna, Austria. ivan.coluzza@univie.ac.at
ABSTRACT
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A general understanding of the complex phenomenon of protein evolution requires the accurate description of the constraints that define the sub-space of proteins with mutations that do not appreciably reduce the fitness of the organism. Such constraints can have multiple origins, in this work we present a model for constrained evolutionary trajectories represented by a markovian process throughout a set of protein-like structures artificially constructed to be topological intermediates between the structure of two natural occurring proteins. The number and type of intermediate steps defines how constrained the total evolutionary process is. By using a coarse-grained representation for the protein structures, we derive an analytic formulation of the transition rates between each of the intermediate structures. The results indicate that compact structures with a high number of hydrogen bonds are more probable and have a higher likelihood to arise during evolution. Knowledge of the transition rates allows for the study of complex evolutionary pathways represented by trajectories through a set of intermediate structures. Related in: MedlinePlus |
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Mentions: So far we have only considered the dependence of the rate constants on two structural parameters that cannot represent the total difference between two proteins. In particular we are missing information regarding how many contacts are in common between the pair of proteins. This information must play a role in the rate, as even for two proteins with the same number of hydrogen bonds and the same number of total contacts, we expect that the differences in the topology will make the population of sequences quite separate in energy. In other words if there are not many common contacts it is difficult to optimize two structures at the same time. A common measure of the similarity between two structures is the number of common native contacts . The dependence of the rate constants from must not alter the detailed balance condition that we verified in the appendix (Section Detailed Balance). The condition of detailed balance requires that for each pair of stepping stones the ratio between the and rate constants is equal to the ratio between the probabilities of observing the two structures, hence the ratio cannot depend on a quantity that cannot be factorized out. The simplest function is then the one resulting from the addition to the functions in Eq. (11) of a new term . In order to maintain detailed balance, the new term must be symmetric under inversion of with . In order to determine the form of we plotted the rate constants in Fig. (5) as a function of the number of common contacts and the function that describes the dependence of the rate constants as a function of the hydrogen bond energy difference and the difference in the total number of contacts. The plot shows that the data have a linear profile along , which supports our assumption for the factorization of the contribution of . Hence we considered the following expression for :(12)where again the parameters and were obtained through a fit of the data in Fig. (5). The expression of in eq. (12), is similar to the sigmoidal function used for . This is not a surprising result, considering the correct function must be close to the maximum rate when the two proteins are very similar (). |
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Affiliation: Department of Physics, University of Vienna, Vienna, Austria. ivan.coluzza@univie.ac.at