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1 (c) M Gerstein, 2006, Yale, gersteinlab.org Seq. Alignment, Struc. Alignment, Threading Cor e.

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Presentation on theme: "1 (c) M Gerstein, 2006, Yale, gersteinlab.org Seq. Alignment, Struc. Alignment, Threading Cor e."— Presentation transcript:

1 1 (c) M Gerstein, 2006, Yale, gersteinlab.org Seq. Alignment, Struc. Alignment, Threading Cor e

2 2 (c) M Gerstein, 2006, Yale, gersteinlab.org Mail Servers and Web Forms Illustration Credits: D Frishman handout Extra

3 3 (c) M Gerstein, 2006, Yale, gersteinlab.org Additional Features of DNA sequences in Genomes

4 4 (c) M Gerstein, 2006, Yale, gersteinlab.org Gene finding composition of codons, nts Splice site finding genome mRNA

5 5 (c) M Gerstein, 2006, Yale, gersteinlab.org Genetic Code Codons with second position pyrimidines encode mostly hydrophobic amino acids (tan), while those with second position purines encode mostly polar amino acids (blue, red, and purple) The genetic code is nonambiguous. Each codon encodes a single amino acid. The only exception is GUG which in some mRNAs is used as a start codon to encode Met The genetic code includes three stop codons, UAG, UAA, and UGA which are termed amber, ochre, and opal codons The genetic code is nearly but not absolutely universal. The genetic code in mitochondria and some ciliates use a slightly modified version of the code The genetic code is highly degenerate (64 codons to encode 20 amino acids) Three amino acids (Arg, Leu, Ser) are each specified by six codons, and many of the other amino acids are specified by two or four codons The arrangement of the codons within the genetic code is not random In most cases mutation of the third nucleotide in the codon would either cause no change in the amino acid (Arg, Val or Leu for example) or would create a fairly conservative change (Phe to Leu or Asp to Glu) (Page adapted from S Strobel, Biochemistry Lecture Notes) CAI Extra

6 6 (c) M Gerstein, 2006, Yale, gersteinlab.org Splicing Splicing must be done accurately. Missplicing by even one nucleotide would result in a frameshift mutation throughout the remainder of the message The splice sites are defined largely by sequences within the intron The intron begins with the sequence GU and ends with AG and is part of a larger consensus sequence at both the5’ and 3’ splice sites (see figure) nucleotides upstream of the 3’ splice site is the branch site which includes an A that serves as the nucleophile in the reaction (Page adapted from S Strobel, Biochemistry Lecture Notes) Extra

7 7 (c) M Gerstein, 2006, Yale, gersteinlab.org Alternative Splicing: Multiple Proteins from One Gene A single transcript can be processed to include or not include specific exons within the gene. This is termed alternative splicing This makes it possible to generate multiple proteins from a single gene For example a single rate gene encodes seven tissue-specific variants of the muscle protein a-tropomyosin through the process of alternative splicing Sex determination in Drosophila is largely controlled by a series of alternative splicing events (Page adapted from S Strobel, Biochemistry Lecture Notes)

8 8 (c) M Gerstein, 2006, Yale, gersteinlab.org Promotors The RNA polymerase recognizes a promoter sequence within the DNA The consensus promoter includes two six base pair regions upstream of the transcription start site (defined as nucleotide +1) The Pribnow box (consensus sequence of TATAAT) is 10 nt upstream There is second element 35 nt upstream (consensus sequence TTGACA) The rates at which genes are transcribed vary directly with the rate that their promoters from stable initiation complexes with the holoenzyme The -10 and -35 regions of the promoter sequence are recognized by the sigma subunit of the RNA polymerase holoenzyme (which also includes two  and two  subunits Without the sigma subunit the RNA polymerase has no affinity for the DNA After entering the elongation phase of transcription, the sigma factor is removed from the polymerase complex Expression of different sigma factors makes it possible for a bacteria to efficiently respond to external stimuli (turn on sporulation genes, heat shock genes, etc.) (Page adapted from S Strobel, Biochemistry Lecture Notes) Extra

