Structure Prediction Methods. Fundamental Questions. Protein Structure Prediction. Protein Folding. The Protein Folding Problem MACGT...

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1 Tertary Structure of Proten Anfnsen s experments, late 1950 s through 1960 s Rbonuclease, an enzyme nvolved n cleavage of nuclec acds. Structure has a combnaton of α and β segments and four dsulfde brdges What are Dsulfde Brdges? Oxdaton Cys- Cys- Cys-S S-Cys Reducton Actve, natve structure BME s reducng agent Urea unfolds protens add β-mercaptoethanol (BME) HS-CH -CH -OH add urea, H N-C-NH Cys58 Cys110 Cys58 Cys110 O Denatured, nactve, random col, many conformatons Many Conformatons Many Conformatons Remove BME Remove urea Remove BME - Natve structure -fully actve - 4 dsulfde bond correct One Conformaton S S S S S S S Mxture of 105 dfferent conformatons, 1% actve S Remove urea One Conformaton 1

2 The Proten Foldng Problem Levnthal s paradox Consder a 100 resdue proten. If each resdue can take only 3 postons, there are = possble conformatons. If t takes s to convert from 1 structure to another, exhaustve search would take years! MACGT...? Gven a partcular sequence of amno acd resdues (prmary structure), what wll the tertary/quaternary structure of the resultng proten be? Proten Structure Predcton and Proten Foldng Proten Structure Predcton Fundamental Questons What s the structure of ths proten? Can be expermentally determned, today we know the structure of ~35,000 protens Can be predcted for some protens, usually n ~1 day on today's computers Proten Foldng: Fast Folders Tme Scale: 80 s 90 s 00 s 00 s 90 s 80 s ps ns µs µs ms sec Foldng MD Smulatons Foldng Experments Proten Foldng How does ths proten form ths structure? The process or mechansm of foldng Lmted expermental characterzaton Why does ths proten form ths structure? Why not some other fold? Why so quckly? -> Levnthal's Paradox: As there are an astronomcal number of conformatons possble, an unbased search would take too long for a proten to fold. Yet most protens fold n less than a second! Trp-cage, desgned mn-proten (0 aa): 4µs β-harpn of C-termnus of proten G (16 aa) : 6µs Engraled homeodoman (En-HD) (61 aa): ~7µs WW domans (38-44 aa): >4µs Fe(II) cytochrome b 56 (106 aa): extrapolated ~5µs B doman of proten A (58 aa): extrapolated ~8µs Structure Predcton Methods Homology Modelng 1 QQYTA KIKGR 11 TFRNE KELRD 1 FIEKF KGR Algorthm Secondary structure (only sequence) Homology modelng (usng related structure) Fold recognton Ab-nto 3D predcton Assumes smlar (homologous) sequences have very smlar tertary structures Basc structural framework s often the same (same secondary structure elements packed n the same way) Loop regons dffer Wde dfferences possble, even among closely related protens

3 Threadng Strateges for Proten Structure Predcton Gven: sequence of proten P wth unknown structure Database of known folds Fnd: Most plausble fold for P Evaluate qualty of such arrangement Places the resdues of unknown P along the backbone of a known structure and determne stablty of sde chans n that arrangement Comparatve Modelng Fold Recognton Ab Into 1. Identfy sequence homologs 1. Representaton as templates 1. Fold classfcaton. Force feld. Use sequence algnment to Method 3D-Profles 3. Global Optmzaton generate model 3. Improves wth data 4. Structure at global mnmum 3. Fll n unalgned regons 5. Can dscover new folds 4. Improves wth data 1. Requres > 5% sequence 1. Needs good number of protens n 1. Computatonally ntensve dentty each fold Drawbacks. Physcal modelng. Loops and sdechan. Crtcally dependent on scorng conformatons are crtcal functon Resoluton < 3 A 3-7 A > 5 A Tme to Compute < Day ~ Day >> Day Complementarty of the Methods X-ray crystallography- hghest resoluton structures; faster than NMR NMR- enables wdely varyng soluton condtons; characterzaton of motons and dynamc, weakly nteractng systems Computaton- fundamental understandng of structure, dynamcs and nteractons; models wthout experment; very fast The proten sequence contans all nformaton needed to create a correctly folded proten. Many protens fold spontaneously to ther natve structure Proten foldng s relatvely fast Chaperones speed up foldng, but do not alter the structure Forces drvng proten foldng It s beleved that hydrophobc collapse s a key drvng force for proten foldng Hydrophobc core Polar surface nteractng wth solvent Mnmum volume (no cavtes) Dsulfde bond formaton stablzes Hydrogen bonds Polar and electrostatc nteractons Four models that could account for the rapd rate of proten foldng durng bologcal proten synthess. - The Framework Model - The Nucleaton Model - The Molten Globule Model - Foldng Funnels Natve state s typcally only 5 to 10 kcal/mole more stable than the unfolded form 3

