2020

• Stochastic Simulation to Visualize Gene Expression and Error Correction in Living Cells, K.Y. Chen*, D.M. Zuckerman, P.C. Nelson, The Biophysicist, 1:3 (2020) https://doi.org/10.35459/tbp.2019.000101.
• EmrE reminds us to expect the unexpected in membrane transport, Michael Grabe*, Daniel Zuckerman*, John Rosenberg*, J. Gen. Physiology, 152:e201912467 (2020)
• A very simple introduction to Bayesian statistics: From coin flips to insight.  Daniel M. Zuckerman, 2020. https://doi.org/10.31219/osf.io/nzwk4
• Seven easy lessons introducing non-equilibrium statistical mechanics and (bio)physical chemistry. Daniel M. Zuckerman, 2020.  https://doi.org/10.31219/osf.io/98ypj
• Key biology you should have learned in physics class: Using ideal-gas mixtures to understand biomolecular machines, Daniel M. Zuckerman*, American Journal of Physics, 88:182-193 (2020).  Editor’s pick.  https://doi.org/10.1119/10.0000634,https://arxiv.org/abs/1906.08392

• Optimizing weighted ensemble sampling of steady states, David Aristoff* and Daniel M. Zuckerman*, Multiscale Modeling and Simulation, 18:646–673 (2020).

• A kinetic mechanism for enhanced selectivity of membrane transport, Paola Bisignano, Michael A. Lee, August George, Daniel M. Zuckerman, Michael Grabe*, John M. Rosenberg*, PLOS Comp. Biol., e1007789. https://doi.org/10.1371/journal.pcbi.1007789.

• A systems-biology approach to molecular machines: Exploration of alternative transporter mechanisms", August George and Daniel M. Zuckerman*, PLOS Computational Biology, PLoS Comput Biol 16(7): e1007884. https://doi.org/10.1371/journal.pcbi.1007884

• Connexin-46/50 in a dynamic lipid environment resolved by CryoEM at 1.9 Å, Jonathan Flores, Bassam Haddad, Kimberly Dolan, Janette Myers, Craig Yoshioka, Jeremy Copperman, Daniel Zuckerman, and Steve Reichow*, Nature Communications, Accepted.

2019

• HER2 Cancer Protrusion Growth Signaling Regulated by Unhindered, Localized Filopodial Dynamics, Wai Yan Lam, Yi Wang, Barmak Mostofian, Danielle Jorgens, Sunjong Kwon, Koei Chin, M. Alexandra Carpenter, Thomas Jacob, Katie Heiser, Anurag Agrawal, Jing Wang, Xiaolin Nan, Young Hwan Chang, Daniel M. Zuckerman, Joe Gray, Marcel Bruchez, Keith A. Lidke, Tania Q. Vu, 2019. biorxiv.org https://doi.org/10.1101/654988

• Computational estimation of ms-sec atomistic folding times, Upendra Adhikari, Barmak Mostofian, Jeremy Copperman, Sundar Raman Subramanian, Andrew A. Petersen, and Daniel M. Zuckerman*, J. Am. Chem. Soc., 141, 6519−6526 (2019).  PMCID: PMC6660137

• Middle-way flexible docking:  Pose prediction using mixed-resolution Monte Carlo in estrogen receptor α, Justin Spiriti, Sundar Raman Subramanian, Rohith Palli, Maria Wu, Daniel M. Zuckerman*, PLOS ONE 14(4): e0215694 (2019). PMCID: PMC6478315

• Statistical uncertainty analysis for small-sample, high log-variance data: Cautions for bootstrapping and Bayesian bootstrapping, Barmak Mostofian and Daniel M. Zuckerman*, J. Chem. Theory Comp., 15:3499-3509 (2019).  PMCID: PMC6754704

• A Suite of Tutorials for the WESTPA Rare-Events Sampling Software [Article v1.0], Anthony T. Bogetti, Barmak Mostofian, Alex Dickson, AJ Pratt1*, Ali S. Saglam, Page O. Harrison, Joshua L. Adelman, Max Dudek, Paul A. Torrillo, Alex J. DeGrave, Upendra Adhikari, Matthew C. Zwier, Daniel M. Zuckerman*, and Lillian T. Chong*, Living Journal of Computational Molecular Science, 1:10607 (2019).

