FAST Prediction of Protein Thermodynamics

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1 FAST Prediction of Protein Thermodynamics Donald Jacobs, Associate Professor Department of Physics and Optical Science

2 Protein Thermodynamic Stability Dependence on thermodynamic/solvent conditions and sequence Temperature induced denaturation R.M. Ballew, et al., PNAS (1996) [co-solute] induced denaturation G. Pappenberger, Nature: Struct. Biol. 8, (2001) examples of how to denature a protein (randomly selected) sequence dependence ph induced denaturation Spector and Raleigh. JMB , (1998) pressure induced denaturation Torrent, et. al., Biochemistry, 38, , (1999)

3 Modeling Protein Thermodynamics Speed versus accuracy tradeoffs G = H - TS calculation of free energy requires entropy thermodynamic models Two modeling paradigms 1. free-energy based 2. energy based greater speed Ising-like thermodynamic models coarse-grain Molecular Mechanics (MM) Go-like all-atom MM all-atom MM in implicit solvent explicit MM MM/QM greater accuracy not tractable QM

4 Modeling Protein Thermodynamics Speed versus accuracy tradeoffs G = H - TS calculation of free energy requires entropy thermodynamic models computational bottlenecks 1. assumes additivity 2. requires long simulations greater speed Ising-like thermodynamic models coarse-grain Molecular Mechanics (MM) Go-like all-atom MM all-atom MM in implicit solvent explicit MM MM/QM greater accuracy not tractable QM

5 Modeling Protein Thermodynamics Speed versus accuracy tradeoffs G = H - TS calculation of free energy requires entropy Free Energy Decomposition (FED) is fast but assumes additivity! greater speed greater accuracy

6 Free Energy Decomposition (FED) The problem of hidden thermodynamics G total = Σ G part parts How to add back the parts? Hidden thermodynamics No unique way to decompose a system into parts. Decomposition of the Free Energy of a System in Terms of Specific Interactions A. E. Mark and W. F van Gunsteren, J. Mol. Biol. 240, 167 (1994) In regard to the detailed separation of free energy components, we must acknowledge that the hidden thermodynamics of a protein will, unfortunately, remain hidden See: Ken A. Dill, Additivity Principles in Biochemistry, The Journal of Biological Chemistry 272, (1997)

7 The Distance Constraint Model (DCM) Restoring the utility of a free energy decomposition (FED) A NEW PERSPECTIVE D.J. Jacobs,et. al., Network rigidity at finite temperature: Relationships between thermodynamic stability, the nonadditivity of entropy, and cooperativity in molecular systems. Physical Reviews E. 68, (2003) The DCM resolves the problem of nonadditivity by explicitly regarding network rigidity as a long-range mechanical interaction between components to identify the independent ones. atomic level molecular structure Constraint Theory and Free Energy Decomposition DCM Distance Constraint Model I never satisfy myself until I can make a mechanical model of a thing. If I can make a mechanical model I can understand it! --- Lord Kelvin

8 Tao of the DCM: Free Energy Reconstitution Network rigidity accounts for non-additivity in conformational entropy Mind your Ps and Qs (P,Q) interdependence G(F) = H (F) "TS(F) ΔH = 0 ΔS = 0 ΔH = - ε ΔS = - δ H(F) = S(F) = " c " c h c p c (F) s c q c (F) ΔH = - ε ΔS = - δ ΔH = - ε + - ε ΔS = - δ Regarding NETWORK RIGIDITY as a mechanical interac?on accounts for NON- ADDITIVITY IN ENTROPY

9 Linking Molecular Structure to Thermodynamics The Gibbs ensemble consists of all accessible constraint networks Globally Rigid MECHANICS Globally Flexible H -TS H -TS H -TS STABLE representing the native state UNSTABLE representing the transition state STABLE representing the unfolded state THERMODYNAMICS

