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Intra-Chain Organisation of Hydrophobic Residues Controls Inter-Chain Aggregation Rates of Amphiphilic Polymers

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Intra-Chain Organisation of Hydrophobic Residues Controls Inter-Chain Aggregation Rates of Amphiphilic Polymers Intra-chain organisation of hydrophobic residues controls inter-chain aggregation rates of amphiphilic polymers The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Citation Varilly, Patrick et al. “Intra-Chain Organisation of Hydrophobic Residues Controls Inter-Chain Aggregation Rates of Amphiphilic Polymers.” The Journal of Chemical Physics 146, 13 (April 2017): 135102 © 2017 Author(s) As Published http://dx.doi.org/10.1063/1.4977932 Publisher American Institute of Physics (AIP) Version Original manuscript Citable link http://hdl.handle.net/1721.1/113639 Terms of Use Creative Commons Attribution-Noncommercial-Share Alike Detailed Terms http://creativecommons.org/licenses/by-nc-sa/4.0/ Intra-chain organisation of hydrophobic residues controls inter-chain aggregation rates of amphiphilic polymers Patrick Varilly,1 Adam P. Willard,2 Julius B. Kirkegaard,3 Tuomas P. J. Knowles,1, 4 and David Chandler5 1)Department of Chemistry, University of Cambridge, Lensfield Road, Cambridge CB2 1EW, UK 2)Department of Chemistry, MIT, Cambridge MA, USA 3)Department Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK 4)Cavendish Laboratory, Department of Physics, University of Cambridge, J J Thomson Avenue, Cambridge CB3 0HF, UK 5)Department of Chemistry, University of California, Berkeley, California 94720, USA Aggregation of amphiphiles through the action of hydrophobic interactions is a common feature in soft condensed mat- ter systems and is of particular importance in the context of biophysics as it underlies both the generation of functional biological machinery as well as the formation of pathological misassembled states of proteins. Here we explore the aggregation behaviour of amphiphilic polymers using lattice Monte-Carlo calculations and show that the distribution of hydrophobic residues within the polymer sequence determines the facility with which dry/wet interfaces can be created and that such interfaces drive the aggregation process. Keywords: Hydrophobic effect — Protein aggregation — Polypeptide sequence Due to their importance in governing self-assembly of bio- grained density field ρ(~r) which is readily simulated with a logical components, hydrophobic interactions and the mecha- discretized binary field ni which tracks the density fluctua- nism of hydrophobic collapse leading to the aggregation of tions resulting in the appearance of cells i with a lower den- hydrophobic species in an aqueous environment have been sity, ”vapour” cells ni = 0, and ”wet” cells within the bulk studied in detail using approaches ranging from spectroscopy solvent with ni = 1. This description is particularly well to atomistic and coarse-grained simulations1–16. The phe- suited for numerical evaluation as the computationally costly nomenon of hydrophobic collapse by its very nature involves short length scale fluctuations characteristic of atomistic mod- the removal of water molecules between adjacent hydropho- els have been treated analytically. Within this picture, there is bic entities in order to allow for them to come together, a cost to create a wet/dry interface, given by nearest neigh- and therefore the creation of an interface with unsatisfied bour interactions of the form nini±1 and an energy associated hydrogen-bonding separating ”wet” solvent from ”dry” aggre- with the solvation of chemical species niµex. This descrip- gated hydrophobes in a manner reminiscent of a liquid-vapour tion is therefore equivalent to a 3D lattice gas system with the phase transition. The picture that has emerged from computa- Hamiltonian: tional studies of the collapse of hydrophobic chains is that it X X is the creation of such interfaces which controls the transition H[fnig; fhig] = [−µ + µexhi]ni + ninj (1) between solvated hydrophobes and their compact aggregated i hi;ji state17–19. In biological systems, hydrophobic units rarely oc- cur in isolation and are most commonly part of a macromolec- where µ is the solvent chemical potential and h::: i indicates ular system. In the present work, we investigate using lattice summation over nearest neighbours on the lattice. The pres- Monte-Carlo calculations the nature of the hydrophobic col- ence of hydrophobic solutes at lattice sites i with hi = 1 re- lapse for polymers with a varying distribution of hydrophobic sults in an excess chemical potential µex. The values of the parameters governing the coarse grained and hydrophilic elements and demonstrate that the clustering 4 of hydrophobic entities is crucial for nucleating the formation water degrees of freedom, µ = 3 − 2:25 · 10 kBT and of dry interfaces driving the eventual aggregation process. = 1:51 kBT , can be determined for a lattice size of l = 0:21 nm through comparisons with the experimental