4 Framework Model Nucleaton Model Elements of Secondary Structure Formed Only MOST Stable Sec. Structure Formed Nucleaton Molten Globule Foldng Funnel Concept Unfolded, many conformatons More Compact Some secondary structural elements formed wth hydrophobc resdues nsde Many Possble Foldng Pathways to Get to Natve State Natve State, one conformaton Thermodynamcs of Proten Foldng Bond stretchng: sec. Elastc vbratons: sec. Rotatons of surface sdechans: sec. Hnge bendng: sec. Rotaton of bured sde chans: sec. Proten foldng: sec. Smulated foldng n 1 µsec; peptde n a box of water Free Energy Funnel Entropy and Enthalpy n Proten Foldng Unfolded Proten H, small, negatve S, large, postve G = H - T S bondng flexblty Folded Proten H, large, negatve S, small, postve Compensaton n entropy and enthalpy for proten Contrbuton of entropy of water molecules released upon foldng S of water s large and postve 4

5 Thermodynamcs of Proten Foldng G foldng =G folded -G unfolded = Natve state (N) C N Denatured state C C N N (H folded -H unfolded )-T(S folded -S unfolded )= H foldng -T S foldng G foldng H foldng -T S foldng Energy - unfolded folded Folded protens are hghly ordered S foldng negatve, so T S foldng s a postve quantty H foldng s a negatve quantty - enthalpy s favored n folded state. Total Gbbs free energy dfference s negatve folded state favoured N Sze of cavty n solvent ~6500Å S chan: sgnfcantly decreased, due to the well defned conformaton Non-bonded nteractons: ntra-molecular Compact structure G D N = H D N T S D N N C C S chan: large, due to the large number of dfferent conformatons Non-bonded nteractons: nter-molecular Non compact structure N Average sze of cavty n solvent :0,500Å C 1) ELECTROLYTE ADDITION - nterference wth the collod state ) INSOLUBLE SALT FORMATION - ProtenTrchloracetate 3) ORGANIC SOLVENTS - ETHANOL - nterferes wth the delectrc constant 4) HEAT DENATURATION - more energy n system (bonds break) 5) ph Factors that dsrupt the Natve state - destroys charge - destroys ablty to nteract wth water Thermodynamc Descrpton of Proten Foldng The natve and unfolded states are n equlbrum, the foldng reacton can be quantfed n terms of thermodynamcs. The equlbrum (N U) between the natve (N) and unfolded (U) states s defned by the equlbrum constant, K, as: K = [U]/[N] = K U The dfference n Gbbs free energy ( G) between the unfolded and natve states s then: G = -RT ln K For K u, a postve G ndcates that the natve state s more stable. 6) DESTRUCTION OF HYDROGEN BONDING - UREA - known H-bond dsrupter The free energy s composed of both enthalpc and entropc contrbutons: G = H - T S where H and S are the enthalpy and entropy change, respectvely, upon unfoldng. Thermal Unfoldng Snce H and S are strongly temperature-dependent, G s better expressed as: G = H 1 C P (T-T 1 ) - T [ S 1 C P ln(t/t 1 )] where the subscrpt 1 ndcates the value of H and S at a reference temperature, T 1, and C p s the specfc heat or heat-capacty change. Most protens denature reversbly allowng thermodynamc analyss. 1) ELECTROLYTE ADDITION - nterference wth the collod state ) INSOLUBLE SALT FORMATION - ProtenTrchloracetate 3) ORGANIC SOLVENTS - ETHANOL - nterferes wth the delectrc constant 4) HEAT DENATURATION - more energy n system (bonds break) 5) ph Factors that dsrupt the Natve state - destroys charge - destroys ablty to nteract wth water 6) DESTRUCTION OF HYDROGEN BONDING - UREA - known H-bond dsrupter 5

