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Theoretically Describing Ensembles of Pathways between Disconnected Markov State ModelsJoseph Persichetti, David Wolfe, Ajeet K. Sharma, and Edward P. O’Brien

Department of Chemistry, Penn State University

References

Introduction and Motivation

Future Work

• Build a Markov State Model for candidate enzyme systems and useQM/MM finite-temperature string method to obtain paths in collectivevariable space. Use umbrella sampling and WHAM to obtain free energyprofiles for these paths.

• Predict trends in the observed rate of an enzyme at differenttemperatures and for different mutations.

(1) Ovchinnikov, V.; Karplus, M. J. Chem. Phys. 2014, 140 (17), 175103.(2) Vanden-Eijnden, E.; Venturoli, M. J. Chem. Phys. 2009, 130 (19), 194103.(3) Dhoke, G. V; Davari, M. D.; Schwaneberg, U.; Bocola, M. ACS Catal. 2015, 5

(6), 3207–3215.(4) Benkovic, S. J.; Hammes, G. G.; Hammes-Schiffer, S. Biochemistry 2008, 47

(11), 3317–3321.(5) Buchete, N.; Hummer, G. J. Phys. Chem. B 2008, 112, 6057-6069(6) Noé, F.; Schütte, C.; Vanden-Eijnden, E.; Reich, L.; Weikl, T. R.; PNAS. 2009,

106 (45), 19011-19016

The high-dimensional space and non-zero temperature at which chemicalreactions take place ensures that for most systems there will be multiple,disparate reaction pathways connecting the ensemble of reactant states tothe ensemble of product states. However, most quantummechanical/molecular mechanical (QM/MM) path-sampling methods formodeling reactions do not account for this fact. We are developing atheoretical framework which combines finite-temperature string method(FTSM) with Master equation modeling to predict observable rates whenconsidering the ensemble of pathways. To begin, we have modeled thealanine, proline and glycine dipeptide using classical path sampling atdifferent temperatures to develop the foundation for our method. In thefuture, we will apply our framework to an enzyme. This will require both QMand MM levels of theory for the FTSM simulations. A subset of the activesite of the enzyme will be modeled with a QM Hamiltonian so that thechemical reaction is accurately represented. The remainder of the enzymeand explicit water will be modeled classically to maintain efficiency andfeasibility.

Question 1) How can we combine the kinetics obtained from Markov

States and chain of states simulations to predict observable rates?

Question 2) How is the observable rate affected by changes in

temperature or mutations to the structure? Can we characterize these

trends with regard to perturbations to the ensemble of pathways?

Chain of States Method Connecting Markov States Rate Prediction from Multiple Pathways

Project Workflow

We have constructed a Markov State Model for alanine, proline and glycinedipeptide in vacuo from long timescale classical MD simulations. Thisprovides the rate (𝜔) of transitioning between Markov states in either thereactants or products. Finite-temperature string method provides the finalpath through dihedral space connecting two Markov states. We then seedumbrella sampling simulations from this path to obtain the free energyprofile describing transitions between reactant and product Markov statesfrom which we obtain the rate (𝑘). Finally, we construct a Master equationfor our system which incorporates kinetic information from disconnectedMarkov State Models to predict the observed rate of transitioning fromreactants to products through an ensemble of pathways.

We analyzed the flux througheach of the observedpathways which havesignificant flux for proline

dipeptide. This analysisshows that methods whichonly consider the pathwaywhich is lowest in energy onlycaptures ~ 20% of the totalflux for this test case. It isnecessary to include allpathways in order to accountfor the total flux fromreactants to products.

The observed trends ofcalculated rate as a functionof temperature reveal that thebest approximation (kNET) ofthe true rate includescontributions from allobserved pathways.

𝒌𝒊𝒋 ∝ 𝐞−∆𝐄‡

𝐤𝐁𝐓

𝐊 =

−𝜶𝟏 𝛚𝟏𝟐 𝒌𝟏𝟑 𝒌𝟏𝟒𝛚𝟐𝟏 −𝜶𝟐 𝒌𝟐𝟑 𝒌𝟐𝟒𝟎 𝟎 −𝜶𝟑 𝛚𝟑𝟒

𝟎 𝟎 𝝎𝟒𝟑 −𝜶𝟒

𝒌𝒐𝒃𝒔 =𝟏

𝛕, 𝛕 = −𝝅𝑻𝑲𝑹

−𝟏𝟏