10 New Modeling Paradigm for Protein Thermodynamics DCM provides high speed and accuracy G = H - TS calculation of free energy requires entropy mdcm (minimal DCM) Jacobs & Dallakyan (2005) Biophysical J. 88:903 Livesay et al. (2004) FEBS Letters 576:468 greater speed FAST (Flexibility And Stability Test) based on an extended DCM that includes all essential enthalpyentropy compensation mechanisms greater accuracy

11 Quantified Stability/Flexibility Relationships (QSFR) Examples of two mechanical response characterizations Backbone flexibility is dependent upon temperature and ph Cooperativity correlation quantifies flexibility/rigidity pairwise couplings For more information, see the following: Livesay, et al. FEBS Letters (2004) 576:468. Jacobs & Dallakyan. Biophysical J. (2005) 88:903. Livesay & Jacobs. Proteins (2006) 62:130. Jacobs, et al. J. Mol. Biol. (2006) 358:882. Livesay, et al. Chem. Cen. J. (2008) 2:17. Mottonen, et al. Proteins (2009) 75:610. Mottonen, et al., Biophysical J (2010) 99:2245.

12 Distance Constraint Model The Heart of the DCM Consists of Three Essential Elements DCM Free Energy Decomposition FED constraint theory Free Energy Reconstitution FER I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated. --- Poul Anderson in New Scientist (1969)

13 Protein Stability is Linked to Solvent Modeling of solvent and conformational interactions solvent protein protein affects solvent solvent affects protein interface

14 The Free Energy Functional FAST models all essential enthalpy-entropy mechanisms FREE ENERGY DECOMPOSITION

15 Free Energy Decomposition (FED) Model highlights: Solvent penetration using transfer free energies Local environments for residues, are classified using 3 solvent states. Empirical Parameterization Partition Coefficient, P(T) P(T) = p rm prc, where p rx = Q rx ( Qrb +Q rm +Q rc ) non-polar env. aqueous env. Temperature (K) buried p rb clathrate p rc mobile p rm

16 Free Energy Decomposition (FED) Model highlights: Electrostatics determined by optimizing pka values base = TRln(10) p rs [ pka(s,r) " ph]p rs G ion G ion # # acid = TRln(10) # p rs [ ph " pka(s,r) ] p rs S ion $ r#titrable prot = "R p rs p rs dp protonation states deprotonated r$base s p dp pr rs = 1" p rs # pr protonated r$acid s [ pr ln(p pr )+ p dp ln(p dp rs )] p pr rs = 10"[pH"pKa(s,r)] rs rs 1+10 "[ph"pka(s,r)] p dp pr rs " p rs " p rb p rc p rm

17 Free Energy Decomposition (FED) Model highlights: Electrostatics determined by optimizing pka values base = TRln(10) p rs [ pka(s,r) " ph]p rs G ion G ion # # acid = TRln(10) # p rs [ ph " pka(s,r) ] p rs S ion $ r#titrable prot = "R p rs p rs dp protonation states deprotonated r$base s p dp pr rs = 1" p rs # pr protonated r$acid s [ pr ln(p pr )+ p dp ln(p dp rs )] p pr rs = 10"[pH"pKa(s,r)] rs rs 1+10 "[ph"pka(s,r)] p dp rs " p " rs p rb p rc p rm

18 Free Energy Decomposition (FED) Covalent bond network defines structural template covalent bonding between different types of residues 2D schematic of the process

19 Free Energy Decomposition (FED) Solvent penetration defines a heterogeneous local environments The solvation state of a protein is defined at the residue level. 2D schematic of the process

20 Free Energy Decomposition (FED) Crosslinking distance constraints couple to solvent penetration DCM Based on the solvation state, all crosslinking interactions are added to the covalent bond network to form a constraint network. 2D schematic of the process

21 Constructing the Free Energy Functional Modeling Essential Mechanisms FED MOLECULAR PARTITION FUNCTION (MPF) Z(T ) = e S o R"# 0 U 0 $ P(E T o )e "(#"# 0 )E de sampled mean energy sampled energy distribution function at reference temperature T o. β o = 1/(R T o ) S o = R # k " k The Schlitter quasi-harmonic approximation is used to estimate S o from the eigenvalues {λ k } of the mass weighted covariance matrix of atomic displacements in XYZ-coordinates. " k = 1 ln [( k 2 BTe 2 /!)# k +1] The {σ k }define a pure entropy spectrum for which there will be 3N - 6 finite values.

