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Explicit Incorporation of Hard and Soft Protein-Protein Interactions Into Models for Crowding Effects in Binary and Ternary Protein Mixtures I. Comparison of Approximate Analytical Solutions With Numerical Simulation Travis Hoppe, and Allen P. Minton J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b07736 • Publication Date (Web): 25 Oct 2016 Downloaded from http://pubs.acs.org on October 26, 2016

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The Journal of Physical Chemistry

Revised submission - J Phys Chem B – Oct 2016

Incorporation of Hard and Soft Protein-Protein Interactions Into Models for Crowding Effects in Binary and Ternary Protein Mixtures. Comparison of Approximate Analytical Solutions with Numerical Simulation

Travis Hoppe1 and Allen P. Minton2*

1

Laboratory of Chemical Physics and 2Laboratory of Biochemistry and Genetics, National

Institute of Diabetes and Digestive and Kidney Diseases, National Institutes of Health, United States Public Health Service, Bethesda MD 20892 USA *Correspondence to Dr. Allen P. Minton, Building 8, Room 226, NIH, Bethesda MD 20892. Tel: (+1) 301-496-3604. Email: [email protected]

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Abstract: In order to better understand how nonspecific interactions between solutes can modulate specific biochemical reactions taking place in complex media, we introduce a simplified model aimed at elucidating general principles. In this model, solutions containing two or three species of interacting globular proteins are modeled as a fluid of spherical particles interacting through square well potentials that qualitatively capture both steric hard core repulsion and longer-ranged attraction or repulsion. The excess chemical potential, or free energy of solvation, of each particle species is calculated as a function of species concentrations, particle radii, and square well interaction range and depth. The results of analytical models incorporating two-body and three-body interactions are compared with the estimates of free energy obtained via Widom insertion into simulated equilibrium square-well fluids. The analytical models agree well with results of numeric simulations carried out for a variety of model parameters and fluid compositions up to a total particle volume fraction of ca. 0.2.


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Introduction The effect of high concentrations of unrelated macromolecules and small organic molecules upon the equilibria and kinetics of specific reactions of biological macromolecules has been the subject of numerous experimental and theoretical studies during the last 40 years

1-5

. A

general thermodynamic description of the effect of nonspecific interactions upon chemical equilibria was first presented by Minton 6 and may be summarized as follows. Consider the following general reversible reaction: n1 X 1 + n2 X 2 + ... + n j X j

n j +1 X j +1 + n j + 2 X j + 2 + ... + nq X q

[1]

with a conventionally defined equilibrium constant

K≡

q

j

∏ c ∏c i

i

i = j +1

i =1

where ci denotes the molar concentration of reactant species i at equilibrium. The value of K depends upon the magnitude of interactions between all solute species, not just those participating in the reaction: j

q

i =1

i = j +1

ln K = ln K 0 (T , P ) + ∑ ln γ i − ∑ ln γ i

[2]

where K0 denotes the true equilibrium constant defined for an ideal solution at a defined temperature and pressure, and γi the thermodynamic activity coefficient of the ith solute species, which depends upon solution composition according to

ln γ i = ∆Gi( nonideal ) (T , P, {c}) RT

[3]

where R denotes the molar gas constant and T the absolute temperature. ∆Gi( nonideal ) (T , P, {c}) denotes the Gibbs free energy of interaction per mole of a molecule of reactant species i with all



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of the other solute molecules in the solution, the composition of which is denoted by the array of solute concentrations {c}. It follows from equations [2] and [3] that in order to estimate the extent to which the presence and amount of non-reactant or “crowder” solutes influence the equilibrium state of a particular test reaction, it is thus necessary to first estimate the free energies of interaction between the crowding species and each reactant species. If kinetic information is additionally sought, an estimate of the free energy of interaction between the crowding species and the transition state is required as well (see appendix to 7). ∆Gi(

nonideal )

is equivalent to the change in

Gibbs free energy per mole accompanying the transfer of a molecule of reactant species i from an infinitely dilute or ideal solution to a solution of composition {c} , sometimes referred to as the excess free energy of solvation. Although a real-world process would take place under constant pressure, since typical protein reactions in solution (unfolding, association) are accompanied by volume changes of less than 1%