9 9 (c) M Gerstein, 2006, Yale, gersteinlab.org References Argos P. (1976) Prediction of the secondary structure of mouse nerve growth factor and its comparison with insulin. Biochemical and Biophysical Research Communications 3: Bairoch A and Apweiler R. (1996) The SWISS-PROT protein sequence data bank and its new supplement TREMBL. Nucleic Acids Res 24: Barton GJ. (1995) Protein secondary structure prediction. Curr Opinion Struct Biol 5: Benner SA, Gerloff DL, and Jenny TF. (1994) Predicting protein crystal structures. Science 265: Benner SA. (1995) Predicting the conformation of proteins from sequences. Progress and future progress. J Mol Recogn 8:9-28. Boyd, D., Schierle, C. & Beckwith, J. (1998). How many membrane proteins are there? Prot. Sci. 7, Crawford IP, Niermann T, and Kirschner K. (1987) Prediction of secondary structure by evolutionary comparison: application to the alpha subunit of thryptophan synthase. Proteins: Struct Func Genet 2: Deleage G and Roux B. (1987) An algorithm for protein secondary structure prediction based on class prediction. Protein Engineering 4: Eigenbrot C, Randal M, and Kossiakoff AA. (1992) Structural Effects Induced by Mutagenesis Affected by Crystal Packing Factors: the Structure of a Disulfide Mutant of Basic Pancreatic Trypsin Inhibitor. Proteins 14:75. Fasman, G. D. & Gilbert, W. A. (1990). The prediction of transmembrane protein sequences and their conformation: an evaluation. Trends Biochem Sci 15, Frishman D, and Argos P. (1995) Knowledge-Base Protein Secondary Structure Assignment. Proteins: Structure, Function, and Genetics 23: Frishman D, and Argos P. (1996) Incorporation of Non-Local Interactions in Protein Secondary Structure Prediction From the Amino Acid Sequence. Protein Engineering 2:in the press. Frishman D, and Argos P. (1997) The Future of Protein Secondary Structure Prediction Accuracy. Folding & Design 2: Frishman, D, and P Argos. (1996) 75% Accuracy in Protein Secondary Structure Prediction. Proteins 1997 Mar;27(3): Garnier J and Levin JM. (1991) The protein structure code: what is its present status. Comput Appl Biosci 7: Garnier, J. (1990). Protein structure prediction. Biochimie 72, Garnier, J., Gibrat, J. F. & Robson, B. (1996a). GOR method for predicting protein secondary structure from amino acid sequence. Meth. Enz. 266,

10 10 (c) M Gerstein, 2006, Yale, gersteinlab.org References Garnier, J., Gibrat, J. F. & Robson, B. (1996b). GOR method for predicting protein secondary structure from amino acid sequence. Methods Enzymol 266, Garnier, J., Osguthorpe, D. & Robson, B. (1978). Analysis of the accuracy and implications of simple methods for predicting the secondary structure of globular proteins. J. Mol. Biol. 120, Geourjon C, and Deléage G. (1995) SOPMA: Significant Improvements in Protein Secondary Structure Prediction by Consensus Prediction From Multiple Sequences. Comput Appl Biosci 11: Gibrat, J., Garnier, J. & Robson, B. (1987). Further developments of protein secondary structure prediction using information theory. J. Mol. Biol. 198, Gilbert RJ. (1992) Protein structure prediction from predicted residue prperties utilizing a digital encoding algorithm. J Mol Graph 10: Holley LH, and Karplus M. (1989) Protein Secondary Structure Prediction With a Neural Network. Proc Natl Acad Sci USA 86: Hunt NG, Gregoret LM, and Cohen FE. (1994) The origins of protein secondary structure. Effects of packing density and hydrogen bonding studied by a fast conformational search. J Mol Biol 241: Kabsch W and Sander C. (1984) On the use of sequence homologies to predict protein structure. Identical pentapeptides can have completely different conformation. Proc Natl Acad Sci USA 81: Kabsch W, and Sander C. (1983) Dictionary of Protein Secondary Structure: Pattern Recognition of Hydrogen-Bonded and Geometrical Features. Biopolymers 22: Kendrew JC, Klyne W, Lifson S, Miyazawa T, Nemethy G, Phillips DC, Ramachandran GN, and Sheraga HA. (1970). Biochemistry 9: King, R. D. & Sternberg, M. J. E. (1996). Identification and application of the concepts important for accurate and reliable protein secondary structure prediction. Prot. Sci. 5, King, R. D., Saqi, M., Sayle, R. & Sternberg, M. J. (1997). DSC: public domain protein secondary structure predication. Comput Appl Biosci 13, Levin J, Pascarella S, Argos P, and Garnier J. (1993) Quantification of secondary structure prediction improvement using multiple alignments. Protein Engineering 6: Levin JM, Robson B, and Garnier J. (1986) An algorithm for secondary structure determination in proteins based on sequence similarity. FEBS Lett 205: Levitt M, and Greer J. (1977) Automatic Identification of Secondary Structure in Globular Proteins. J Mol Biol 114: Lim VI. (1974) Algorithms for prediction of alpha-helical and beta-structural regions in globular proteins. J Mol Biol 88: Livingstone CD, Barton GJ (1996). Identification of functional residues and secondary structure from protein multiple sequence alignment. Methods Enzymol 266: Lupas A, Koster AJ, Walz J, and Baumeister W. (1994) Predicted secondary structure of the 20 S proteasome and model structure of the putative peptide channel. FEBS Lett 354: Matthews B. (1975) Comparison of the predicted and observed secondary structure of T4 phage lysozyme. Biochem Biophys Acta 405: Mehta PK, Heringa J, and Argos P. (1995) A simple and fast approach to prediction of protein secondary structure from multiply aligned sequences with accuracy above 70%. Protein Science 4:

