Proteins aggregate and precipitate from high concentration solutions in a wide variety of problems of natural and technological interest. Consequently, there is a broad interest in developing new ways to model the thermodynamic and kinetic aspects of protein stability in these crowded cellular or solution environments. We use a coarse-grained modeling approach to study the effects of different crowding agents on the conformational equilibria of proteins and the thermodynamic phase behavior of their solutions. At low to moderate protein concentrations, we find that crowding species can either stabilize or destabilize the native state, depending on the strength of their attractive interaction with the proteins. At high protein concentrations, crowders tend to stabilize the native state due to excluded volume effects, irrespective of the strength of the crowder-protein attraction. Crowding agents reduce the tendency of protein solutions to undergo a liquid-liquid phase separation driven by strong protein-protein attractions. The aforementioned equilibrium trends represent, to our knowledge, the first simulation predictions for how the properties of crowding species impact the global thermodynamic stability of proteins and their solutions.
Issue Section:
Research Papers
1.
Benedek
, G. B.
, Pande
, J.
, Thurston
, G. M.
, and Clark
, J. I.
, 1999, “Theoretical and Experimental Basis for the Inhibition of Cataract
,” Prog. Retin Eye Res.
1350-9462, 18
, pp. 391
–402
.2.
Harper
, J.
, and Lansbury
, P. T.
, 1997, “Models of Amyloid Seeding in Alzheimer's Disease and Scrapie: Mechanistic Truths and Physiological Consequences of the Time-Dependent Solubility of Amyloid Proteins
,” Annu. Rev. Biochem.
0066-4154, 66
, pp. 385
–407
.3.
Fink
, A. L.
, 1998, “Protein Aggregation: Folding Aggregates, Inclusion Bodies, and Amyloids
,” Fold. Des.
, 3
, pp. R9
–R23
. 1359-02784.
Dobson
, C. M.
, 2001, “The Structural Basis of Protein Folding and Its Links with Human Disease
,” Philos. Trans. R. Soc. London, Ser. B
0962-8436, 356
, pp. 133
–145
.5.
Eisenberg
, D.
, Nelson
, R.
, Sawaya
, M. R.
, Balbirnie
, M.
, Sambashivan
, S.
, Ivanova
, M. I.
, Madsen
, A. O.
, and Riekel
, C.
, 2006, “The Structural Biology of Protein Aggregation Diseases: Fundamental Questions and Some Answers
,” Acc. Chem. Res.
0001-4842, 39
, pp. 568
–575
.6.
DeYoung
, L. R.
, Dill
, K. A.
, and Fink
, A. L.
, 1993, “Aggregation and Denaturation of Apomyoglobin in Aqueous Urea Solutions
,” Biochemistry
0006-2960, 32
, pp. 3877
–3886
.7.
Wetzel
, R.
, 1994, “Mutations and Off-Pathway Aggregation of Proteins
,” Trends Biotechnol.
0167-7799, 12
, pp. 193
–198
.8.
Kendrick
, B. S.
, Carpenter
, J. F.
, Cleland
, J. L.
, and Randolph
, T. W.
, 1998, “A Transient Expansion of the Native State Precedes Aggregation of Recombinant Human Interferon-γ
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 95
, pp. 14142
–14146
.9.
Goldberg
, M. E.
, Rudolph
, R.
, and Jaenicke
, R.
, 1991, “A Kinetic Study of the Competition Between Renaturation and Aggregation During the Refolding of Denatured-Reduced Egg White Lysozyme
,” Biochemistry
0006-2960, 30
, pp. 2790
–2797
.10.
Safar
, J.
, Roller
, P. P.
, Gajdusek
, D. C.
, and Gibbs
, C. J.
, 1994, “Scrapie Amyloid (Prion) Protein has the Conformational Characteristics of an Aggregated Molten Folding Intermediate
,” Biochemistry
0006-2960, 33
, pp. 8375
–8383
.11.
Chi
, E. Y.
, Krishnan
, S.
, Randolph
, T.
, and Carpenter
, J.
, 2003, “Physical Stability of Proteins in Aqueous Solutions: Mechanism and Driving Forces in Nonnative Protein Aggregation
,” Pharm. Res.
0724-8741, 20
, pp. 1325
–1336
.12.
Roberts
, C. J.