8

, the assumption of constant volume is quite

good, permitting substitution of the more readily calculated Helmholtz free energy of solvation

∆Ai( nonideal ) for the Gibbs free energy of solvation without significant loss of accuracy. Estimation of the free energy of solute-solute interaction in solution is based upon the use of models for the potential of mean force (PMF) or effective potential acting between solute molecules in solution 9. In early theoretical treatments focusing on the role of excluded volume or steric repulsion between macromolecular solutes it was assumed that the PMF could be approximated by a purely repulsive hard particle potential acting between equivalent convex particles representing each macrosolute species 3,6. The influence of small molecules (solvent, electrolytes) was taken into account implicitly as modulating the PMF acting between 4 ACS Paragon Plus Environment

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macrosolutes. In order to take into account experimentally detectable longer-ranged repulsive and attractive electrostatic interactions between macrosolutes, an equivalent hard particle model was introduced in which the size of the equivalent particle could be increased to simulate long range repulsion or decreased to simulate long range attraction that partially compensates for steric repulsion 10,11. More recently the influence of nonspecific attractive interactions between crowders and probe macromolecules has received increasing attention (see for example 12,13). Depending upon the system studied, it has been reported that high concentrations of added macromolecules can either stabilize or destabilize proteins with respect to unfolding and can either facilitate or inhibit macromolecular associations 14-17. It became evident that the equivalent hard particle model cannot account for a net attractive interaction between a test macrosolute and background species. Simple semi-empirical models containing explicit rather than implicit contributions of attractive interactions to the total free energy of interaction between a test macrosolute and a single background species were therefore proposed 10,15,18-21. In parallel, Brownian dynamics and Monte Carlo simulations were performed for model systems containing both hard core exclusion and weak attraction between probe and a single species of crowding molecules 18,22-24. Many investigations of these phenomena are motivated by a desire to understand the role of comparable nonspecific interactions in complex biological media. In recent years attempts have been made by experimentalists to study the behavior of labeled trace molecules that have been introduced into intact cells 12,25-27. In contrast to a controlled medium in vitro, the composition of the local environment (or environments) of the probe molecules in an intact cell is poorly defined, rendering the interpretation of observed behavior uncertain.



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At the same time, efforts have been made to study the equilibrium and transport behavior of dilute probe molecules by means of Brownian dynamic computer simulations of models for cytoplasm containing a large variety of macromolecules and small organic molecules with assumed potentials of interaction between crowding and probe molecules 13,28,29. As complex and computation-intensive as these simulations are, they fail to capture global features of the cellular interior such as the high degree of reversible macromolecular association, the presence of membranes and other static structures with which soluble molecules may interact, and the heterogeneous nature of local environments that vary with time and location within the cell. Moreover, the complexity of the simulated system and the corresponding computational burden makes it unfeasible to systematically study the effect of varying composition and assumed potentials of interaction between the various molecular species. The limited experimental information currently available indicates that the effect of multiple background species (mixed crowders) upon reactions of trace species in vitro may be far from additive in the effects of each background species individually 30-33. It is evident that in order to attain a satisfactory understanding of crowding effects in media as complex as cytoplasm, the effect of multiple crowding species upon the free energy of interaction between a test species and its solution environment must be better understood. Until now, the only analytical treatments available for systematic exploration of mixed crowder effects have been an equivalent hard particle model 30 that does not allow for the presence of net attractive interactions between crowder and reactant species and a modified equivalent hard particle model with a first-order attractive correction to the tracer-crowder interaction 34, to be discussed below. We propose to employ simplified analytical models explicitly incorporating attractive as well as repulsive interactions between all macromolecular solutes to systematically explore the 6 ACS Paragon Plus Environment