6 Solvng Proten Structures Only knds of technques allow one to get atomc resoluton pctures of macromolecules X-ray Crystallography (frst appled n Kendrew & Perutz) NMR Spectroscopy (frst appled n Ernst & Wuthrch) Structure Functon Structure Mechansm Structure Orgns/Evoluton Structure-based Drug Desgn Solvng the Proten Foldng Problem Ab Into Predcton Predctng the 3D structure wthout any pror knowledge Used when homology modellng or threadng have faled (no homologues are evdent) Equvalent to solvng the Proten Foldng Problem Stll a research problem Ab Into Foldng Two Central Problems Samplng conformatonal space ( ) The energy mnmum problem The Samplng Problem (Solutons) Lattce models, off-lattce models, smplfed chan methods The Energy Problem (Solutons) Threadng energes, packng assessment, topology assessment Problems n Proten Foldng Two key questons: Evaluaton how can we tell a correctly-folded proten from an ncorrectly folded proten? H-bonds, electrostatcs, hydrophobc effect, etc. Derve a functon, see how well t does on real protens Optmzaton once we get an evaluaton functon, can we optmze t? Smulated annealng/monte Carlo Interacton Covalent bonds Ionc Hydrogen bond Hydrophobc van der Waals Approx. bond strength n kj/mole > 00 (rangng up to 900) 0-40 ~5-0 ~ 8 ~ 4 Evaluaton of Proten Folds Emprcal potental functons Resdue-based: spatal relatonshps among resdues Stereochemstry-based: molecular nteractons (covalent, electrostatc, etc.) wth coeffcents Ab-nto potental functons Procheck, etc. Full molecular dynamcs Very computatonally expensve Polypeptdes Represented by a range of approaches or approxmatons ncludng: all atom representatons n cartesan space all atom representatons n dhedral space smplfed atomc versons n dhedral space tube/cylnder/rbbon representatons lattce models AMBER (Asssted Model Buldng wth Energy Refnement ) force feld 3 Vn Etotal = Kr ( r req ) Kθ ( θ θ eq ) [ cos( n )] 1 ω bonds angles dhedrals 1 atoms a b atoms qq j j r r 1 6 < < j ε r 6

7 The hydrophobc zpper effect: Lattce Models Scorng Lattce Models H/P model scorng: count noncovalent hydrophobc nteractons. Ken Dll ~ 1997 Sometmes: Penalze for bured polar or surface hydrophobc resdues Fold Optmzaton Smple lattce models (HP-models) Two types of resdues: hydrophobc and polar -D or 3-D lattce The only force s hydrophobc collapse Score = number of H H contacts 3.5Å A Smple D Lattce Lattce Foldng Buld a n x m matrx (a D array) Choose an arbtrary pont as your N termnal resdue (start resdue) Add or subtract 1 from the x or y poston of the start resdue Check to see f the new pont (resdue) s off the lattce or s already occuped Evaluate the energy Go to step 3 and repeat untl done Lattce Algorthm Red = hydrophobc Blue = hydrophlc If Red s near empty space E = E1 If Blue s near empty space E = E-1 If Red s near another Red E = E-1 If Blue s near another Blue E = E0 If Blue s near Red E = E0 7

8 More Complex Lattces What can we do wth lattce models? For smaller polypeptdes, exhaustve search can be used Lookng at the best fold, even n such a smple model, can teach us nterestng thngs about the proten foldng process More realstc models 1.45 A Hgher resoluton lattces (45 lattce, etc.) Off-lattce models Local moves Optmzaton/search methods and φ/ψ representatons Greedy search Graph theoretcal methods Monte Carlo, smulated annealng, etc. Non-Lattce Models Non-Lattce Models Res H C R H 3.5 Å Wth a more realstc off-lattce model, we need a better energy functon to evaluate a conformaton (fold). Theoretcal force feld: G = G van der Waals G h-bonds G solvent G coulomb 1.53 Å C 1.3 Å N 1.00 Å Emprcal force felds O 1.4 Å 1.47 Å C Res 1 R H Energy Terms Covalent r q φ Bondng Terms: bond stretch Harmonc Potental Stretchng K r (r -r j ) Noncovalent r van der Waals A /r 6 -B /r 1 Bendng K θ (θ - θ j ) r Coulomb q q j /4πεr Torsonal K φ (1 cos(nφ j )) r H-bond C /r 10 -D /r 1 Most often Harmonc 1 V ( bond kr r r = 0) bonds Morse Potental for dssocaton studes V Morse = bonds D e Two new parameters: D: dssocaton energy a: wdth of the potental well a( r r0 ) [ 1] Vbond D Vmorse r 0 bond length Morse Potental r 0 D bond length 8