22 Constructing the Free Energy Functional Modeling Essential Mechanisms FED MOLECULAR PARTITION FUNCTION (MPF) Z(T o ) = e S o R"# 0 U 0 " P(E T o )de =1 sampled mean energy S o = R # k " k The Schlitter quasi-harmonic approximation is used to estimate S o from the eigenvalues {λ k } of the mass weighted covariance matrix of atomic displacements in XYZ-coordinates. " k = 1 ln [( k 2 BTe 2 /!)# k +1] The {σ k }define a pure entropy spectrum for which there will be 3N - 6 finite values.

23 Constructing the Free Energy Functional Modeling Essential Mechanisms FED MOLECULAR PARTITION FUNCTION (MPF) g int " #RT ln[ e #$ 0 U % 0 P int (E)e #($#$ 0 )E de] int " interaction ( ) # G int (T ) = g int int "T ( R $ k "RT ln Z int (T ) & ' 3N "6 % k=1 ) + * molecular free energy for a specified interaction

24 Constructing the Free Energy Functional Modeling Essential Mechanisms FED Z int (T ) = N states MOLECULAR PARTITION FUNCTION (MPF) " int Z j (T ) = e j=1 #$G int N + % 3N "6 (. G int (T ) = $ states g int int - j "T' R $ # kj * +TRln(p j ) 0 p j j=1, & k=1 ) / molecular free energy for a specified interaction p j = Z int j Z int subdividing configuration space j=1 j=2 j=4 j=3

25 Constructing the Free Energy Functional Modeling Essential Mechanisms FED G = G res + G lnk + G ihb + G pck cnf cnf cnf cnf cnf Conformational components define a distance constraint network generic form: N, & G int " = % states. g int "j #T ( R j=1 -. ' 3N int #6 % k=1 int $ "jk int N % k i=1 W int int "ki q "jki ) + +TRln(p "j * int ) / 1 01 p int "j Accounts for the many-body mechanical interactions Accounts for nonadditivity in conformational entropy

26 Constructing the Free Energy Functional Modeling Essential Mechanisms FED G = G res + G lnk + G ihb + G pck cnf cnf cnf cnf cnf Conformational components define a distance constraint network generic form: N, & G int " = % states. g int "j #T ( R j=1 -. ' 3N int #6 % k=1 int $ "jk int N % k i=1 W int int "ki q "jki ) + +TRln(p "j * int ) / 1 01 p int "j A set of distance constraints are used to model each entropy mode. N k int i=1 int W "ki normalization condition: # = 1 The probability for a distance constraint to be independent as determined by graphrigidity calculations.

27 Constructing the Free Energy Functional Modeling Essential Mechanisms FED G = G res + G lnk + G ihb + G pck cnf cnf cnf cnf cnf Conformational components define a distance constraint network generic form: N, & G int " = % states. g int "j #T ( R j=1 -. ' p "j 3N int #6 % k=1 int $ "jk int = exp #$g "j int N % k i=1 W int int "ki q "jki ) + +TRln(p "j * ( int + & % int & "jk w int i "ki q int ) k "jki int Z " The probability for an interaction to form depends on all other interactions int ) / 1 01 p int "j

28 Free Energy Landscape Constraint Networks are Defined by Solvent Macrostates FER F LOW T Intermediate T HIGH T For a given template structure U c U h Clathrate Mobile Mobile