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effects of varying the size, number, and composition of multiple background species, and the effective potentials of interaction between background and reactant species, upon the equilibria governing a variety of prototypical chemical reactions. We believe that such models can serve to elucidate qualitative properties of such systems that are generally valid and independent of the details of molecular structure and precise nature of the intermolecular interaction potentials. The simplest model of the potential of mean force acting between spherical particles of species i and j, denoted by Uij, that allows for the presence of both hard (excluded volume) and longer-ranged soft interactions is the well-known square-well (SW) potential: ∞  U ij ( rij ) = ε ij 0 

rij < ri + rj ri + rj ≤ rij < ri + rj + ∆ ij

[4]

ri + rj + ∆ ij ≤ rij

where ri and rj denote the “hard” radii of spheres representing the ith and jth solute species, rij the distance between the centers of the two spheres, and ∆ij the range of the square-well potential of mean force acting between the two species. In the present report we shall present exact and approximate analytically soluble models that permit rapid calculation of the excess chemical potential, or solvation free energy, of each solute species in fluids containing mixtures of up to three spherical solute species interacting through additive square well potentials. We further demonstrate that these analytical models are able to reproduce semiquantitatively the results of accurate numerical calculations of the excess chemical potential of each of the solute species in SW fluids containing up to three distinct solute species, interacting via five distinct potentials of binary interaction, at total volume fractions of up to 0.2.



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Analytical calculation of excess chemical potential The logarithm of the activity coefficient of solute species i is defined as that part of the chemical potential of species i deriving from solute-solute interaction, or the difference between the free energy per mole associated with placing a molecule of species i into a solution of unit volume containing other solutes at number densities {ρ } and that of placing it into the same volume of solvent. According to the solution theory of McMillan and Mayer 9, ln γ i may be expanded in powers of the number densities of all solute species:

ln γ i = ∑ Bij ρ j + ∑ Bijk ρ j ρ k + ... j

[5]

i, j

where each of the coefficients is a known function of the potential of mean force acting between 2, 3, ... solute molecules. The number of terms required to describe the system accurately depends upon the particle densities. For spherically symmetric potentials of mean force the twobody and three-body interaction coefficients are given by

Bij =

80π 2 Bijk = 9

4π 3

∞

∫ f (r ) r ij

ij

2

[6]

drij

0

∞∞∞

∫ ∫ ∫ f (r ) f (r ) f (r ) r r r ij

ik

jk

ij ik jk

drij drik drjk

[7]

0 0 0

where f ( rij ) ≡ 1 − exp  −U ij ( rij ) kT  , and U ij ( rij ) denotes the potential of mean force acting between solute molecules of species i and j at a distance rij between the centers of the two interacting particles 35. Equation [7] is based upon the assumption that multibody potentials of


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interaction are linearly additive in the constituent pairwise potentials, i.e., U ijk ( rij , rik , rjk ) = U ij ( rij ) + U ik ( rik ) + U jk ( rjk ) .

The evaluation of the integral appearing in equation [6] is trivial, but analytical evaluation of the triple integral appearing in equation [7] is prohibitively difficult for all but hard sphere potentials, and was accomplished for square well potentials only by Kihara 36,37. The resulting equations are reproduced in Appendix 1. In the results section, values of ln γ i calculated for various combinations of composition, particle size, and SW parameters using equations [4], [A1], and [A2] are compared with those obtained via numerical simulation. We shall refer to this calculation as the Kihara model in acknowledgement of his elegant solution of the three body SW problem. It will be seen that at higher total volume fractions, the results of Kihara calculations are systematically lower than those calculated numerically. This is not due to inaccuracy of the interaction coefficients, but rather a result of truncating equation [3] after the second term on the right-hand side. In order to extend the validity of the exact analytical model to higher volume fractions, the following approximation was made. We partition the excess free energy into contributions from steric repulsion (“hard”) and longer-ranged (“soft”) interactions:

ln γ i = ln γ ihard + ln γ isoft

[6]

