Entropy in Biomolecular Systems
- Richard Henchman (University of Manchester, United Kingdom)
- Bojan Zagrovic (University of Vienna, Austria)
- Michel Cuendet (Swiss Institute of Bioinformatics, Lausanne, Switzerland and Weill Cornell Medical College, New York, USA)
- Chris Oostenbrink (University of Natural Resources and Life Sciences, Austria)
Venue: Max F. Perutz Laboratories, University of Vienna, Dr. Bohrgasse 9, Vienna, A-1030, Austria.
Registration Fee: 350 EUR
Workshop registration 21 April 2014
Registration payment deadline 30 April 2014
Summary: This workshop brings together the world's experts to address the challenges of determining the entropy of biomolecular systems, either by experiment or computer simulation. Entropy is one the main driving forces for any biological process such as binding, folding, partitioning and reacting. Our deficient understandng of entropy, however, means that such important processes remain controversial and only partially understood. Contributions of water, ions, cofactors, and biomolecular flexibility are actively examined but yet to be resolved. The state-of-the-art of each entropy method will be presented and explained, highlighting its capabilities and deficiencies. This will be followed by intensive discussion on the main areas that need improving, leading suitable actions and collaborations to address the main biological and industrial questions.
Introduction and Motivation
Entropy and enthalpy comprise the two main driving forces for any physical, chemical or biological process. Taken together, these two quantities constitute the free energy, which determines the extent of any process and, as a free energy barrier, the rate at which it occurs. Historically, a significant part of the effort in molecular simulation has been dedicated to the calculation of free energy differences including a long-standing series of CECAM workshops related to this topic. However, separating entropic and enthalpic contributions is key to understanding the nature of phenomena such as the hydrophobic effect, protein folding or ligand binding. Energy quantifies a system’s molecular interactions and entropy quantifies its structural variation. While energy can be readily obtained from a simulation, the determination of entropy, and, in particular, its conformational and solvent components, is much more challenging than calculating a free energy difference. This is to a large extent due to the fact that for the calculation of absolute and relative entropies one in principle needs full coverage of the phase space. This being essentially impossible because of the astronomical size of phase space, ingenious ways are required to make this feasible. There are numerous techniques that qualitatively probe a system’s flexibility and structural heterogeneity as measured in a simulation or experimentally. However, determining the quantitative and complete link between structure and entropy remains a major unsolved problem. This is especially so for biomolecules, which, together with the surrounding aqueous medium, display considerable structural heterogeneity. Solving this problem will substantially improve our ability to understand the intricacy and subtlety of biological molecular function as well as rationally design our own molecular systems.
The recent years have seen the emergence of a number of promising computational and experimental techniques to address entropy-related issues in various biologically relevant processes. However, major challenges of a fundamental nature exist on both fronts, as detailed below, before entropy calculations can become routinely used in biomolecular systems. Surprisingly, despite the fundamental nature of entropy and the inadequacy of current approaches, very few scientific meetings have been entirely dedicated to its determination and its role in biomolecular systems. By bringing together theorists, computational scientists and experimentalists, the proposed workshop “Entropy in Biomolecular Systems” aims to fill this gap and play a catalytic role in delineating the main challenges in the field, enabling cross-talk across the theory/experiment divide and stimulating active collaborations. We believe that such a meeting would play a transformative role in establishing biomolecular entropy, as well as entropy calculations more generally, as a rich, propulsive, and multidisciplinary field of research.
The important and extremely diverse roles played by entropic effects in different biologically relevant processes are being recognized with an ever-increasing frequency. Here, we mention three examples. Firstly, in the area of rational drug design, it has long been recognized that solvent entropy and conformational entropy changes upon binding of a ligand to a biomolecule represent an important and yet still largely uncharted space of design possibilities (1-4). Understanding the basic quantitative principles that define this space may propel rational drug design to completely new levels of success. Secondly, in protein folding, chaperones can be understood as agents, which decrease the entropy of the unfolded state by reducing the number of conformations available to the protein. Any quantitative assessment of the chaperone’s role must accurately take this entropic contribution into account (5). Finally, dynamically driven allostery in which ligand binding affects protein behavior at distant sites is now seen as the main mechanism underlying signaling in the absence of clear structural changes (6, 7). Again, quantifying conformational entropy changes is key to understanding allostery, as well as many other biomolecular processes underlying biological function.