9 Bondng Terms: angle bendng Bondng Terms: Torsons Harmonc Potental Most often Harmonc V angle 1 = k θ ( θ θ angles 0 ) CHARMM force feld s Urey-Bradley angle term: V UB Vangle θ 0 angle 1 = kub ( s s0 ) Ths UB term s only found n CHARMM UB force feld to optmze the ft to vbratonal spectra. s: the 1,3-dstance. Torson energy: rotaton about a bond (dhedral angles) U torson Vn = [1 cos( nφ δ )] j torsons -j-k-l φ k l Vn: force constant n: perodcty of the angle ( determnes how many peaks and wells n the potental, often from 1-6 ) δ: phase of the angle (often 0º or 180º) Mackerell et al. J. Phys. Chem. B 10, 3586, 1998 Bondng Terms: Improper Torsons Non-bonded Terms Coulomb Potental Improper torson s not a regular torson angle. It s used to descrbe the energy of out-of-plane motons. It s often necessary for planar groups, such as sp hybrdzed carbons n carbonyl groups and n aromatc rngs, because the normal torson terms descrbed above s not suffcent to mantan the planarty (ω~0). or U U mproper mproper V o = [1 cos(ω 180 )] mproper = mproper kw ( ω ω ) 0 j k -j-k-l ω l Electrostatc nteractons (Coulomb s Law) 1 qq j Velec = 4πε < j r Lennard-Jones nteractons 1 6 σ < σ VLJ = 4ε 1 6 j r r Combnaton Rules for LJ 1 ε = εε σ ( ) j = σ σ j σ = Velec VLJ σ σ j ~1/r par dstance LJ Potental par dstance r/sgma 1-4 Non-bonded Interactons Non-bonded exclusons 1- and 1-3 nteractons excluded 1-4 nteractons partally excluded 1-4 nteracton scalngs OPLSAA scales by 0.5 for both electrostatc and LJ AMBER94 scales 0.5 for LJ and 1/1. for electrostatc nteracton CHARMM has specal 1,4-terms j Even though they are non-bonded nteractons, 1-4 terms are often calculated along wth bonded terms. k l The hydrophobc effect The free energy gan from buryng a hydrophobc group s proportonal to the surface area bured 9

10 Lnear relaton between the solvent accessble surface area and the transfer free energy of amno acds Accessble Surface Area G transfer = -γ ASA γ = 0.05 cal/å Accessble Surface Area QHTAWCLTSEQHTAAVIWDCETPGKQNGAYQEDCA HHHHHHCCEEEEEEEEEEECCHHHHHHHCCCCCCC Solvent Probe Reentrant Surface Accessble Surface Van der Waals Surface QHTAWCLTSEQHTAAVIWDCETPGKQNGAYQEDCAMD BBPPBEEEEEPBPBPBPBBPEEEPBPEPEEEEEEEEE Accessble Surface Area Calculatons Force Felds: Typcal Energy Functons DSSP - Database of Secondary Structures for Protens (swft.embl-hedelberg.de/dssp) Connolly Molecular Surface Home Page Naccess Home Page ASA Parallelzaton Proten Structure Database U = bonds angles torsons mproper elec LJ 1 kr ( r r0 ) 1 kθ ( θ θ0) Vn [1 cos( nφ δ )] V ( mproper torson) q q r A [ r j 1 B ] 6 r Bond stretches Angle bendng Torsonal rotaton Improper torson (sp) Electrostatc nteracton Lennard-Jones nteracton 10