, 2003, “Kinetics of Irreversible Protein Aggregation: Analysis of Extended Lumry–Eyring Models and Implications for Predicting Protein Shelf Life
,” J. Phys. Chem. B
1089-5647, 107
, pp. 1194
–1207
.13.
Fawzi
, N. L.
, Chubukov
, V.
, Clark
, L. A.
, Brown
, S.
, and Head-Gordon
, T.
, 2005, “Influence of Denatured and Intermediate States of Folding on Protein Aggregation
,” Protein Sci.
0961-8368, 14
, pp. 993
–1003
.14.
Roberts
, C. J.
, Darrington
, R. T.
, and Whitley
, M. B.
, 2003, “Irreversible Aggregation of Recombinant Bovine Granulocyte-Colony Simulating Factor (bg-csf) and Implications for Predicting Protein Shelf Life
,” J. Pharm. Sci.
0022-3549, 92
, pp. 1095
–1111
.15.
Weiss
, W. F.
, IV, Hodgdon
, T. K.
, Kaler
, E. W.
, Lenhoff
, A. M.
, and Roberts
, C. J.
, 2007, “Nonnative Protein Polymers: Structure, Morphology, and Relation to Nucleation and Growth
,” Biophys. J.
0006-3495, 93
, pp. 4392
–4403
.16.
Andrews
, J. M.
, and Roberts
, C. J.
, 2007, “A Lumry–Eyring Nucleated Polymerization Model of Protein Aggregation Kinetics 1: Aggregation With Pre-Equilibrated Unfolding
,” J. Phys. Chem. B
1089-5647, 111
, pp. 7897
–7913
.17.
Roberts
, C. J.
, 2007, “Non-native Protein Aggregation Kinetics
,” Biotechnol. Bioeng.
0006-3592, 98
, pp. 927
–938
.18.
DeYoung
, L. R.
, Fink
, A. L.
, and Dill
, K. A.
, 1993, “Aggregation of Globular Proteins
,” Acc. Chem. Res.
0001-4842, 26
, pp. 614
–620
.19.
Georgiou
, G.
, Valax
, P.
, Ostermeier
, M.
, and Horowitz
, P. M.
, 1994, “Folding and Aggregation of TEM β-Lactamase: Analogies With the Formation of Inclusion Bodies of Escherichia coli
,” Protein Sci.
0961-8368, 3
, pp. 1953
–1960
.20.
Horowich
, A.
, 2002, “Protein Aggregation in Disease: A Role for Folding Intermediates Forming Specific Multimeric Interactions
,” J. Clin. Invest.
0021-9738, 110
, pp. 1221
–1232
.21.
Nephew
, J. B.
, Nihei
, T. C.
, and Carter
, S. A.
, 1998, “Reaction-Induced Phase Separation Dynamics: A Polymer in a Liquid Crystal Solvent
,” Phys. Rev. Lett.
0031-9007, 80
, pp. 3276
–3279
.22.
Tran-Cong
, Q.
, and Harada
, A.
, 1996, “Reaction-Induced Ordering Phenomena in Binary Polymer Mixtures
,” Phys. Rev. Lett.
0031-9007, 76
, pp. 1162
–1165
.23.
Kyu
, T.
, and Lee
, J. H.
, 1996, “Nucleation Initiated Spinodal Decomposition in a Polymerizing System
,” Phys. Rev. Lett.
0031-9007, 76
, pp. 3746
–3749
.24.
Williams
, R. J. J.
, Rozenberg
, B. A.
, and Pascault
, J. -P.
, 1997, “Reaction-Induced Phase Separation in Modified Thermosetting
,” Adv. Polym. Sci.
0065-3195, 128
, pp. 95
–156
.25.
Luo
, K.
, 2006, “The Morphology and Dynamics of Polymerization-Induced Phase Separation
,” Eur. Polym. J.
0014-3057, 42
, pp. 1499
–1505
.26.
Wang
, X.
, Okada
, M.
, Matsushita
, Y.
, Furukawa
, H.
, and Han
, C. C.
, 2005, “Crystal-Like Array Formation in Phase Separation Induced by Radical Polymerization
,” Macromolecules
0024-9297, 38
, pp. 7127
–7133
.27.
Kimura
, K.
, Kohama
, S. -i.
, and Yamazaki
, S.
, 2006, “Morphology Control of Aromatic Polymers in Concert With Polymerization
,” Polym. J.
, 38
, pp. 1005
–1022
. 0032-389628.
Wang
, Y.
, and Annunziata
, O.