Despite such clear and compelling motivation, theoretical treatment of conformational and solvent entropies in particular is still largely underdeveloped, and the same goes for experimental measurement of these quantities. Part of the problem is a lack of recognition of the importance of entropy and the inadequacy of the current methods for its determination. One only has to look at the advanced and sophisticated theories that have been developed for electronic structure calculations of energy to realize how inferior is our theory for entropy. When it comes to the prediction of entropies, a combination of theory and computer simulation is required in all but the simplest cases of ideal gases, polymers or crystals. Depending on the problem at hand, the key quantity in question may be either the relative entropy between initial and final states of a given process or the absolute entropy of a given state. Changes in entropy can be derived using computer simulation from the free energy and enthalpy or from the temperature derivative of the free energy (8). While these methods are widely studied and widely used, they require sampling along a pathway between the two states, something that becomes computationally unfeasible for complex systems such as biomolecules. Moreover, they do not give the separation of entropy into its structural components, and most importantly its conformational and solvent parts.
The two traditional methods to calculate the absolute conformational entropy of a molecule are normal mode analysis (9-11) and quasi-harmonic analysis (12-16). Both of these approaches assume that the conformational state is restricted to a single basin of the potential energy surface, which is locally approximated by a harmonic oscillator. Normal mode analysis derives the associated force constants from the Hessian matrix. This is the preferred method in the electronic structure community because it only requires a single optimized structure (11). On the other hand, quasi-harmonic analysis derives force constants from the variance-covariance matrix of the displacements measured in constant temperature molecular dynamics simulations (12-16). Both normal mode and quasi-harmonic methods fail if the system is inherently anharmonic or if non-linear correlations exists between atomic motions, both conditions being frequently met in large biomolecules. The contributions of the surrounding solvent and ions have commonly been treated using continuum and empirical terms, which do not adequately capture local solvation effects (17, 18). Finally, while changes in total entropy are experimentally accessible in many cases using calorimetric techniques such as isothermal titration calorimetry, they remain silent when it comes to entropy components (19).
Over the past twenty years, a number of new methods have been proposed both computationally and experimentally to treat these inadequacies (16, 20). This is particularly true for biomolecular systems because of their central scientific importance and because of the conspicuous shortcomings of the traditional methods for treating entropy in such systems. On the simulation front, recent methodological developments account for the multiple conformational states, the anharmonic nature of these states, the presence of correlated motions and the entropy of the solvent (15, 21-32). In particular, different corrections have been developed to go beyond the quasi-harmonic approximation including, for example, those in third moments of coordinates (21) or in pairwise, linear correlations (15, 27). Most importantly, mutual information expansion techniques have been developed to allow for a treatment of supra-linear correlations, albeit with notorious convergence challenges (15, 22, 26, 30). For example, the minimally coupled subspace approach combines full correlation analysis, adaptive kernel density estimation and mutual information expansion to give accurate conformational entropy estimates at least for small peptides (31). Finally, important advances have also been made when it comes to the treatment of solvent entropy. From cell theory-based approaches (23, 25) to morphometric and integral equation theory (24, 29) to permutation reduction techniques (28), different methods have been developed in this context. While all of these approaches represent welcome advances in conformational and solvent entropy treatment, numerous computational and conceptual bottlenecks still prevent their wider usage. One of the main goals of the meeting proposed herein would be to define, categorize and prioritize these bottlenecks and delineate strategies that might lead to their resolution.