11 Whch Force Feld to Use? Most popular force felds: CHARMM, AMBER and OPLSAA OPLSAA(000): Probably the best avalable force feld for condensed-phase smulaton of peptdes. Work to develop parameterzaton that wll nclude broader classes of drug-lke molecules s ongong. GB/SA solvaton energes are good. MMFF: An excellent force feld for bopolymers and many drug-lke organc molecules that do not have parameters n other force felds. AMBER*/OPLS*: Good force felds for bopolymers and carbohydrates; many parameters were added n MacroModel whch extend the scope of ths force feld to a number of mportant organc functonal groups. GB/SA solvaton energes range from moderate (AMBER*) to good (OPLS*). AMBER94: An excellent force feld for protens and nuclec acds. However, there are no extensons for non-standard resdues or organc molecules, also there s a alpha-helx tendency for protens. AMBER99 fxes ths helx problem to some degree, but not completely. MM*/MM3*: Excellent force felds for hydrocarbons and molecules wth sngle or remotely spaced functonal groups. GB/SA solvaton energes tend to be poor relatve to those calculated wth other force felds. CHARMM: Good general purpose force feld for protens and nuclec acds. A bt weak for drug-lke organc molecules. GROMOS96: Good general purpose force feld for protens, partcularly good for free energy perturbatons due to soft-core potentals. Weak for reproducng solvaton free energes of organc molecules and small peptdes. Force Feld Parameterzaton Equlbrum bond dstances and angles: X-ray crystallography Bond and angle force constants: vbratonal spectra, normal mode calculatons wth QM Dhedral angle parameters: dffcult to measure drectly expermentally; ft to QM calculatons for rotatons around a bond wth other motons fxed Atom charges: ft to expermental lqud propertes, ESP charge fttng to reproduce electrostatc potentals of hgh level QM, X-ray crystallographc electron densty Lennard-Jones parameters: often most dffcult to determne, ft to expermental lqud propertes, ntermolecular energy fttng Applcatons NMR or X-ray structure refnement Proten structure predcton Proten foldng knetcs and mechancs Conformatonal dynamcs Global optmzaton DNA/RNA smulatons Membrane protens/lpd layers smulatons Delectrc constant NH 3 ε water = 80 Partal Charges -O-C 0.4 C = O N - H 0. O ε vacuum = 1 ε proten nteror = -10 -O-C NH 3 O ε water, salt > O H H If you know the poston of every partal charge (ncludng water), you do not need a delectrc constant Delectrc constant NH 3 ε water = 80 -O-C O ε vacuum = 1 ε proten nteror = -10 -O-C NH 3 O ε water, salt > 80 a Dpole - Monopole Interactons q θ q- dpole r r 1 r q o Dpole - Dpole Interactons monopole U = Σ U (n) = U 1 U U = 1/4πε o (q/r 1 -q/r ) = q/4πε o (r 1 -r / r 1 r ) f r >> a then r -r 1 ~ a cos θ and r 1 r = r U = qa/4πεo (cos θ/ r ) U s a functon of θ and r. If you rotate around the dpole axs, there s no change n the value of U ε 10 ε 80 The electrostatc potental at any pont relatve to fxed known charges even n the presence of moble charges usng the Posson - Boltzmann Equaton E = -µ a µ b / εr 3 E = -µ a µ b / εr 3 µ = dpole moment = Zd water = 1.85 D peptde bond = 3.5 D retnal = 15 D Interacton energy s dependent on orentaton and dstance 11

12 Na Na Cl- Cl- Emprcal Force Felds and Molecular Mechancs -10 kcal/mol descrbe nteracton of atoms or groups substrate NH 3 -O-C O NH 3 -O-C O enzyme the parameters are emprcal,.e. they are dependent on others and have no drect ntrnsc meanng Examples: GROMOS96 (van Gusteren) CHARMM (M. Karplus) AMBER (Kollman) unfavorable - ordered water favorable - van der Waals - free water hydrophobc effect = ~50 cal/mol/å Example for a (very) smple Force Feld: ν = bonds angles torsons k k ( l l ) ( θ θ ) VN,0,0 ( 1 cos( nω γ )) 1 6 N N σ σ qq j = 4πε 1 j 1 r r 4πε 0r = Complete Energy Functon: p 1 H = kr ( r req) m atoms bond rotaton [ V (1 e 0 H bond AMBER (Asssted Model Buldng wth Energy Refnement ) force feld 3 Vn Etotal = Kr ( r req ) Kθ ( θ θ eq ) [ cos( n )] 1 ω bonds bond stretch vn [1 cos( nφ γ )] ' a( r r0 ) ) V ] 0 angles 0 S bond dhedrals 1 atoms a b atoms qq j j r r 1 6 < < j ε r bond angle bendng [ V (1 e non bonded A B [ 1 6 r r ' a( r r0 ) ) V ] qq j ] ε r 1 kθ ( θ θeq) 0 Sources of force parameters: Bonds, VdW, Electrostatc (for amno acds, nucleotdes only): AMBER: J. Am. Chem. Soc. 117, CHARMM: J. Comp. Chem. 4, H-bonds (Morse potental): Nuclec Acds Res. 0, Bophys. J. 66, Electrostatc parameters of organc molecules need to be computed ndvdually by usng specal software (such as Gaussan) 1