, 2008, “Liquid-Liquid Phase Transition of Protein Aqueous Solutions Isothermally Induced by Protein Cross-Linking
,” Langmuir
0743-7463, 24
, pp. 2799
–2807
.29.
San Biagio
, P. L.
, and Palma
, M.
, 1991, “Spinodal Lines and Flory–Huggins Free-Energies for Solutions of Human Hemoglobin HbS and HbA
,” Biophys. J.
0006-3495, 60
, pp. 508
–512
.30.
Sciortino
, F.
, Prasad
, K. U.
, Urry
, D. W.
, and Palma
, M. U.
, 1993, “Self-Assembly of Biopolymeric Structures From Solutions: Mean-Field Critical Behavior and Flory–Huggins Free-Energy of Interactions
,” Biopolymers
0006-3525, 33
, pp. 743
–752
.31.
ten Wolde
, P. R.
, and Frenkel
, D.
, 1997, “Enhancement of Protein Crystal Nucleation by Critical Density Fluctuations
,” Science
0036-8075, 277
, pp. 1975
–1978
.32.
Galkin
, O.
, and Vekilov
, P. G.
, 2000, “Control of Protein Crystal Nucleation Around the Metastable Liquid-Liquid Phase Boundary
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 97
, pp. 6277
–6281
.33.
Serrano
, M. D.
, Galkin
, O.
, Yau
, S. T.
, Thomas
, B. R.
, Nagel
, R. L.
, Hirsch
, R. E.
, and Vekilov
, P. G.
, 2001, “Are Protein Crystallization Mechanisms Relevant to Understanding and Control of Polymerization of Deoxyhemoglobin S?
,” J. Cryst. Growth
0022-0248, 232
, pp. 368
–375
.34.
Chen
, Q.
, Vekilov
, P. G.
, Nagel
, R. L.
, and Hirsch
, R. E.
, 2004, “Liquid-Liquid Phase Separation in Hemoglobin: Distinct Aggregation Mechanisms of the β6 Mutants
,” Biophys. J.
0006-3495, 86
, pp. 1702
–1712
.35.
Vaiana
, S. M.
, Rotter
, M. A.
, Emanuele
, A.
, Ferrone
, F. A.
, and Palma-Vittorelli
, M. B.
, 2005, “Effect of T-R Conformational Change on Sickle-Cell Hemoglobin Interactions and Aggregation
,” Proteins
0887-3585, 58
, pp. 426
–438
.36.
Gliko
, O.
, Neumaier
, N.
, Pan
, W.
, Weichun
, H.
, Haase
, I.
, Fischer
, M.
, Bacher
, A.
, Weinkauf
, S.
, and Vekilov
, P. G.
, 2005, “A Metastable Prerequisite for the Growth of Lumazine Synthase Crystals
,” J. Am. Chem. Soc.
0002-7863, 127
, pp. 3433
–3438
.37.
Vaiana
, S. M.
, Palma-Vittorelli
, M. B.
, and Palma
, M. U.
, 2003, “Time Scale of Protein Aggregation Dictated by Liquid-Liquid Demixing
,” Proteins
0887-3585, 51
, pp. 147
–153
.38.
Kashchiev
, D.
, Vekilov
, P. G.
, and Kolomeisky
, A. B.
, 2005, “Kinetics of Two-Step Nucleation of Crystals
,” J. Chem. Phys.
0021-9606, 122
, p. 244706
.39.
Cheung
, J. K.
, and Truskett
, T. M.
, 2005, “Coarse-Grained Strategy for Modeling Protein Stability in Concentrated Solutions
,” Biophys. J.
0006-3495, 89
, pp. 2372
–2384
.40.
Shen
, V. K.
, Cheung
, J. K.
, Errington
, J. R.
, and Truskett
, T. M.
, 2006, “Coarse-Grained Strategy for Modeling Protein Stability in Concentrated Solutions ii: Phase Behavior
,” Biophys. J.
0006-3495, 90
, pp. 1949
–1960
.41.
Cheung
, J. K.
, Shen
, V. K.
, Errington
, J. R.
, and Truskett
, T. M.
, 2007, “Coarse-Grained Strategy for Modeling Protein Stability in Concentrated Solutions iii: Directional Protein Interactions
,” Biophys. J.
0006-3495, 92
, pp. 4316
–4324
.42.
Fields
, G. B.
, Alonso
, D.
, Stigter
, D.