The hard contribution is calculated using the scaled particle theory of hard sphere mixtures 38, which has been found to provide results in good agreement with the results of Monte Carlo simulations of binary mixtures of hard spheres at total volume fractions of up to 0.4 (unpublished results). The soft contribution is calculated via the following modification of equation [1]:



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ln γ isoft = ∑ Bijsoft ρ j + ∑ Bijksoft ρ j ρ k j

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[7]

i, j

where the soft two and three body interaction coefficients are obtained by subtracting the hard core contribution to the corresponding total interaction coefficient. Analytical expressions for

ln γ ihard , Bijsoft and Bijksoft are presented in Appendix 2. The approximate model described by equations [6],[7],[A3], [A4] and [A5] will be henceforth referred to as the hybrid model. In the results section we present a comparison of results obtained via numerical simulation, the Kihara model, and the hybrid model. Compositions of binary fluid mixtures are specified by φtot , the total fraction of volume occupied by spherical particles, and f1, the mole fraction of species 1. These are related to the number densities of species 1 and 2 by the following relations:

φ1 = φtot

f1r13 f1r13 + (1 − f1 ) r23

φ2 = φtot − φ1 φi ρi = 4π ri 3 3

[8]

where ri denotes the hard core radius of the ith species. We note that the values of ln γi calculated numerically and analytically for specified values of φtot and f1 depend only on the relative sizes (radii) of interacting species and the range of SW interaction potentials, and not on their absolute values.


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Description of numerical simulations To validate the analytical results derived in the following sections, a series of molecular dynamics simulations of SW fluids were performed over a range of SW parameters. Since the square-well potential acting between each pair of spherical particles is both isotropic and piecewise continuous, the equations of motion can be propagated exactly. This was accomplished using the discrete molecular dynamics (DMD) software DynamO 39. All systems were initialized as point particles in random positions to ensure no overlap. For all simulations, there were a total of N=4000 molecules, where the number density of each species in a binary mixture of particles was determined by the specified mole fraction of the first species. The maximum total volume fraction explored was φ = 0.3 , and interaction potentials were chosen to be below those leading to a phase separation 40. During an initial phase the spheres were given randomized velocities and slowly grown to their final radius. Once the system had been packed to the target density, the square-well potential was switched on and the system was coupled to an Andersen thermostat until the desired temperature was reached. After a final equilibration at the target temperature and density, a data collection phase was initiated. In this phase, snapshots of the particle configurations were saved at regular intervals. To calculate the excess chemical potential, we employed the Widom test insertion method 41,42. Here the chemical potential is estimated by repeated attempts of a random insertion of a test particle into the saved configurational snapshots. For each attempt, the change in energy due to the insertion of the test particle, ∆Et , is calculated. If the test particle overlaps the hardsphere radius of another particle in the system, the energy is infinite. If the test particle does not overlap, the energy is an integer combination of square well interaction energies, which



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facilitates the data collection process. The chemical potential is estimated as the average over all insertion attempts according to

µ − µ0 = − kT ln exp ( − ∆Et kT )

V ,T

[9]

where µ0 denotes the chemical potential of an ideal gas of N particles at temperature T in the same volume V, and k denotes the Boltzmann constant. Test insertions are repeated until the average indicated in equation [2] becomes constant to within 1% of the reported value. The full range of parameters explored, along with the measured excess chemical potential for each species, is given in the supplementary information.