13 Concept of energy scale s Important for molecular modelng Energy Mnmzaton E = f(x) E s a functon of coordnates ether cartesan or nternal At mnmum the frst dervatves are zero and the second dervatves are all postve de = 0 dx d E 0 dx a mult dmensonal energy landscape Energy coord x Potental Energy Surface (PES) coord y Systematc Searchng explore the whole PES Stochastc Searchng fnd all low energy mnma by generatng startng conformaton wth random changes of rotatable dhedral angles (sometmes combned wth random perturbaton of the Cartesan coordnates) followed by mnmzaton Monte Carlo Smulatons generate a Boltzmann dstrbuted ensemble of conformatons, can estmate macroscopc thermodynamc propertes Molecular Dynamcs Smulates the tme dependent moton of the molecular system, can estmate macroscopc thermodynamc propertes Smulated Annealng Playng wth the temperature (T) n ether MD or MC smulatons to speed up search for low energy mnma Dstance geometry method for generatng conformatons that satsfy expermental constrants Systematc Searchng Molecular Mechancs - Energy Mnmzaton The energy of the system s mnmzed. The system tres to relax Typcally, the system relaxes to a local mnmum (LM). 13

14 Conformatonal Samplng Md-energy lower energy lowest energy hghest energy Treat Proten molecule as a set of balls (wth mass) connected by rgd rods and sprngs Rods and sprngs have emprcally determned force constants Hgh Energy Steepest Descent & Conjugate Gradents Low Energy Overhead Vew Mnmzaton Methods Sde Vew Energy surfaces for protens are complex hyperdmensonal spaces Bggest problem s overcomng local mnmum problem Smple methods (slow) to complex methods (fast) Monte Carlo Method Steepest Descent Conjugate Gradent Frequently used for energy mnmzaton of large (and small) molecules Ideal for calculatng mnma for complex (.e. non-lnear) surfaces or functons Both use dervatves to calculate the slope and drecton of the optmzaton path Both requre that the scorng or energy functon be dfferentable (smooth) Steepest Descent Conjugate Gradent Mnmzaton Hgh Energy Hgh Energy Low Energy Low Energy Makes small locally steep moves down gradent Includes nformaton about the pror hstory of path The steepest descent method uses the frst dervatve to determne the drecton towards the mnmum. 14

15 Monte Carlo Algorthm Monte Carlo Mnmzaton Generate a conformaton or algnment (a state) Calculate that state s energy or score If that state s energy s less than the prevous state accept that state and go back to step 1 If that state s energy s greater than the prevous state accept t f a randomly chosen number s < e -E/kT where E s the state energy otherwse reject t Go back to step 1 and repeat untl done Low Energy Hgh Energy Performs a progressve or drected random search Molecular Dynamcs (MD) In molecular dynamcs, energy s suppled to the system, typcally usng a constant temperature (.e. constant average knetc energy). Molecular Dynamcs (MD) Use Newtonan mechancs to calculate the net force and acceleraton experenced by each atom. Each atom s treated as a pont wth mass m and fxed charge q Determne the force F on each atom: r r d r r F = m = ν ( R) dt Use postons and acceleratons at tme t (and postons from t - δ t) to calculate new postons at tme t δ t Intal veloctes (v ) usng the Boltzmann dstrbuton at the gven temperature v = (m /πkt) 1/ exp (- m v /kt) Molecular dynamcs (MD) smulatons A determnstc method based on the soluton of Newton s equaton of moton F = m a for the th partcle; the acceleraton at each step s calculated from the negatve gradent of the overall potental, usng F = - grad V -= - V In molecular dynamcs forces are derved from a potental energy functon V, whch depend on the partcle coordnates: The problem of modellng a materal can therefore be restated as that of fndng a potental functon for that materal. Molecular dynamcs (MD) smulatons V = Σ k (energes of nteractons between and all other resdues k located wthn a cutoff dstance of R c from ) Dervatve of V wth respect to the poston vector r = (x, y, z ) T at each step a x ~ - V/ x a y ~ - V/ y a z ~- V/ z Non-Bonded Interacton Potentals Electrostatc nteractons of the form E k (es) = q q k /r k Van der Waals nteractons E (vdw) = - a k /r k 6 b k /r k 1 Bonded Interacton Potentals Bond stretchng E (bs) = (k bs /) (l l 0 ) Bond angle dstorton E (bad) = (k θ /) (θ θ 0 ) Bond torsonal rotaton E (tor) = (k φ /) f(cosφ ) 15