, and Dill
, K. A.
, 1992, “Theory for the Aggregation of Protein Copolymers
,” J. Phys. Chem.
0022-3654, 96
, pp. 3974
–3981
.43.
Zimmerman
, S. B.
, and Minton
, A. P.
, 1993, “Macromolecular Crowding: Biochemical, Biophysical and Physiological Consequences
,” Annu. Rev. Biophys. Biomol. Struct.
1056-8700, 22
, pp. 27
–65
.44.
Smith
, A. V.
, and Hall
, C. K.
, 2001, “Protein Refolding Versus Aggregation: Computer Simulations on an Intermediate-Resolution Protein Model
,” J. Mol. Biol.
0022-2836, 312
, pp. 187
–202
.45.
Dima
, R. I.
, and Thirumalai
, D.
, 2002, “Exploring Protein Aggregation and Self-Propagation Using Lattice Models: Phase Diagrams and Kinetics
,” Protein Sci.
0961-8368, 11
, pp. 1036
–1049
.46.
Braun
, F. N.
, 2002, “Adhesion and Liquid-Liquid Phase Separation in Globular Protein Solutions
,” J. Chem. Phys.
0021-9606, 116
, pp. 6826
–6830
.47.
Hall
, D.
, and Minton
, A. P.
, 2003, “Macromolecular Crowding: Qualitative and Semiquantitative Successes, Quantitative Challenges
,” Biochim. Biophys. Acta
0006-3002, 1649
, pp. 127
–139
.48.
Kinjo
, A. R.
, and Takada
, S.
, 2003, “Competition Between Protein Folding and Aggregation With Molecular Chaperones in Crowded Solutions: Insight From Mesoscopic Simulations
,” Biophys. J.
0006-3495, 85
, pp. 3521
–3531
.49.
Jang
, H.
, Hall
, C. K.
, and Zhou
, Y.
, 2004, “Thermodynamics and Stability of a β-Sheet Complex: Molecular Dynamics Simulations on Simplified Off-Lattice Protein Models
,” Protein Sci.
0961-8368, 13
, pp. 40
–53
.50.
Nguyen
, H. D.
, and Hall
, C. K.
, 2004, “Molecular Dynamics Simulations of Spontaneous Fibril Formation by Random-Coil Peptides
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 101
, pp. 16180
–16185
.51.
Nguyen
, H. D.
, and Hall
, C. K.
, 2004, “Phase Diagrams Describing Fibrillization by Polyalanine Peptides
,” Biophys. J.
0006-3495, 87
, pp. 4122
–4134
.52.
Sear
, R. P.
, 2004, “Solution Stability and Variability in a Simple Model of Globular Proteins
,” J. Chem. Phys.
0021-9606, 120
, pp. 998
–1005
.53.
Cheung
, M. S.
, Klimov
, D.
, and Thirumalai
, D.
, 2005, “Molecular Crowding Enhances Native State Stability and Refolding Rates of Globular Proteins
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 102
, pp. 4753
–4758
.54.
Dill
, K. A.
, 1985, “Theory for the Folding and Stability of Globular Proteins
,” Biochemistry
0006-2960, 24
, pp. 1501
–1509
.55.
Dill
, K. A.
, Alonso
, D. O. V.
, and Hutchinson
, K.
, 1989, “Thermal Stability of Globular Proteins
,” Biochemistry
0006-2960, 28
, pp. 5439
–5449
.56.
Errington
, J. R.
, 2003, “Direct Calculation of Liquid-Vapor Phase Equilibria From Transition Matrix Monte Carlo Simulation
,” J. Chem. Phys.
0021-9606, 118
, pp. 9915
–9925
.57.
Shen
, V. K.
, and Errington
, J. R.
, 2005, “Determination of Fluid-Phase Behavior Using Transition-Matrix Monte Carlo: Binary Lennard-Jones Mixtures
,” J. Chem. Phys.
0021-9606, 122
, p. 064508
.58.
Errington
, J. R.
, and Shen
, V. K.
, 2005, “Direct Evaluation of Multi-Componenet Phase Equilibria Using Flat Histogram Methods
,” J. Chem. Phys.
0021-9606, 123
, p. 164103
.59.
Klimov
, D. K.
, Newfield
, D.
, and Thirumalai
, D.
, 2002, “Simulations of β-Hairpin Folding Confined to Spherical Pores Using Distributed Computing
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 99
, pp. 8019
–8024
.60.