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Results and Discussion Example 1: Binary mixtures of equally sized spheres with repulsive self-interaction and attractive hetero-interaction. Numeric simulations were carried out for spheres of radius 1 interacting via SW potentials with interaction range ∆ ≡ ∆11 = ∆ 22 = ∆12 = 0.2 and interaction strength

ε ≡ ε11 = ε 22 = −ε12 = 0, 0.5, 1.0, 1.5 and 2.0 kT at multiple total volume fractions up to 0.3 and multiple values of f1 covering the entire range between 0 and 1. Analytical calculations were performed for the same compositions. A subset of the results are plotted in Figures 1A-C, and a complete set of the data abstracted in Figure 1B may be viewed in the supplementary file

Rotating Fig 1B.mpg. In accordance with intuition, the free energy of interaction of each species with its solution environment is most positive when the solution consists mostly or primarily of itself, with which it interacts repulsively. Likewise, the free energy of interaction of each species with its solution environment is most negative when the solution consists mostly or primarily of the other species, with which it has an attractive soft interaction. Quantitatively, we observe quantitative or semiquantitative agreement of both analytical models with the results of simulation at total volume occupancies up to ca. 0.2.

Example 2: Binary mixtures of unequally sized spheres with asymmetric interactions. SW parameters for this simulation were obtained by fitting light scattering data obtained from mixtures of ovalbumin (species 1) and ribonuclease (species 2) at pH 7 (Wu and Minton 2015) with a square well interaction model (manuscript in preparation). Note that at pH 7,



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ovalbumin carries a net negative charge and ribonuclease carries a net positive charge. Thus soft homo-interactions are repulsive and soft hetero-interactions are attractive. While the parameter values given below are not uniquely determined by the data, they do account quantitatively for the experimentally observed dependence of scattering intensity upon solution composition over a range of total protein concentration up to 40 g/l. ri and ∆ij are specified in Ångstroms, and εij in units of kT: r1 = 23.9, r2 = 16.9, ∆11 = 29, ∆22 = 10, ∆12 = 8.3, ε 11 = ε 22 = −ε12 = 0.5. A subset of results calculated numerically and analytically are plotted in Figures 2A-B. Complete sets of the data abstracted in Figures 2A-B may be viewed in the supplementary files Rotating Fig 2A.mpg and Rotating Fig 2B.mpg. As in the preceding case, the total free energy of interaction of each species with its solution environment is lowest when the solution is composed primarily of the other species, with which it has an attractive soft interaction. The result of both model calculations are in good agreement with the results of numeric simulation up to total volume occupancies of ca. 0.2.

Example 3: Excess chemical potential of a third trace species in a mixture of two crowder species. The primary motivation for developing an analytical description of the combined effect of soft and hard solute-solute interactions in multicomponent square well fluids is to provide a simple model for qualitative exploration of the effect of mixed crowders on a variety of prototypical biochemical reactions involving dilute reactants and products in crowded solutions. In principle, the present model would enable us to estimate the activity coefficient of any solute species in a solution containing an arbitrary number of crowder species over a finite range of


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composition. In the example below we calculate the activity coefficient of trace species 3 (ϕ3 ~ 0), as a function of the concentrations of two crowder species. A numeric calculation was carried out using the following square well parameters: r1 = 20, r2 = 25, r3 = 25, all ∆ ij = 5, ε ≡ ε11 = ε 22 = −ε12 = −ε13 = −ε 23 = 1.5 kT . Since species 3 is presumed to be infinitely dilute, no self-interaction need be specified. However, the analytical calculations are applicable to any number of species at arbitrary concentration, and ε33 and ∆33 may be specified when the concentration of species 3 is significant. Partial results are plotted in

Figures 3A-C, and complete sets of the data abstracted in these figures may be viewed in the supplementary files Rotating Fig 3A.mpg, Rotating Fig 3B.mpg, and Rotating Fig 3C.mpg. As in the previous two examples, species 1 and 2 have a repulsive soft self-interaction and an attractive soft hetero-interaction. Trace species 3 has a soft attractive interaction with both crowding species. The results of analytical calculations of the free energies of interaction of crowding species 1 and 2 with the solution environment are qualitatively similar to those obtained in the previous two examples and in fairly good quantitative agreement with the results of numeric simulation up to a total volume fraction of ca 0.2. A novel result is obtained for the interaction of trace species 3 with the solution environment: at total crowder volume fractions of up to ca. 0.2, the total free energy of interaction of species 3 with the solution environment is very small (compare the units of ln γ in Fig 3C with those in Figs 3A and 3B), and almost independent of the mole fraction of component 1. This result is due to accidental nearcancellation of substantial but nearly equal contributions of net repulsive and net attractive interactions of species 3 with species 1 and 2. It is of special interest to note that the cancelling effects are well captured by both analytical models, but especially well by the hybrid model.



