Keller, B.

Noé, F.

Prinz, J.-H.

English

Repository: Freie Universität Berlin (FU), Math Department (fu_mi_publications)

Resolving the apparent gap in complexity betweensimulated and measured kinetics of biomoleculesBettina G. Keller, Jan-Hendrik Prinz, and Frank Noe´Freie Universita¨t Berlin,Arnimallee 6, 14195 Berlin, GermanyE-mail: bettina.keller@fu-berlin.de, jan-hendrik.prinz@fu-berlin.de, frank.noe@fu-berlin.deMolecular simulations of biomolecules often reveal a complex picture of the their kinetics,whereas kinetic experiments typically seem to indicate considerably simpler two- or three-statekinetics. Markov state models (MSM) provide a tool to link between simulation and experi-ment, and to resolve this apparent contradiction.1 IntroductionMolecular simulations of large biomolecules typically reveal a complex picture of the free-energy surface with many kinetically relevant states1–3. Complementary to this, advancedexperimental techniques allow probing the equilibrium kinetics of biomolecules directly.This can either be done by perturbation techniques, such as temperature or pressure jump,or by equilibrium experiments in which the measured signal is autocorrelated to obtainkinetic information, as e.g. in fluorescence correlation spectroscopy. These measurementscan be described by dynamical fingerprints, i.e. densities of relaxation timescales whereeach peak corresponds to an exponential relaxation process. In many cases, single- ordouble-peaked fingerprints are found, suggesting that a two- or three-state model may pro-vide a satisfactory description of the biomolecule studied4, 5.We sketch an approach combining Markov state models (MSM)6, 3 of the simulateddynamics with dynamical fingerprints, which allows addressing the following questions:(i) Is the largest relaxation timescales observed always due to the folding process? (ii) Cana given experiment detect all relaxation processes that are present in the dynamics of themolecule? (iii) Are the processes observed in perturbation experiments the same as thoseobserved in equilibrium experiments? We illustrate our findings using a four-state modelof a protein folding equilibrium.2 TheoryIn MSMs of conformational kinetics, the conformational space of the molecule is dis-cretized into N states. The kinetics between the states is described by transition probabil-ity tij of going from a state i to a state j within a time step τ , which are summarized in atransition matrix T(τ). The entire information of the kinetics of the system is contained inthis matrix. In particular, each (left) eigenvector li of the transition matrix represents oneof the kinetic processes of the system. For a detailed review of the theory of MSM andtheir application to the simulation of biomolecules see Ref. 6 and 3.Kinetic experiments yield time series µ(t), either of the observed signal a directly (per-turbation experiments), or of the autocorrelation of the signal (equilibrium experiments).1Figure 1. Sketch of a protein folding equilibrium. The arrows represent possible transitions between confor-mational states. Their thickness corresponds to the transition probability. The yellow stars represent possiblechromophore attachment points.From physical principles, we expect that in both cases µ(t) is a noisy realization of a sumof multiple exponential functionsµ(t) =∫t′dt′γ(t′) exp(− tt′)≈N∑iγi exp(− tt′i). (1)γ(t′) is inverse the Laplace transform of µ(t) and is called the dynamical fingerprint7.The approximation in Eq. 1 results from the fact that any experimental signal is time-discretized, and that in an MSM-representation of the kinetics the number of processeswhich contribute to the dynamical fingerprint is limited to the number of states N . Therelaxation timescales ti of the system are linked to the eigenvalues λi of the transitionmatrix by t′i = −τ/ lnλi One can derive analytical expressions7, 8 for the amplitudes γiof equilibrium experiments γpi,ai = 〈a, li〉2 and of perturbation experiments γp(0),ai =〈a, li〉 〈p′(0), li〉. p′(0) is the excess probability density with p′j(0) = pj(0)/pij , where piis the equilibrium density. a is the observable vector which associates each states in theMSM with a signal strength of the observable a.3 Model system and resultsEach of the secondary structure elements of the protein folding model in Fig. 1 can foldand unfold in a single distinct step. This leads to a MSM of the conformational equilibriumwith N = 4 metastable states. The transition probabilities are represented by the thicknessof the arrows in Fig. 1. None of the eigenvectors of the MSM (Fig. 2.a) reflects our notionof folding. Hence, the folding process may not be contained in a kinetic process and canthus not always be associated with a single folding rate.The yellow stars in Fig. 1 mark possible chromophore attachment points. We chooseour observable vectors to