• Transient probability currents provide upper and lower bounds on non-equilibrium steady-state currents in the Smoluchowski picture, Jeremy Copperman, David Aristoff*, Dmitrii Makarov*, Gideon Simpson*, and Daniel M. Zuckerman*, Journal of Chemical Physics, 151:174108 (2019); https://doi.org/10.1063/1.5120511.  PMCID: PMC7043855

• High-throughput single-particle tracking reveals nested membrane domains that dictate KRas^G12D diffusion and trafficking, Yerim Lee, Carey Phelps, Tao Huang, Barmak Mostofian, Lei Wu, Ying Zhang, Young Hwan Chang, Philip J. S. Stork, Joe W. Gray, Daniel M. Zuckerman*, Xiaolin Nan*, eLife, 8:e46393 (2019).  PMCID: PMC7060040

• Computational estimation of ms-sec atomistic folding times, Upendra Adhikari, Barmak Mostofian, Jeremy Copperman, Sundar Raman Subramanian, Andrew A. Petersen, and Daniel M. Zuckerman*, J. Am. Chem. Soc., 141, 6519−6526 (2019).  PMCID: PMC6660137

• Middle-way flexible docking: Pose prediction using mixed-resolution Monte Carlo in estrogen receptor α, Justin Spiriti, Sundar Raman Subramanian, Rohith Palli, Maria Wu, Daniel M. Zuckerman*, PLOS ONE 14(4): e0215694 (2019). PMCID: PMC6478315
• Statistical uncertainty analysis for small-sample, high log-variance data: Cautions for bootstrapping and Bayesian bootstrapping, Barmak Mostofian and Daniel M. Zuckerman*, J. Chem. Theory Comp., 15:3499-3509 (2019).

2018

• Clinically Observed Estrogen Receptor Alpha Mutations within the Ligand-Binding Domain Confer Distinguishable Phenotypes, Shanhang Jia, Mark T. Miedel, Marilyn Ngo, Ryan Hessenius, Ning Chen, Peilu Wang, Amir Bahreini,   Zheqi Li, Zhijie Ding, Tong Ying Shun, Daniel M. Zuckerman, D. Lansing Taylor, Shannon L. Puhalla, Adrian V. Lee, Steffi Oesterreich, Andrew M. Stern, Oncology, 94:176–189 (2018). PMCID: PMC5828968
• Systematic testing of belief-propagation estimates for absolute free energies in atomistic peptides and proteins, Donovan-Maiye, Rory; Langmead, Christopher; Zuckerman*, Daniel, J. Chem. Theory Comp., 14:426–443 (2018).  PMCID: PMC5933972
• Escape of a small molecule from inside T4 lysozyme by multiple pathways, A. Nunes-Alves, D. M. Zuckerman*, and G. M. Arantes*, Biophysical Journal, 114: 1058-1066 (2018).  PMCID: PMC5883560
• Supramolecular self assembly of nanodrill-like structures for intracellular delivery, N Ashwanikumara, Justin S. Plaut, Barmak Mostofian, Siddharth Patel, Peter Kwaka, Conroy Sun, Kerry McPhail, Daniel M. Zuckerman, Sadik C. Esener, Gaurav Sahay, J. Controlled Release, 282:76-89 (2018).  PMCID: PMC6008205
• Best Practices for Foundations in Molecular Simulations [Article v1. 0].  Braun* E, Gilmer* J, Mayes* HB, Mobley* DL, Monroe* JI, Prasad* S, Zuckerman* DM.  Living Journal of Computational Molecular Science. 1:5957 (2018). View on LiveCoMS
• Structure of native lens connexin-46/50 intercellular channels by CryoEM, Janette B. Myers, Bassam G. Haddad, Susan E. O’Neill, Dror S. Chorev, Craig C. Yoshioka, Carol V. Robinson, Daniel M. Zuckerman and Steve L. Reichow,* Nature, 564: 372–377 (2018).  PMCID: PMC6309215

2017

• Weighted Ensemble Simulation: Review of Methodology, Applications, and Software, Daniel M. Zuckerman and Lillian T. Chong, Annual Review of Biophysics, 46:43-57 (2017)
• Biophysical Comparison of ATP-Driven Proton Pumping Mechanisms Suggests a Kinetic Advantage for the Rotary Process Depending on Coupling Ratio, Ramu Anandakrishnan* and Daniel M. Zuckerman*, PLOS ONE, 12(3): e0173500 (2017). 
• Path-sampling strategies for simulating rare events in biomolecular systems, Lillian T Chong*, Ali S Saglam and Daniel M. Zuckerman*, Current Opinion in Structural Biology, 43:88–94 (2017).