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It may be seen from the results presented above and in the supplementary files that the analytical Kihara and hybrid calculations can qualitatively and even semi-quantitatively reproduce the numerically calculated results obtained under a variety of conditions, particularly for total volume fractions less than 0.2. Since the average partial specific volumes of polypeptides is ~0.73 cm3/g, 43 this limit would correspond to a maximum total protein concentration of ~270 g/l. In general, the hybrid model seems to be slightly more accurate, but its accuracy declines as attractive interactions become stronger. This is to be expected, as within the hybrid model the contribution of soft attractive interactions to the total free energy of interaction is calculated only to three body interactions (eq [7]), while scaled particle theory calculation of the contribution of steric repulsion to total free energy is approximately valid with respect to higher order multibody interactions as well. However, within its assessed range of applicability, the calculation of the excess chemical potential of a third tracer species using the hybrid approximation is surprisingly accurate given that the overall result represents a small difference between two large and opposing effects. The analytical models presented here (Hoppe-Minton or HM models for brevity) may be compared to an analytical model previously presented by Kim and Mittal (KM model) that explicitly incorporates net attractive tracer-crowder interactions 18. Both KM and HM models represent macromolecular solutes as particles interacting via spherically symmetrical potentials of mean force. Both the KM and the hybrid HM models partition the crowder-tracer interaction into steric-repulsive and longer ranged attractive contributions, and calculate the steric-repulsive part of the solvation free energy using hard sphere fluid theory. The models presented here differ significantly from that of KM in three ways: (1) In the KM model, the attractive part of the tracer-crowder potential of mean force is represented as the product of a parametric 16 ACS Paragon Plus Environment

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coefficient, the surface area of the tracer and the concentration of crowder, analogous to a model attractive potential proposed independently by Minton 19. An attractive potential of this functional form (i.e. linear in crowder concentration) can at best account for only two-body interactions, limiting its validity to initial deviations from ideal behavior. In contrast, the models proposed here explicitly take into account three-body as well as two-body interactions, extending the validity of the analytical treatment to higher concentrations. (2) The models proposed here, unlike that of KM, take into account crowder-crowder interactions as well as tracer-crowder interactions. In a subsequent report we shall demonstrate that crowder-crowder interactions can significantly modulate tracer-crowder interactions and therefore should not be neglected. (3) The models presented here were developed specifically in order to study the effect of mixtures of multiple species of crowders upon tracer reactions, and rigorously treat all attractive and repulsive interactions in a square well fluid to at least second order. In order to generalize the treatment of KH to multiple crowders it would be necessary to assume that the effects of individual crowders are additive in the mixture. The present work and a number of experimental studies 31-33 demonstrate that the assumption of additivity is approximately true only when all intermolecular interactions are predominantly repulsive, a condition that cannot be assumed to prevail generally. The analytical models described here are presently being used to systematically explore the effects of individual and mixed crowders upon equilibria of several prototypical macromolecular reactions, including conformational isomerization, limited self-association, hetero-association, and condensation. It is anticipated that the results of these continuing studies will add to our knowledge and understanding of the effects of crowding in heterogeneous media.



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Conclusion The excess chemical potential of each of up to three solute species interacting with each other in solution via a square well potential of mean force has been calculated via approximate analytical expressions and by direct numerical simulation. It is found that the results of the analytical expressions are in semi-quantitative agreement with the results of numeric simulation over a wide range of solute composition and model potential interaction parameters, subject to a maximum total solute volume fraction of ca. 0.2.