On the other hand, advances in the nuclear magnetic resonance (NMR) techniques, most notably relaxation techniques of multinuclear and multidimensional NMR, have recently led to a revolutionary new view of the role of dynamics in general and conformational entropy, in particular, in biomolecular processes (7, 33-39). Specifically, most progress has been made with NMR by deriving conformational entropy from generalized order parameters of isolated bond vectors or methyl symmetry axes (33-36, 39). Importantly, however, NMR analyses must rely on dynamical models in order to give a parametric relationship between different NMR observables, physical rearrangements and conformational entropy (33-36, 39). The absolute entropies obtained in this way very much depend on the particulars of the potential energy function used. What is more, the extremely important correlated motions are typically ignored in such approaches, primarily because they are inaccessible to most experiments. On the other hand, molecular dynamics simulations not only help interpret NMR relaxation experiments in terms of entropic contributions, but also give a detailed, atomistic picture of such contributions even on their own (40). As both simulation and experiment converge on calculating the same quantities, the proposed meeting will provide an exciting and valuable platform for comparing the approximations and assumptions of the two approaches, exploring the potential synergies, and making new methodologies and capabilities available to the wider community.
University of Vienna, Dr. Bohrgasse 9, Vienna, A-1030, Austria
1. Young T, Abel R, Kim B, Berne BJ, & Friesner RA (2007) Motifs for molecular recognition exploiting hydrophobic enclosure in protein-ligand binding. P Natl Acad Sci USA 104(3):808-813.
2. Kirchmair J, Spitzer GM, & Liedl KR (2011) Consideration of water and solvation effects in virtual screening. Virtual screening: principles, challenges, and practical guidelines, (Wiley-VCH Verlag).
3. Fenley AT, Muddana HS & Gilson MK (2012) Entropy-enthalpy trasduction caused by conformational shifts can obscure the forces driving protein-ligand binding. P Natl Acad Sci USA 109(49):20006-20011.
4. Baron R & McCammon JA (2013) Molecular Recognition and Ligand Association. Ann Rev Phys Chem 64: 151-175.
5. England J, Lucent D, & Pande V (2008) Rattling the cage: computational models of chaperonin-mediated protein folding. Curr Opin Struc Biol 18(2):163-169.
6. Smock RG & Gierasch LM (2009) Sending signals dynamically. Science 324(5924):198-203.
7. Tzeng SR & Kalodimos CG (2009) Dynamic activation of an allosteric regulatory protein. Nature 462(7271):368-U139.
8. Peter C, Oostenbrink C, van Dorp A, & van Gunsteren WF (2004) Estimating entropies from molecular dynamics simulations. J Chem Phys 120(6):2652-2661.
9. Go N & Scheraga HA (1969) Analysis of contribution of internal vibrations to statistical weights of equilibrium conformations of macromolecules. J Chem Phys 51(11):4751-&.
10. Levy RM, Karplus M, Kushick J, & Perahia D (1984) Evaluation of the configurational entropy for proteins - application to molecular-dynamics simulations of an alpha-helix. Macromolecules 17(7):1370-1374.
11. Jensen F (2006) Introduction to Computational Chemistry (Wiley-Blackwell).
12. Karplus M & Kushick JN (1981) Method for estimating the configurational entropy of macromolecules. Macromolecules 14(2):325-332.
13. Edholm O & Berendsen HJC (1984) Entropy estimation from simulations of non-diffusive systems. Mol Phys 51(4):1011-1028.
14. Andricioaei I & Karplus M (2001) On the calculation of entropy from covariance matrices of the atomic fluctuations. J Chem Phys 115(14):6289.
15. Baron R, van Gunsteren WF, & Hünenberger PH (2006) Estimating the configurational entropy from molecular dynamics simulations: Anharmonicity and correlation corrections to the quasi-harmonic approximation. Trends in Phys Chem 11:87-122.
16. Polyansky AA, Zubac R, & Zagrovic B (2012) Estimation of conformational entropy in protein-ligand Interactions: a computational perspective. Methods Mol Biol 819:327-353.
17. Still WC, Tempczyk A, Hawley RC, & Hendrickson T (1990) Semianalytical treatment of solvation for molecular mechanics and dynamics. J Am Chem Soc 112(16):6127-6129.
18. Srinivasan J, Cheatham TE, Cieplak P, Kollman PA, & Case DA (1998) Continuum solvent studies of the stability of DNA, RNA, and phosphoramidate - DNA helices. J Am Chem Soc 120(37):9401-9409.