2016

• Entire-Dataset Analysis of NMR Fast-Exchange Titration Spectra: A Mg(2+) Titration Analysis for HIV-1 Ribonuclease H Domain, Karki, Ichhuk; Christen, Martin; Spiriti, Justin; Slack, Ryan; Oda, Masayuki; Kanaori, Kenji; Zuckerman, Daniel;Ishima, Rieko, J. Phys. Chem. B, 120:12420–12431 (2016).
• Efficient Atomistic Simulation of Pathways and Calculation of Rate Constants for a Protein–Peptide Binding Process: Application to the MDM2 Protein and an Intrinsically Disordered p53 Peptide, Matthew C. Zwier, Adam J. Pratt, Joshua L. Adelman, Joseph W. Kaus, Daniel M. Zuckerman, and Lillian T. Chong, J. Phys. Chem. Lett., 7:3440–3445 (2016). PMCID: PMC5008990
• Biophysical comparison of ATP synthesis mechanisms shows a kinetic advantage for the rotary process, Ramu Anandakrishnan, Zining Zhang, Rory M. Donovan, and Daniel M. Zuckerman*, Proceedings of the National Academy of Sciences, 113: 11220–11225 (2016).
• Accurate estimation of protein folding and unfolding times: Beyond Markov state models, Suárez, Ernesto; Adelman, Joshua; Zuckerman, Daniel*, Journal of Chemical Theory and Computation, 12:3473–3481 (2016). PMCID: PMC5022777
• Unbiased Rare Event Sampling in Spatial Stochastic Systems Biology Models Using a Weighted Ensemble of Trajectories, Rory M. Donovan, Jose-Juan Tapia, Devin P. Sullivan, James R. Faeder, Robert F. Murphy, Markus Dittrich, Daniel M. Zuckerman*, PLoS Computational Biology, 12(2): e1004611 (2016). PMCID: PMC4741515

2015

• Tabulation as a high-resolution alternative to coarse-graining protein interactions: Initial application to virus capsid subunits, Justin Spiriti and Daniel M. Zuckerman*, Journal of Chemical Physics, 143: 243159 (2015). PMCID: PMC4698120
• Estimating First Passage Time Distributions from Weighted Ensemble Simulations and non-Markovian Analyses, Ernesto Suarez, Adam J. Pratt, Lillian T. Chong, and Daniel M. Zuckerman*, Protein Science, 25:67-78 (2015). PMCID: PMC4815309
• Structural Integrity of the Ribonuclease H domain in HIV-1 Reverse Transcriptase, Slack, Ryan L, Spiriti, Justin M, Ahn, Jinwoo. Parniak, Michael A, Zuckerman*, Daniel M. [co-corresponding author], Ishima, Rieko, Proteins: Structure, function, and bioinformatics, 83:1526–1538 (2015). PMCID: PMC4509971.
• WESTPA: An interoperable, highly scalable software package for weighted ensemble simulation and analysis, Zwier, Matthew;Adelman, Joshua;Kaus, Joseph;Pratt, Adam;Wong, Kim;Rego, Nicholas;Suárez, Ernesto;Lettieri, Steven;Wang, David;Grabe, Michael;Zuckerman, Daniel;Chong, Lillian, J. Chem. Theory Comput., 11, 800–809 (2015). PMCID: PMC4573570.

2014

• Tunable Coarse Graining for Monte Carlo Simulations of Proteins via Smoothed Energy tables: Direct and Exchange Simulations, Justin Spiriti and Daniel M. Zuckerman*, J. Chem. Theory Comput., 10, 5161–5177 (2014). PMCID: PMC423037.
• Simultaneous computation of dynamical and equilibrium information using a weighted ensemble of trajectories, Ernesto Suárez, Steven Lettieri, Matthew C. Zwier, Carsen A. Stringer, Sundar Raman Subramanian, Lillian T. Chong, and Daniel M Zuckerman*, J. Chem. Theory Comput., 10, 2658−2667 (2014). PMCID: PMC4168800.