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SUPPLEMENTARY MATERIAL Rotating Figure 1B, Rotating Figure 2A, Rotating Figure 2B, Rotating Figure 3A, Rotating Figure 3B, Rotating Figure 3C

ACKNOWLEDGEMENTS This research was supported by the Division of Intramural Research of the National Institute of Diabetes and Digestive and Kidney Diseases, and utilized the computational resources of the NIH HPC Biowulf cluster (http://hpc.nih.gov).



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APPENDIX 1 - Two and three body interaction coefficients for a square well potential Let σ ij = ri + rj , λij = ri + rj + ∆ij , and xij = exp ( ε ij kT ) − 1 . Substitution of equation [1] into equation [4] yields:

Bij =

4π 3 σ ij − xij λij3 − σ ij3  3

{

}

[A1]

Results obtained by Kihara 36,37 are equivalent to:

{

}

Bijk = (1 2 ) I 0 −  xij I1.1 + xik I1.2 + x jk I1.3  +  xij xik I 2.1 + xij x jk I 2.2 + xik x jk I 2.3  − xij xik x jk I 3 [A2] where I 0 = W (σ ij , σ ik , σ jk ) I1.1 = W ( λij , σ ik , σ jk ) − I 0 I1.2 = W (σ ij , λik , σ jk ) − I 0 I1.3 = W (σ ij , σ ik , λ jk ) − I 0 I 2.1 = W ( λij , λik , σ jk ) − I 0 − I1.1 − I1.2 I 2.2 = W ( λij , σ ik , λ jk ) − I 0 − I1.1 − I1.3 I 2.3 = W (σ ij , λik , λ jk ) − I 0 − I1.2 − I1.3 I 3 = W ( λij , λik , λ jk ) − I 0 − ∑ ( I1.t + I 2.t ) t

and


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 a 6 + b6 + c 6 + 18a 2b 2 c 2 + 16 ( a 3b 3 + b3c 3 + a 3c 3 )  W ( a , b, c ) = (π 18 )   −9 a 4 ( b 2 + c 2 ) + b 4 ( a 2 + c 2 ) + c 4 ( a 2 + b 2 )    when a < b + c 2

{

}

(16π 9 ) b c 2

3 3

when a ≥ b + c

APPENDIX 2 - Hybrid model The excess chemical potential of the ith species of particle is calculated according to text equation [6]. The contribution of hard repulsive (steric) interactions between spherical particles to the excess chemical potential is calculated to all orders of concentration according to the scaled particle theory of hard sphere mixtures 38: 2 kT ln γ ihard = − ln (1 − ξ3 ) +  6ξ2 (1 − ξ3 )  ri + 12ξ1 (1 − ξ3 ) + 18ξ22 (1 − ξ3 )  ri 2   2 3 + 8ξ0 (1 − ξ3 ) + 24ξ1ξ2 (1 − ξ3 ) + 24ξ23 (1 − ξ3 )  ri3  

[A3]

where n

ξl ≡ (π 6 ) ∑ ρ i ( 2ri )

l

i =1

and n is the number of species of spherical particles. The contribution of longer ranged “soft” interactions is calculated according to text equation [7], with Bijsoft = Bij − 4πσ ij3 3

[A4]

Bijksoft = Bijk − I 0 2

[A5]