Ping
, G.
, Yuan
, J. M.
, Mallieres
, M.
, Dong
, H.
, Sun
, Z.
, Wei
, Y.
, Li
, F. Y.
, and Lin
, S. H.
, 2003, “Effects of Confinement on Protein Folding and Stability
,” J. Chem. Phys.
0021-9606, 118
, pp. 8042
–8048
.61.
Takagi
, F.
, Koga
, N.
, and Takada
, S.
, 2003, “How Protein Thermodynamics and Folding Mechanisms are Altered by the Chaperonin Cage: Molecular Simulations
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 100
, pp. 11367
–11372
.62.
Friedel
, M.
, Sheeler
, D. J.
, and Shea
, J. -E.
, 2003, “Effects of Confinement on the Thermodynamics and Kinetics of Folding of a Minimalist β-Barrel Protein
,” J. Chem. Phys.
0021-9606, 118
, pp. 8106
–8113
.63.
Rathore
, N.
, Knotts
, T. A.
, IV, and de Pablo
, J. J.
, 2006, “Confinement Effects on the Thermodynamics of Protein Folding: Monte Carlo Simulations
,” Biophys. J.
0006-3495, 90
, pp. 1767
–1773
.64.
Cheung
, M. S.
, and Thirumalai
, D.
, 2007, “Effects of Crowding and Confinement on the Structures of the Transition State Ensemble in Proteins
,” J. Phys. Chem. B
1089-5647, 111
, pp. 8250
–8257
.65.
Ueda
, Y.
, Taketomi
, H.
, and Go
, N.
, 1978, “Studies on Protein Folding, Unfolding, and Fluctuations by Computer Simulation. ii. A Three-Dimensional Lattice Model of Lysozyme
,” Biopolymers
0006-3525, 17
, pp. 1531
–1548
.66.
Petsev
, D. N.
, Wu
, X.
, Galkin
, O.
, and Vekilov
, P. G.
, 2003, “Thermodynamic Functions of Concentrated Protein Solutions From Phase Equilibria
,” J. Phys. Chem. B
1089-5647, 107
, pp. 3921
–3926
.67.
Privalov
, P. L.
, 1979, “Stability of Proteins: Small Globular Proteins
,” Adv. Protein Chem.
0065-3233, 33
, pp. 167
–241
.68.
Cheung
, J. K.
, Shah
, P.
, and Truskett
, T. M.
, 2006, “Heteropolymer Collapse Theory for Protein Folding in the Pressure-Temperature Plane
,” Biophys. J.
0006-3495, 91
, pp. 2427
–2435
.69.
Stigter
, D.
, Alonso
, D. O. V.
, and Dill
, K. A.
, 1991, “Protein Stability: Electrostatic Compact Denatured States
,” Proc. Natl. Acad. Sci. U.S.A.
0027-8424, 88
, pp. 4176
–4180
.70.
Alonso
, D. O. V.
, Dill
, K. A.
, and Stigter
, D.
, 1991, “The Three States of Globular Proteins: Acid Denaturation
,” Biopolymers
0006-3525, 31
, pp. 1631
–1649
.71.
Singh
, J. K.
, Kofke
, D. A.
, and Errington
, J. R.
, 2003, “Surface Tension and Vapor-Liquid Phase Coexistence of the Square-Well Fluid
,” J. Chem. Phys.
0021-9606, 119
, pp. 3405
–3412
.72.
Shen
, V. K.
, and Errington
, J. R.
, 2004, “Metastability and Instability in the Lennard-Jones Investigated via Transition-Matrix Monte Carlo
,” J. Phys. Chem. B
1089-5647, 108
, pp. 19595
–19606
.73.
Singh
, J. K.
, and Kofke
, D. A.
, 2004, “Molecular Simulation Study of Effect of Molecular Association on Vapor-Liquid Interfacial Properties
,” J. Chem. Phys.
0021-9606, 121
, pp. 9574
–9580
.74.
Singh
, J. K.
, and Errington
, J. R.
, 2006, “Calculation of Phase Coexistence Properties and Surface Tensions of n-Alkanes Using Grand-Canonical Transition-Matrix Monte Carlo Simulation and Finite-Size Scaling
,” J. Phys. Chem. B
1089-5647, 110
, pp. 1369
–1376
.75.
Singh
, J. K.
, Adhikari
, J.
, and Kwak
, S. K.