2013

• Efficient stochastic simulation of chemical kinetics networks using a weighted ensemble of trajectories, Rory M. Donovan, Andrew J. Sedgewick, James R. Faeder, and Daniel M. Zuckerman*, J. Chem. Phys. 139, 115105, (2013). PMCID: PMC3790806
• All Pittsburgh students should learn computer programming, Daniel M. Zuckerman, Pittsburgh Post-Gazette Sept. 8, 2013.

2012

• Tunable, mixed-resolution modeling using library-based Monte Carlo and graphics processing units, Mamonov, Artem; Lettieri, Steven; Ding, Ying; Sarver, Jessica; Palli, Rohith; Cunningham, Timothy; Saxena, Sunil; Zuckerman*, Daniel, J. Chem. Theory Comp., 8, 2921−2929 (2012), PMCID: PMC3496292.  
• Accelerating molecular Monte Carlo simulations using distance and orientation dependent energy tables: tuning from atomistic accuracy to smoothed "coarse-grained" models, Steven Lettieri and Daniel M. Zuckerman*, J. Comp. Chem. 33:268-275 (2012). PMCID: PMC3408236.

2011

• Simulations of the alternating access mechanism of the sodium symporter Mhp1, Joshua L. Adelman, Amy L. Dale, Matthew C. Zwier, Divesh Bhatt, Lillian T. Chong, Daniel M. Zuckerman, Michael Grabe, Biophysical Journal, 101:2399-2407 (2011), PMCID: PMC3218348
• Beyond microscopic reversibility: Are observable non-equilibrium processes precisely reversible?, Divesh Bhatt and Daniel M. Zuckerman*, J. Chem. Theory Comp., 7:2520-2527 (2011). PMCID: PMC3159166
• Extending fragment-based free energy calculations with library Monte Carlo simulation: Annealing in interaction space, Steven Lettieri, Artem Mamonov, and Daniel M. Zuckerman*, J. Comp. Chem., 32:1135-1143 (2011). PMCID PMC3390976
• Rapid sampling of all-atom peptides using a library-based polymer-growth approach, Artem B. Mamonov, Xin Zhang, and Daniel M. Zuckerman*, J. Comp. Chem., 32:396-405 (2011). PMCID PMC3005036
• Equilibrium Sampling in Biomolecular Simulation, DM Zuckerman, Annual Review of Biophysics, 40:41-62 (2011). PMCID: PMC4434601

2010

• Heterogeneous path ensembles for conformational transitions in semi-atomistic models of adenylate kinase, Divesh Bhatt and Daniel M. Zuckerman*, J. Chem. Theory Comp., 6:3527-3539 (2010). PMCID: PMC3108504 
• Automated sampling assessment for molecular simulations using the effective sample size, Xin Zhang, Divesh Bhatt, and Daniel M. Zuckerman*, J. Chem. Theory Comp., 6:3048-3057 (2010). PMCID: PMC3017371
• Steady state via weighted ensemble path sampling, Divesh Bhatt, Bin Zhang, and Daniel M Zuckerman*, J. Chem. Phys., 133:014110 (2010). PMCID: PMC2912933.
• Efficient Equilibrium Sampling of All-Atom Peptides Using Library-Based Monte Carlo, Ying Ding, Artem Mamonov, Daniel M. Zuckerman*, J. Phys. Chem. B, 114:5870-5877 (2010). PMCID: PMC2882875
• The "weighted ensemble" path sampling method is statistically exact for a broad class of stochastic processes and binning procedures, Bin Zhang, David Jasnow, and Daniel M. Zuckerman*, J. Chem. Phys., 132:054107 (2010). PMCID: PMC2830257

2009

• Absolute free energies and equilibrium ensembles of dense fluids computed from a non-dynamic growth method, Divesh Bhatt and Daniel M. Zuckerman*, J. Chem. Phys., 131:214110 (2009). PMCID: PMC2802520
• A general library-based Monte Carlo technique enables equilibrium sampling of semi-atomistic protein models, Mamonov, Artem;Bhatt, Divesh;Cashman, Derek;Ding, Ying;Zuckerman*, Daniel M., J. Phys. Chem. B, 113: 10891–10904 (2009). PMCID: PMC2766542
• Absolute free energies estimated by combining pre-calculated molecular fragment libraries, Xin Zhang, Artem Mamonov, and Daniel M. Zuckerman*, J. Comp. Chem., 30:1680-1691 (2009). PMCID: PMC2783641
• Resampling improves the efficiency of a "fast-switch" equilibrium sampling protocol, E. Lyman and D.M. Zuckerman*, J. Chem. Phys., 130:081102 (2009). PMCID: PMC2671214
• Quantifying uncertainty and sampling quality in biomolecular simulations, Alan Grossfield* and Daniel M. Zuckerman*, Annual Reports in Computational Chemistry, 5:23-48 (2009). PMCID: PMC2865156