and



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REFERENCES (1) Gnutt, D.; Ebbinghaus, S. The macromolecular crowding effect - from in vitro into the cell. Biol. Chem. 2016, 397, 37-44. (2) Hall, D.; Minton, A. P. Macromolecular crowding: qualitative and semiquantitative successes, quantitative challenges. Biochim Biophys Acta 2003, 1649, 127-139. (3) Minton, A. P. The effect of volume occupancy upon the thermodynamic activity of proteins: some biochemical consequences. Mol. Cell. Biochem. 1983, 55, 119-140. (4) Zhou, H.-X.; Rivas, G.; Minton, A. P. Macromolecular crowding and confinement: biochemical, biophysical, and potential physiological consequences. Annu. Rev. Biophys. 2008, 37375-397. (5) Zimmerman, S. B.; Minton, A. P. Macromolecular crowding: biochemical, biophysical, and physiological consequences. Annu. Rev. Biophys. Biomol. Struct. 1993, 22, 2765. (6) Minton, A. P. Excluded volume as a determinant of macromolecular structure and reactivity. Biopolymers 1981, 20, 2093-2120. (7) Minton, A. P. Effects of excluded surface area and adsorbate clustering on surface adsorption of proteins II. Kinetic models. Biophys. J. 2001, 80, 1641-1648. (8) Kahn, P. C.; Schwanwede, J. M.; Ippolito, A. M.; Mihalyfi, B. Volume changes of globular protein association. Biophys. J. 1980, 32, 86-87. (9) McMillan, W. G., Jr.; Mayer, J. E. The statistical thermodynamics of multicomponent systems. J. Chem. Phys. 1945, 13, 276-305. (10) Minton, A. P. A molecular model for the dependence of the osmotic preessure of bovine serum albumin upon concentration and pH. Biophys. Chem. 1995, 57, 65 - 70. (11) Minton, A. P.; Edelhoch, H. Light scattering of bovine serum albumin solutions: extension of the hard particle model to allow for electrostatic repulsion. Biopolymers 1982, 21, 451-458. (12) Phillip, Y.; Schreiber, G. Formation of protein complexes in crowded environments - from in vitro to in vivo. FEBS Letters 2013, 587, 1046-1052. (13) Wang, Y.; Sarkar, M.; Smith, A. E.; Krois, A. S.; Pielak, G. J. Macromolecular crowding and protein stability. J. Am. Chem. Soc. 2012, 134, 16614-16618. (14) Aden, J.; Wittung-Stafshede, P. Folding of an unfolded protein by macromolecular crowding in vitro. Biochemistry 2014, 53, 2271-2277. 22 ACS Paragon Plus Environment

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FIGURE CAPTIONS

Fig 1. Dependence of the excess chemical potential of species 1 upon total volume fraction in a fluid mixture containing a mole fraction of species 1 equal to 1 (black curves and symbols), 0.52 (blue curves and symbols) and 0.1 (red curves and symbols). Panels A, B and C present results obtained with ε = 1.0, 1.5 and 2.0 kT respectively. Symbols represent the results of numeric simulation, dashed curves the results of the Kihara calculation, and solid curves the results of the hybrid calculation. A full comparison of the calculations sampled in Panel B may be viewed in Supplementary File “Rotating Fig 1B.mpg”.

Fig 2. Excess chemical potential of species 1 (A) and species 2 (B) calculated as a function of total fractional volume occupancy and for f1 = 1 (black symbols and curves), 0.5 (blue symbols and curves) and 0 (red symbols and curves). Symbols represent the results of numeric simulation, dashed curves the results of the Kihara calculation, and solid curves the results of the hybrid calculation. A full comparison of the calculations sampled in Panels A and B may be viewed in Supplementary Files “Rotating Fig 2A.mpg” and “Rotating Fig 2B.mpg”.

Fig 3. Excess chemical potential of each of three species in a fluid containing varying concentrations of species 1 and 2 and a trace amount of species 3, plotted as a function of total volume occupancy and mole fraction of species 1 equal to 0 (black), .46 (blue) and 1 (red). Symbols represent the results of numeric simulation, dashed curves the results of the Kihara calculation, and solid curves the results of the hybrid calculation. A full comparison of the 26 ACS Paragon Plus Environment

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calculations sampled in Panels A, B, and C may be viewed in Supplementary Files “Rotating Fig 3A.mpg”, “Rotating Fig 3B.mpg”, and “Rotating Fig 3C.mpg”.



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Figure 1


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Figure 2



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Figure 3


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