resemble FRET constructs, i.e. two chromophores attached to themolecule with strong signal if they are close to each other, weak signal if they are furtherapart (Fig. 2.c).2Figure 2. Markov model and experimental setup for the protein folding model.Figure 3. Amplitudes of the dynamical fingerprints of a variety of experimental setups for the protein foldingmodel. (A) Equilibrium experiment, dynamical fingerprint of the autocorrelation function of observables a, b,and c. (B) Perturbation experiment, dynamical fingerprint of the decay signal of the observable c combined withinitial distributions p1(0) and p2(0), respectively.Fig. 3.A show the dynamical-fingerprint amplitudes of equilibrium experiments usingeach of the three possible observable vectors a, b, and c. The scalar product of observablevector a with eigenvectors l3 and l4, is close to zero and therefore the corresponding exper-iment is insensitive to these processes. This is in line with the intuition that an experimentin which the chromophores are attached at site 1 and 2 will be most sensitive to conforma-tional changes of the α-helix. Conversely, observable b has little overlap with eigenvectorsl2, and the corresponding experiment is insensitive to this kinetic process representing theα-helix folding. Only with observable c all relevant kinetic processes can be observed.This shows that a single experiment will typically be insensitive to some of the processespresent in the system and explains why kinetics appear often simpler in experiments thanin simulations.3A perturbation experiment cannot reveal a kinetic process which would be invisible inan equilibrium experiment with the same observable. This is easily seen by consideringthat the amplitude is given as γp(0),ai = 〈a, li〉 〈p′(0), li〉. If the first factor is zero, i.e. theprocess is not detectable in an equilibirum experiment, the corresponding amplitude inthe fingerprint of the perturbation experiment will also be zero - independent of the initialdistribution p(0). However, not all kinetic processes are involved in relaxing a particularinitial distribution to the equilibrium distribution and therefore the second factor can be-come zero, too. This is shown in Fig. 3.B, where 〈p′1(0), l3〉 ≈ 0 and 〈p′2(0), l4〉 ≈ 0,respectively (see also Fig. 2.a and see Fig. 2.c). Consequently, these perturbation experi-ments are insensitive to the third and fourth kinetic process respectively, even though theyare conducted with an observable which sensitive to all processes in the system.Given a MSM of the conformational kinetics of biomolecule, our approach can also beused to suggest optimal attachment points for the chromophores7, 8.AcknowledgmentsFunding from the German Science Foundation (DFG) through grant number NO 825/2 andthrough research center MATHEON is gratefully acknowledged.References1. S. Muff and A. Caflisch, Kinetic analysis of molecular dynamics simulations revealschanges in the denatured state and switch of folding pathways upon single-point mu-tation of a beta-sheet miniprotein, Proteins-Struct., Funct., Bioinf., 70, no. 4, 1185–1195, 2008.2. F. Noe´, C. Schu¨tte, E. Vanden-Eijnden, L. Reich, and T. R. Weikl, Constructingthe equilibrium ensemble of folding pathways from short off-equilibrium simulations,Proc. Natl. Acad. Sci. USA, 106, no. 45, 19011–19016, 2009.3. G. R. Bowman, K. A. Beauchamp, G. Boxer, and V. S. Pande, Progress and chal-lenges in the automated construction of Markov state models for full protein systems,J. Chem. Phys., 131, no. 12, 124101, 2009.4. Marcus Ja¨ger, Houbi Nguyen, Jason C. Crane, Jeffery W. Kelly, and Martin Gruebele,The folding mechanism of a beta-sheet: the WW domain, J. Mol. Biol., 311, no. 2,373–393, 2001.5. Hannes Neuweiler, Marc Lo¨llmann, So¨ren Doose, and M. Sauer, Dynamics of Un-folded Polypeptide Chains in Crowded Environment Studied by Fluorescence Corre-lation Spectroscopy, J. Mol. Biol., 365, 856–869, 2007.6. J-H. Prinz, H. Wu, M. Sarich, B. Keller, M. Fischbach, M. Held, J. D. Chodera,C. Schu¨tte, and F. Noe´, Markov model and molecular kinetics: generation and vali-dation, J. Chem. Phys., 134, 174105, 2011.7. F. Noe´, S. Doose, I. Daidone, M. Lollmann, M. Sauer, J. D. Chodera, and J. C. Smith,Dynamical fingerprints for probing individual relaxation processes in biomoleculardynamics with simulations and kinetic experiments, Proc. Natl. Acad. Sci. USA, 108,no. 12, 4822–4827, 2011.8. B. G. Keller, J.-H. Prinz, and F. Noe´, Markov models and dynamical fingerprints:unraveling the complexity of molecular kinetics., Chem. Phys., submitted 2011.4

Resolving the apparent gap in complexity between simulated and measured kinetics of biomolecules

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Resolving the apparent gap in complexity between simulated and measured kinetics of biomolecules

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