2008

• A black-box re-weighting analysis can correct flawed simulation data, F. Marty Ytreberg and Daniel M. Zuckerman*, Proc. Nat. Acad. Sci. USA 105:7982-7987 (2008). PMCID: PMC2786942
• DM Zuckerman, "Principles and Practicalities of Canonical Mixed-Resolution Sampling of Biomolecules" (book chapter) in Coarse-Graining of Condensed Phase and Biomolecular Systems, edited by GA Voth (Taylor and Francis, 2008).

2007

• Efficient and verified simulation of a path ensemble for conformational change in a united-residue model of calmodulin, Bin W. Zhang, David Jasnow, and Daniel M. Zuckerman*, Proc. Nat. Acad. Sci. USA 104:18043-18048 (2007).
• Demonstrated convergence of the equilibrium ensemble for a fast united-residue protein model, F. M. Ytreberg, S. Kh. Aroutiounian, D. M. Zuckerman*, J. Chem. Theory Comp. 3:1860-1866 (2007).
• On the structural convergence of biomolecular simulations by determination of the effective sample size, E. Lyman and D. M. Zuckerman, J. Phys. Chem. B. 111:12876-12882 (2007). PMCID: PMC2538559
• Annealed importance sampling of peptides, Edward Lyman and Daniel M. Zuckerman*, J. Chem. Phys. 127:065101 (2007).
• Transition-event durations in one-dimensional activated processes, Bin W. Zhang, David Jasnow, and Daniel M. Zuckerman*, J. Chem. Phys. 126:074504 (2007). 

2006

• Comparison of free energy methods for molecular systems, Ytreberg, F.M., R.H. Swendsen, and D.M. Zuckerman*, J. Chem. Phys. 125: 184114-11 (2006).
• A second look at canonical sampling of biomolecules using replica exchange simulation, E. Lyman and D. M. Zuckerman*, J. Chem. Theory Comp., 2:1200-1202 (2006). PMCID: PMC2586297
• Ensemble-based convergence analysis of biomolecular trajectories, E. Lyman and D. M. Zuckerman*, Biophysical J., 91:164-172 (2006).
• Resolution exchange simulation with incremental coarsening, E. Lyman and D. M. Zuckerman*, J. Chem. Theory Comp. 2:656-666 (2006).
• Simple estimation of absolute free energies for biomolecules, F. M. Ytreberg and D. M. Zuckerman*, J. Chem. Phys. 124:104015 (2006).
• Resolution Exchange Simulation, E. Lyman, F.M. Ytreberg and D. M. Zuckerman*, Phys. Rev. Lett. 96:028105 (2006). 

2005

• Peptide Conformational equilibria computed via a single-stage shifting protocol, F. M. Ytreberg and D. M. Zuckerman*, J. Phys. Chem. B., 109:9096-9103 (2005). 

2004

• Efficient use of nonequilibrium measurement to estimate free energy differences for molecular systems, F. M. Ytreberg and D. M. Zuckerman*, J. Computational Chemistry, 25:1749-1759 (2004).
• Single-ensemble nonequilibrium path-sampling estimates of free energy differences, F. M. Ytreberg and D. M. Zuckerman*, J. Chem. Phys., 120:10876-10879 (2004).  
• Tools for Channels: Moving Towards Molecular Calculations of Gating and Permeation in Ion Channel Biophysics, Thomas B. Woolf, D. M. Zuckerman, N. Lu, H. Jang, Journal of Molecular Graphics and Modelling, 22:359-368 (2004).
• Fast Simulation Protocol for Protein Structural Transitions: Modeling of the Relationship of Structure and Function, Arun K. Setty and D.M. Zuckerman* in Proteins as Materials, edited by V.P. Conticello, A. Chilkoti, E. Atkins, and D.G. Lynn (Mater. Res. Soc. Symp. Proc. 826E, Warrendale, PA , 2004)
• Simulation of an ensemble of conformational transitions in a united-residue model of calmodulin, D.M. Zuckerman*, J. Phys. Chem. B 108:5127-5137 (2004). 
• Systematic finite-sampling inaccuracy in free energy differences and other nonlinear quantities, D.M. Zuckerman and T. B. Woolf, Journal of Statistical Physics, 114:1303-1323 (2004).