16 Molecular dynamcs (MD) smulatons The Verlet algorthm The most wdely used method of ntegratng the equatons of moton s that ntally adopted by Verlet [1967].The method s based on postons r(t), acceleratons a (t), and the postons r(t -δt) from the prevous step. The equaton for advancng the postons reads as r(tδt) = r(t)-r(t-δt) δt a(t) The veloctes do not appear at all. They have been elmnated by addton of the equatons obtaned by Taylor expanson about r(t): r(tδt) = r(t) δt v(t) (1/) δt a(t)... r(t-δt) = r(t) - δt v(t) (1/) δt a(t)- The veloctes are not needed to compute the trajectores, but they are useful for estmatng the knetc energy (and hence the total energy). They may be obtaned from the formula v(t)= [r(tδt)-r(t-δt)]/δt Trajectory fle: Durng molecular dynamcs (and energy mnmzaton) the coordnates (and veloctes) are saved at regular ntervals. Such a fle s called a trajectory fle. Water Models A recent revew lsted 46 dstnct models, so ndrectly ndcatng ther lack of success n quanttatvely reproducng the propertes of real water. They may, however, offer useful nsght nto water's behavor. Implct Solvent Models Water molecules are not ncluded as molecules, but represented by an extra potental on the solvent accessble surface. only 50% slower than vacuum calculatons Explct Solvent Models Water molecules are explctly ncluded as ndvdual molecules. Force Felds for water molecules are not trval... Computatonally expensve... Models types a, b and c are all planar whereas type d s almost tetrahedral ~10 tmes faster than explct water MD Perodc Boundary Condtons (PBC) Buldng peptdes usng Z matrces Perodc boundary condtons are used to smulate solvated systems or crystals. In solvated systems, PBC prevents that the solvent "evaporates n slco" dstance angle dhedral connectvty N H H H (end of fle) (1 means optmze, 0 means keep constant, -1 means vary accordng to a desgnated pattern) 16

17 PDB Representaton PDB Representaton contd. HETATM 1 C HETATM C HETATM 3 C HETATM 4 H HETATM 5 H HETATM 6 H HETATM 7 H HETATM 8 H HETATM 9 H HETATM 10 H HETATM 11 H contnued... (blue ndcates data columns not utlzed/recognzed by all software) CONECT CONECT CONECT CONECT 4 1 CONECT 5 1 CONECT 6 CONECT 7 CONECT 8 3 CONECT 9 3 CONECT 10 3 CONECT 11 1 END Atom types (AMBER) Bond Parameters Angle Parameters Torson Parameters 17

18 Improper Torsons Van der Waals (LJ) Parameters 1 6 σ < σ = 4ε 1 j r r VLJ 6 Atomc Partal Charges Typcal Tme Scales... Bond stretchng: Elastc vbratons: Rotatons of surface sdechans: Hnge bendng: Rotaton of bured sde chans: Proten foldng: sec sec sec sec sec sec. Tmescale n MD: A Typcal tmestep n MD s 1 fs (10-15 sec) (deally 1/10 of the hghest frequency vbraton) Ab nto proten foldng smulaton Physcal tme for smulaton 10 4 seconds Typcal tme-step sze seconds Number of MD tme steps Atoms n a typcal proten and water smulaton 3,000 Approxmate number of nteractons n force calculaton 10 9 Machne nstructons per force calculaton 1000 Total number of machne nstructons 10 3 BlueGene capacty (floatng pont operatons per second) 1 pentaflop (10 15 ) 18

19 AMBER: J. Am. Chem. Soc. 117,