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: An important property of the rate constants is that at equilibrium they must satisfy the system of equations characteristic of the underlying Markov model. In practice the following set of equations must be satisfied for each pair and (13)where and are the equilibrium probabilities. According to the expression of the rate constant that we fitted on the data (Eq. (11)) the ratio of the rate constants (or the difference between the logarithms) should give an expression that is factorisable into two functions that will depend only on the properties of and . We can easily obtain the general solution to the system of equations (13) by expressing the probabilities in the following form:(14)in the appendix it is demonstrated explicitly that this form solves the system of equations in eq. (13). To further prove the validity of this construction we performed an independent fit of the logarithm of the ratio of the and rate constants calculated for each and . If our theory is correct the data should be optimally fitted by a plane where , , and and . In Fig. (6) we plot the ratio fitted with the function , the points fall nicely on the plane and the optimized values of and are equal to the expected values to within the experimental error. |
View Article: PubMed Central - PubMed
Affiliation: Department of Physics, University of Vienna, Vienna, Austria. ivan.coluzza@univie.ac.at