bulk values for the isothermal compressibility and the surface tension γ = /(2l2) = 0:07N m−1 of water at room temperature arXiv:1610.00183v1 [physics.bio-ph] 1 Oct 2016 I. OFF-LATTICE SOLUTES IN A LATTICE GAS SOLVENT and 1 atm pressure. The value of µcoex = 3 represents the MODEL chemical potential of the solvent at phase coexistence with the vapour phase, and the small difference µ − µcoex kBT Hydrophobic assembly is characterised by the expulsion highlights the fact that water is close to phase coexistence of water molecules from aggregates of hydrophobic entities. under standard conditions. It has been shown that this de- This effect can be captured by considering the evolution of scription of water reproduces faithfully the key properties as- the local water density field. The short length scale density sociated with hydrophobic interactions, in particular the char- fluctuations contributing to the local density are characterised acteristic solvation free energy changes with increasing so- by rapid relaxation and follow to a very good approxima- lute sizes. This coarse-grained water description has previ- tion Gaussian statistics20. These short scale fluctuations can ously been used to study the collapse of a single hydrophobic therefore be integrated analytically17,21,22, resulting in a coarse chain17, and we extend this approach here to cover the aggre- 2 gation of amphiphilic chains. A B 1.0 Å With the specific parameterization given above, the lattice 4.0 Å solvent used here is below the roughening transition for the three-dimensional Ising model. As a result, there can be lat- tice artifacts due to a tendency of an interface to align with the orientation of the underlying lattice vectors23. A true liquid- vapor interface would not exhibit this behavior. Vaikun- tanathan and Geissler have recently demonstrated that this tendency can give rise to inaccurate solvation free energies for hydrophobic solutes that are nanometer sized or larger24. C This inaccuracy is most significant for aspherical or irregu- 6 5 0 larly shaped solutes, but grows less pronounced as the rough- ening transition is approached from below. In the case pre- sented here the model is only slightly below the roughening 6 0 0 transition, which is located at ≈ 1:64kBT , and so lattice ar- tifacts are only expected to manifest on length scales of about ) 5 5 0 1 − 2`. Since the critical nucleus size for hydrophobic pep- 3 tide aggregation is about 1nm, or ≈ 5`, we expect any lattice (Å artifacts associated with being below the roughening transi- tot V 5 0 0 tion to be negligible for the results presented here. Indeed, the mechanism for collapse found for a hydrophobic chain with 17 the lattice solvent model we employ is consistent with that 4 5 0 found for same hydrophobic chain with an atomistic solvent model.18 For more generally shaped solutes, effects of lat- tice artifacts may best be avoided by adopting Vaikuntanathan 4 0 0 and Geisslers related lattice model,24,25 which is slightly more 2 4 6 8 1 0 1 2 complicated than that of Ref. (17). d(Å) For the hydrophobic segments, the excess chemical poten- −3 FIG. 1. Shown in A, the fine grid on which the overlap volume tial is given by µex = cphobv, where cphob = 60 kBT nm is taken to be the reversible work required to accommodate a is calculated. This is mapped onto the lattice on which the field ni is defined as shown in B. In C, the total volume excluded by two hydrophobe of volume v. Idealised hydrophilic segments are spheres of radius R = 4:20 A˚ when a distance d apart, as calculated water like and as such the excess chemical potential due to exactly (Equation (4)) and by the numerical approximation scheme the presence of hydrophobic solutes vanishes µex = 0. Fur- with spacing 1A.˚ thermore, weak depletion forces act between two hydropho- bic particles and originate from the reduction of volume from which the solute excludes solvent molecules. of Heff [fnig] is then simply a (non-trivial) bookkeeping ex- The solvent degrees of freedom in the model, fnig, can ercise. In describing our scheme, we treat cell indices a as be efficiently sampled using a Metropolis Monte Carlo al- vectors that can be added and subtracted. We denote by xa gorithm. By contrast, sampling the solute degrees of free- the coordinates of the corner of cell a with lowest Cartesian dom is more complex. The principal problem is to calcu- components. For a solvent-excluding sphere of radius R cen- late the overlap volume va between the excluded volume v tered at x0, we can pre-calculate the overlap volumes v^s of all and a given fine cell a. The volume v is typically a union cells s by any method, such as Monte Carlo integration. We of overlapping spheres, one for each excluded volume asso- do this once at the beginning of a simulation. ciated with a solute particle, here a polymer segment. Pre- Generically, the center r of a solvent-excluding sphere viously in Refs. (17) and (10), an interpolation scheme was will not coincide with a cell corner.
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