19. Ghai R, Falconer RJ, & Collins BM (2012) Applications of isothermal titration calorimetry in pure and applied researchusurvey of the literature from 2010. J Mol Recognit 25(1):32-52.
20. Chang CE, Chen W, & Gilson MK (2005) Evaluating the accuracy of the quasiharmonic approximation. J Chem Theory Comput 1(5):1017-1028.
21. Rojas OL, Levy RM, & Szabo A (1986) Corrections to the quasiharmonic approximation for evaluating molecular entropies. J Chem Phys 85(2):1037-1043.
22. Matsuda H (2000) Physical nature of higher-order mutual information: Intrinsic correlations and frustration. Phys Rev E 62(3):3096-3102.
23. Henchman RH (2003) Partition function for a simple liquid using cell theory parametrized by computer simulation. J Chem Phys 119(1):400-406.
24. Roth R, Harano Y, & Kinoshita M (2006) Morphometric approach to the solvation free energy of complex molecules. Phys Rev Lett 97(7).
25. Henchman RH (2007) Free energy of liquid water from a computer simulation via cell theory. J Chem Phys 126(6).
26. Killian BJ, Kravitz JY, & Gilson MK (2007) Extraction of configurational entropy from molecular simulations via an expansion approximation. J Chem Phys 127(2):-.
27. Numata J, Wan M, & Knapp EW (2007) Conformational entropy of biomolecules: Beyond the quasi-harmonic approximation. Genome Inform Ser 18:192-205.
28. Reinhard F & Grubmuller H (2007) Estimation of absolute solvent and solvation shell entropies via permutation reduction. J Chem Phys 126(1).
29. Amano KI, Yoshidome T, Harano Y, Oda K, & Kinoshita M (2009) Theoretical analysis on thermal stability of a protein focused on the water entropy. Chem Phys Lett 474(1-3):190-194.
30. Killian BJ, et al. (2009) Configurational entropy in protein-peptide binding: computational study of Tsg101 ubiquitin E2 variant domain with an HIV-derived PTAP nonapeptide. J Mol Biol 389(2):315-335.
31. Hensen U, Lange OF, & Grubmuller H (2010) Estimating absolute configurational entropies of macromolecules: the minimally coupled subspace approach. PLoS One 5(2):e9179.
32. Polyansky AA, Kuzmanic A, Hlevnjak M & Zagrovic B (2012) On the contribution of linear correlations to quasi-harmonic conformational entropy in proteins. J Chem Theo Comput 8:3820-29.
33. Akke M, Bruschweiler R, & Palmer AG (1993) NMR order parameters and free-energy - an analytical approach and its application to cooperative Ca2+ binding by calbindin-D(9k). J Am Chem Soc 115(21):9832-9833.
34. Li Z, Raychaudhuri S, & Wand AJ (1996) Insights into the local residual entropy of proteins provided by NMR relaxation. Protein Sci 5(12):2647-2650.
35. Yang D & Kay LE (1996) Contributions to conformational entropy arising from bond vector fluctuations measured from NMR-derived order parameters: application to protein folding. J Mol Biol 263(2):369-382.
36. Frederick KK, Marlow MS, Valentine KG, & Wand AJ (2007) Conformational entropy in molecular recognition by proteins. Nature 448(7151):325-U323.
37. Henzler-Wildman KA, et al. (2007) A hierarchy of timescales in protein dynamics is linked to enzyme catalysis. Nature 450(7171):913-U927.
38. Henzler-Wildman KA, et al. (2007) Intrinsic motions along an enzymatic reaction trajectory. Nature 450(7171):838-U813.
39. Marlow MS, Dogan J, Frederick KK, Valentine KG, & Wand AJ (2010) The role of conformational entropy in molecular recognition by calmodulin. Nat Chem Biol 6(5):352-358.
40. van Gunsteren WF, et al. (2006) Biomolecular modeling: goals, problems, perspectives. Angew Chem Int Edit 45(25):4064-4092.