2003

• Molecular dynamics simulations of ionic concentration gradients across model bilayers (Sachs J.N., Petrache H.I., Zuckerman D.M., Woolf T.B.) J. Chem. Phys. 118: 1957-1969 (2003). 

2002

• Theory of a Systematic Computational Error in Free Energy Differences (D.M. Z. and T.B. Woolf) Phys. Rev. Lett. 89 180602 (2002).
• Hydrophobic matching mechanism investigated by molecular dynamics simulations (Petrache H.I., Zuckerman D.M., Sachs J.N, Killian J.A., Koeppe R.E., Woolf T.B.) Langmuir 18: 1340-1351 (2002). 
• Overcoming finite-sampling errors in fast-switching free energy estimates: Extrapolative analysis of a molecular system (D.M. Z. and T .B. Woolf) Chem. Phys. Lett. 351, 445-453 (2002).
• Transition Events in Butane Simulations: Similarities Across Models (D.M. Z. and T.B. Woolf) J. Chem. Phys. 116,2586-2591 (2002). 
• Rapid Determination of Multiple Reaction Pathways in Molecular Systems: The Soft-Ratcheting Algorithm (D.M. Zuckerman and T .B. Woolf) arXiv:physics/0209098 (2002).

2001

• Asymmetric Primitive Model Electrolytes: Debye-Huckel Theory, Criticality and Energy Bounds (D.M. Z., M.E. Fisher and S. Bekiranov) Phys. Rev. E 64,011206 (2001). 

2000

• Efficient dynamic importance sampling of rare events in one dimension (D.M. Z. and T .B. Woolf) Phys. Rev. E 63, 016702 (2000). 

1999

• Electrostatics of Membrane Systems -Complex, Heterogeneous Environments (T.B. Woolf, A. Grossfield, and D.M. Z.) in "Simulation and Theory of Electrostatic Interactions in Solution", Eds. L.R. Pratt and G. Hummer (American Inst. of Physics, 1999) 
• Dynamic reaction paths and rates through importance-sampled stochastic dynamics (D.M. Z. and T.B. Woolf) J. Chem. Phys. 111,9475-9484 (1999). 
• Letter to the editor regarding postdoctoral welfare, Daniel M. Zuckerman, Science 286:411 (1999). DOI: 10.1126/science.286.5439.411c

1998

• Exact Thermodynamic Formulation of Chemical Association (M.E. Fisher and D.M. Z.) J. Chem. Phys. 109, 7961-7981 (1998). 
• Chemical Association via Exact Thermodynamic Formulations (M.E. Fisher and D.M. Z.) Chem. Phys. Lett. 293, 461-468 (1998). 
• Critique of Primitive Model Electrolyte Theories using Thermodynamic Bounds (M. E. Fisher, D.M. Z., and B. P. Lee) in "Strongly Coupled Coulomb Systems", Proc. Conf. held in Boston College, August 1997, Eds. G. J. Kalman, K. Blagoev and J. M. Rommel (Plenum Publ. Corp., 1998) 
• Vesicle-Vesicle Adhesion by Mobile Lock-and-Key Molecules: Debye-Huckel Theory and Monte Carlo Simulation (D.M. Z. and R. F. Bruinsma) Phys. Rev. E 57,964-977 (1998). 

1997

• Critique of Primitive Model Electrolyte Theories (D.M. Z., M.E. Fisher, and B.P. Lee) Phys. Rev. E 56, 6569-6580 (1997). 
• Forces between Surfaces with Weakly End-Adsorbed Polymers (J.I. Martin, Z.-G. Wang, D. Z., R. Bruinsma, and P. Pincus) J. Phys. II (France) 7, 1111-1121 (1997). 

1995

• Statistical Mechanics of Membrane Adhesion by Reversible Molecular Bonds (D. Zuckerman and R. Bruinsma), Phys. Rev. Lett. 74, 3900-3903 (1995).