, 2006, “Vapor-Liquid Phase Coexistence Curves for Morse Fluids
,” Fluid Phase Equilib.
0378-3812, 248
, pp. 1
–6
.76.
Errington
, J. R.
, Truskett
, T. M.
, and Mittal
, J.
, 2006, “Excess-Entropy-Based Anomalies for a Waterlike Fluid
,” J. Chem. Phys.
0021-9606, 125
, p. 244502
.77.
Mittal
, J.
, Errington
, J. R.
, and Truskett
, T. M.
, 2006, “Thermodynamics Predicts how Confinement Modifies the Dynamics of the Equilibrium Hard-Sphere Fluid
,” Phys. Rev. Lett.
0031-9007, 96
, p. 177804
.78.
MacDowell
, L. G.
, Shen
, V. K.
, and Errington
, J. R.
, 2006, “Nucleation and Cavitation of Spherical, Cylindrical, and Slablike Droplets and Bubbles in Small Systems
,” J. Chem. Phys.
0021-9606, 125
, p. 034705
.79.
Errington
, J. R.
, 2003, “Evaluating Surface Tension Using Grand-Canonical Transition-Matrix Monte Carlo Simulation and Finite-Size Scaling
,” Phys. Rev. E
1063-651X, 67
, p. 012102
.80.
Shen
, V. K.
, and Errington
, J. R.
, 2006, “Determination of Surface Tension in Binary Mixtures Using Transition-Matrix Monte Carlo
,” J. Chem. Phys.
0021-9606, 124
, p. 024721
.81.
Shen
, V. K.
, Mountain
, R. D.
, and Errington
, J. R.
, 2007, “Comparative Study of the Effect of the Effect of Tail Corrections on Surface Tension Determined by Molecular Simulation
,” J. Phys. Chem. B
1089-5647, 111
, pp. 6198
–6207
.82.
Errington
, J. R.
, and Kofke
, D. A.
, 2007, “Calculation of Surface Tension via Area Sampling
,” J. Chem. Phys.
0021-9606, 127
, p. 174709
.83.
Cichowski
, E. C.
, Schmidt
, T. R.
, and Errington
, J. R.
, 2005, “Determination of Henry's Law Constants Through Transition Matrix Monte Carlo Simulation
,” Fluid Phase Equilib.
0378-3812, 236
, pp. 58
–65
.84.
Errington
, J. R.
, and Wilbert
, D. W.
, 2005, “Prewetting Boundary Tensions From Monte Carlo Simulation
,” Phys. Rev. Lett.
0031-9007, 95
, p. 226107
.85.
Errington
, J. R.
, 2004, “Prewetting Transitions for a Model Argon on Solid Carbon Dioxide System
,” Langmuir
0743-7463, 20
, pp. 3798
–3804
.86.
Grzelak
, E. M.
, and Errington
, J. R.
, 2008, “Computation of Interfacial Properties via Grand Canonical Transition Matrix Monte Carlo Simulation
,” J. Chem. Phys.
0021-9606, 128
, p. 014710
.87.
Rosch
, T. W.
, and Errington
, J. R.
, 2007, “Investigation of the Phase Behavior of an Embedded Charge Protein Model Through Molecular Simulation
,” J. Phys. Chem. B
1089-5647, 111
, pp. 12591
–12598
.88.
Chen
, H.
, and Sholl
, D. S.
, 2006, “Efficient Simulation of Binary Adsorption Isotherms Using Transition-Matrix Monte Carlo
,” Langmuir
0743-7463, 22
, pp. 709
–716
.89.
Chen
, H.
, and Sholl
, D. S.
, 2007, “Examining the Accuracy of Ideal Adsorbed Solution Theory Without Curve-Fitting Using Transition Matrix Monte Carlo Simulations
,” Langmuir
0743-7463, 23
, pp. 6431
–6437
.90.
Paluch
, A. S.
, Shen
, V. K.
, and Errington
, J. R.
, 2008, “Comparing the Use of Gibbs Ensemble and Grand-Canonical Transition-Matrix Monte Carlo Methods to Determine Phase Equilibria
,” Ind. Eng. Chem. Res.
0888-5885, 47
, pp. 4533
–4541
.91.
Ferrenberg
, A. M.
, and Swendsen
, R. H.
, 1988, “New Monte Carlo Technique for Studying Phase Transitions
,” Phys. Rev. Lett.
0031-9007, 61
, pp. 2635
–2638
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