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Analytic markovian rates for generalized protein structure evolution.

Coluzza I, MacDonald JT, Sadowski MI, Taylor WR, Goldstein RA - PLoS ONE (2012)

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, University of Vienna, Vienna, Austria. ivan.coluzza@univie.ac.at

ABSTRACT
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.

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Plot of the logarithm of the rate constantssampled according to Eq. 8 as a function of the differencesandof the Hydrogen bond energy and of the number of total contacts respectively. The simulations data are fitted with the function  from Eq. 11. The points falls quite nicely on the surface indicating that the simple form of the rate in Eq. 11 captures the major trend of the simulation points. It is important to stress that for equally compact structures with the same number of hydrogen bonds the rate may still be influenced by important structural differences not included in this fit.
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pone-0034228-g004: Plot of the logarithm of the rate constantssampled according to Eq. 8 as a function of the differencesandof the Hydrogen bond energy and of the number of total contacts respectively. The simulations data are fitted with the function from Eq. 11. The points falls quite nicely on the surface indicating that the simple form of the rate in Eq. 11 captures the major trend of the simulation points. It is important to stress that for equally compact structures with the same number of hydrogen bonds the rate may still be influenced by important structural differences not included in this fit.

Mentions: The next step in our study is to perform the design of the set of stepping stones. During the design simulation we sample the rate constants according to the Eqs. (8) using the Metropolis scheme described in the method section. Our objective is to correlate these rates with some variables measures that describe the structural difference between each pair of stepping stone. From the rates described by equation (7) it is evident that there is a strong dependence on the difference in energy between two structures for each sequence, hence in order to capture the fundamental structural differences between each stepping stone pair it is natural to select quantities such as the total hydrogen bond energy of structure and for structure , and the total number of residue contacts in structure and the number of contacts in . Another educated guess that we can make is that because the committor is a function only of the energy difference, we can expect the rate to behave similarly, this is also verified by distribution of the rates plotted as a function of the difference in the hydrogen bond energy and the difference in the number of contacts (Fig. (4)), if we remember that the plot is in log scale then the surface follows a step like function very similar to one that represents the committor function . This indicates that the jumps follow an on/off transition process and also that we can extract a universal function that relates the rates with the difference in hydrogen bonds and in the number of contacts in the following way for the rates:(11)where the values for , , and have been obtained by fitting to the simulation data. We have listed the final values for the parameters in Table 1. In Fig. (4) we plot the logarithm of the measured rates and the corresponding fitted rates surfaces from Eq. 11. The small errors over the parameter values of the plot show that there is good agreement between the predicted profile and the simulation data. This demonstrates the validity of our prediction of universal dependence of the rates on the structural variables.


Analytic markovian rates for generalized protein structure evolution.

Coluzza I, MacDonald JT, Sadowski MI, Taylor WR, Goldstein RA - PLoS ONE (2012)

Plot of the logarithm of the rate constantssampled according to Eq. 8 as a function of the differencesandof the Hydrogen bond energy and of the number of total contacts respectively. The simulations data are fitted with the function  from Eq. 11. The points falls quite nicely on the surface indicating that the simple form of the rate in Eq. 11 captures the major trend of the simulation points. It is important to stress that for equally compact structures with the same number of hydrogen bonds the rate may still be influenced by important structural differences not included in this fit.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC3367531&req=5

pone-0034228-g004: Plot of the logarithm of the rate constantssampled according to Eq. 8 as a function of the differencesandof the Hydrogen bond energy and of the number of total contacts respectively. The simulations data are fitted with the function from Eq. 11. The points falls quite nicely on the surface indicating that the simple form of the rate in Eq. 11 captures the major trend of the simulation points. It is important to stress that for equally compact structures with the same number of hydrogen bonds the rate may still be influenced by important structural differences not included in this fit.
Mentions: The next step in our study is to perform the design of the set of stepping stones. During the design simulation we sample the rate constants according to the Eqs. (8) using the Metropolis scheme described in the method section. Our objective is to correlate these rates with some variables measures that describe the structural difference between each pair of stepping stone. From the rates described by equation (7) it is evident that there is a strong dependence on the difference in energy between two structures for each sequence, hence in order to capture the fundamental structural differences between each stepping stone pair it is natural to select quantities such as the total hydrogen bond energy of structure and for structure , and the total number of residue contacts in structure and the number of contacts in . Another educated guess that we can make is that because the committor is a function only of the energy difference, we can expect the rate to behave similarly, this is also verified by distribution of the rates plotted as a function of the difference in the hydrogen bond energy and the difference in the number of contacts (Fig. (4)), if we remember that the plot is in log scale then the surface follows a step like function very similar to one that represents the committor function . This indicates that the jumps follow an on/off transition process and also that we can extract a universal function that relates the rates with the difference in hydrogen bonds and in the number of contacts in the following way for the rates:(11)where the values for , , and have been obtained by fitting to the simulation data. We have listed the final values for the parameters in Table 1. In Fig. (4) we plot the logarithm of the measured rates and the corresponding fitted rates surfaces from Eq. 11. The small errors over the parameter values of the plot show that there is good agreement between the predicted profile and the simulation data. This demonstrates the validity of our prediction of universal dependence of the rates on the structural variables.

Bottom Line: 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.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.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, University of Vienna, Vienna, Austria. ivan.coluzza@univie.ac.at

ABSTRACT
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.

Show MeSH