29 Free Energy Reconstruction (FER) Self-consistent process is used to solve the free energy functional FER adaptive grid {p rc, p rb, p rm } {p pr rs, p dp rs } {p c,q c } interpolations Clathrate Mobile Mobile

30 Network Rigidity Calculations Accounts for non-additivity in conformational entropy conformational entropies renormalize nonadditive % c $ c G cnf " G c # Z c with q c =1 c bare parameters Z = e S 0 /k B "# 0 U 0 $ P 0 (E)e "(#"# 0 )E de from Z c and {q c } calculate {p c } road to additivity I have yet to see any problem, however complicated, which, when you looked at it in the right way, did not become still more complicated. --- Poul Anderson in New Scientist (1969) Jacobs, US Patent #

31 Self-consistent Network Rigidity Calculation Local environments and nonadditivity in conformational entropy conformational entropies renormalize nonadditive % c $ c G cnf " G c # Z c road to additivity with q c =1 c bare parameters Z = e S 0 /k B "# 0 U 0 based on network rigidity and from {p c } calculate {q c } S o new " q new S o $ P 0 (E)e "(#"# 0 )E de from Z c and {q c } calculate {p c }

32 Self-consistent Network Rigidity Calculation Local environments and nonadditivity in conformational entropy conformational entropies renormalize nonadditive % c $ c G cnf " G c # Z c road to additivity with q c =1 c bare parameters Z = e S 0 /k B "# 0 U 0 based on network rigidity and from {p c } calculate {q c } S o new " q new S o $ P 0 (E)e "(#"# 0 )E de from Z c and {q c } calculate {p c }

33 Self-consistent Network Rigidity Calculation Local environments and nonadditivity in conformational entropy conformational entropies renormalize nonadditive % c $ c G cnf " G c # Z c road to additivity with q c =1 c bare parameters Z = e S 0 /k B "# 0 U 0 $ P 0 (E)e "(#"# 0 )E de self consistent constraint theory additive G cnf = G c " $ c # c Z c Z S = e o /K B "# o U o $ P o (E)e "(#"# o )E de

34 Flexibility And Stability Test (FAST) DCM = FED + FER input protein structure parameter-library constraint topology FED renormalized parameters FER mechanical response Flexibility thermodynamic response Stability

35 Stability Curves Results for three typical macrostates 162 residues 2.51 minutes ph=7 37 residues 0.56 minutes ph=7

36 Scalability of FAST Efficient parallelized sparse-hierarchical-adaptive grid methods FAST calculation of multi-dimensional free energy landscape ph FAST performance characteristics Example: Protein size: 150 residues domain: (150 K T 400 ΔT=1K (2 ph ΔpH=0.1 wall time: < 6hrs using 50 (2.3 GHz CPUs) Scales nearly linear with # of atoms Folded phase boundary line depending on: sequence bound ligand solute concentrations pressure temperature etc T

37 Conclusions and Acknowledgements Newtonian Mechanics Network rigidity is a fundamental mechanical property that directly links protein stability, flexibility and dynamics Boltzmann Statistics Principal InvesBgators Don Jacobs Dennis Livesay Post Doctoral Fellows Tong Li Current Students Deeptak Verma Wei Song Jenny Farmer Aaron Brettin Chris Singer Chuck Herrin Collaborators Chris Yengo (PSU) Jörg Rösgen (PSU) Irina Nesmolova Former: Research Associates Oleg Vorov Jim Mottonen Former: Postdoctoral Fellows Chuanbin Du Hui Wang Andrei Istomin Greg Wood Sargis Dallakayan Work supported by: NIH R01 GM , S10 SRR MedImunne, Inc. Charlo@e Research InsBtute (CRI) Center for Biomedical Engineering and Science (CBES) Former: Students Alicia Heckathorne Dang Huynh Jeremy Hules Moon Lee Shelley Green Mike Fairchild Shira Stav Luis Gonzalez Charles David