11 11 (c) M Gerstein, 2006, Yale, gersteinlab.org References Muggleton S, King RD, and Sternberg MJE. (1992) Protein Secondary Structure Prediction Using Logic-Based Machine Learning. Protein Engineering 5: Nishikawa K, and Ooi T. (1986) Amino Acid Sequence Homology Applied to the Prediction of Protein Secondary Structures, and Joint Prediction With Existing Methods. Biochimica Et Biophysica Acta 871: Pauling L, Corey RB, and Branson HR. (1951) The structure of proteins: two hydrogen-bonded helical configurations of the polypeptide chain. Proc Natl Acad Sci USA 37: Persson, B. & Argos, P. (1997). Prediction of membrane protein topology utilizing multiple sequence alignments. J Protein Chem 16, Presnell SR, Cohen BI, and Cohen FE. (1992) A segment-based approach to protein secondary structure prediction. Biochemistry 31: Ptitsyn OB and Finkelstein AV. (1983) Theory of protein secondary structure and algorithm of its prediction. Biopolymers 22: Qian N, and Sejnowski TJ. (1988) Predicting the Secondary Structure of Globular Proteins Using Neural Network Models. J Mol Biol 202: Rackovsky S. (1993) On the nature of the protein folding code. Proc Natl Acad Sci U S A 90: Ramakrishnan C, and Soman KV. (1982) Identification of Secondary Structures in Globular Proteins - a New Algorithm. Int J Pept Protein Res 20: Rao S, Zhu Q-L, Vaida S, and Smith T. (1993) The local information content of the protein structural database. FEBS Lett 2: Rice CM, Fuchs R, Higgins DG, Stoehr PJ, and Cameron G N. (1993) The EMBL data library. Nucleic Acids Res 21: Richards FM, and Kundrot CE. (1988) Identification of Structural Motifs From Protein Coordinate Data: Secondary Structure and First-Level Supersecondary Structure. Proteins: Struct Func Genet 3: Robson B, and Garnier J. (1993) Protein Structure Prediction. Nature 361:506. Rost B, and Sander C. (1993) Prediction of Protein Secondary Structure at Better Than 70% Accuracy. J Mol Biol 232: Rost B, Sander C, and Schneider R. (1994) Redefining the goals of protein secondary structure prediction. J Mol Biol 235: Rost B, Sander C. (1993) Improved prediction of protein secondary structure by use of sequence profiles and neural networks. Proc Natl Acad Sci USA 90: Rost, B., Schneider, R. & Sander, C. (1993). Progress in protein structure prediction? Trends Biochem Sci 18, Rumelhart DE, Hinton GE, and Williams R. (1986) Learning representations by back-propagating errors. Nature 323: Salamov AA, and Solovyev VV. (1995) Prediction of Protein Secondary Structure by Combining Nearest-Meighbour Akgorithms Amd Multiple Sequence Alignments. J Mol Biol 247: Salamov AA, and Solovyev VV. (1997) Protein Secondary Structure Prediction Using Local Alignments. Journal of Molecular Biology 268: Sayle RA, and Milner-White EJ. (1995) RASMOL: Biomolecular Graphics for All. Trends in Biochemical Sciences 20:

12 12 (c) M Gerstein, 2006, Yale, gersteinlab.org References Sayle RA, and Milner-White EJ. (1995) RASMOL: Biomolecular Graphics for All. Trends in Biochemical Sciences 20: Sklenar H, Etchebest C, and Lavery R. (1989) Describing Protein Structure: a General Algorithm Yielding Complete Helicoidal Parameters and a Unique Overall Axis. Proteins: Struct Func Genet 6: Solovyev VV and Salamov AA. (1994) Predicting alpha-helix and beta-strand segments of globular proteins. Comput Appl Biosci 10: Stolorz P, Lapedes A, and Xia Y. (1992) Predicting Protein Secondary Structure Using Neural Net and Statistical Methods. J Mol Biol 225: Sumpter BG, Getino C, and Noid DW. (1994) Theory and applications of neural computing in chemical science. Ann Rev phys Chem 45: Sumpter BG, Getino C, and Noid DW. (19949 Theory and applications of neural computing in chemical science. Ann Rev phys Chem 45: Taylor WR and Thornton JM. (1984) Recognition of super-secondary structure in proteins. J Mol Biol 173: Thompson JD, Higgins DG, and Gibson TJ. (1994) CLUSTAL W: improving the sensitivity of progressive multiple sequence alignment through sequence weighting, position-specific gap penalties and weight matrix choice. Nucleic Acids Res 22: Thornton JM, Flores TP, Jones DT, and Swindells MB. (1991) Prediction of progress at last. Nature 354: von Heijne, G. (1992). Membrane protein structure prediction. Hydrophobicity analysis and the positive-inside rule. J Mol Biol 225, Wasserman PD. (1989) Neural Computing. Theory and Practice. New York. Zhang X, Mesirov JP, and Waltz DL. (1992) Hybrid System for Protein Secondary Structure Prediction. J Mol Biol 225: Zvelebil MJ, Barton GJ, Taylor WR, and Sternberg MJ. (1987) Prediction of Protein Secondary Structure and Active Sites Using the Alignment of Homologous Sequences. J Mol Biol 195:

13 13 (c) M Gerstein, 2006, Yale, gersteinlab.org Main Points Sec struc prediction Computing propensities, generalizing this to a more elaborate stat model (GOR) How you evaluate prediction, testing and training Parametric v heuristic Protein features: conservation pattern, near termini Threading and how its related to dynamic programming

14 14 (c) M Gerstein, 2006, Yale, gersteinlab.org A B C D E MM Markov Models Path: A D B C Probability = Init(A)*0.5*0.3*0.4 Extra

15 15 (c) M Gerstein, 2006, Yale, gersteinlab.org More HMMs R: 0.1 G: 0.2 B: 0.7 R: 0.1 G: 0.1 B: 0.8 R: 0.4 G: 0.4 B: Probability of a given sequence = Sum probability over ALL paths giving that sequence e.g. observed: RRR Extra

16 16 (c) M Gerstein, 2006, Yale, gersteinlab.org Example: simple fully interconnected model (N=3) Extra

17 17 (c) M Gerstein, 2006, Yale, gersteinlab.org Scoring by Brute Force method: Extra

18 18 (c) M Gerstein, 2006, Yale, gersteinlab.org End of class 2005,04.04

19 19 (c) M Gerstein, 2006, Yale, gersteinlab.org Over-training Cross Validation: Leave one out, seven-fold Credits: Munson, 1995; Garnier et al., fold Training Set (determine parms) Testing Set (see how it does) Validation Set Predictions from actual run

20 20 (c) M Gerstein, 2006, Yale, gersteinlab.org Multiple Sequence Methods Average GOR over multiple seq. Alignment The GOR method only uses single sequence information and because of this achieves lower accuracy (65 versus >71 %) than the current "state-of- the-art" methods that incorporate multiple sequence information (e.g. King & Sternberg, 1996; Rost, 1996; Rost & Sander, 1993). Illustration Credits: Livingston & Barton, 1996

21 21 (c) M Gerstein, 2006, Yale, gersteinlab.org Multiple Structure Alignment Building up a Fold Library (structure clusters)

22 22 (c) M Gerstein, 2006, Yale, gersteinlab.org Analysis: What can be Calculated from Simulation?

23 23 (c) M Gerstein, 2006, Yale, gersteinlab.org Average over simulation Deceptive Instantaneous Snapshots (almost anything can happen) Simple thermodynamic averages  Average potential energy  T ~ = ½ m Some quantities fixed, some fluctuate in different ensembles  NVE protein MD (“microcanonical”)  NVT liquid MC (“canonical”)  NPT more like the real world

24 24 (c) M Gerstein, 2006, Yale, gersteinlab.org Energy and Entropy Energy  At each point i (with coordinates x i ) on the pot. energy surface there is a well-defined “energy” U(x i ) Probability of occurrence  P i = exp(-U i /kT)/Q  The boltzmann distribution  Q = Sum over all P i, to normalize probabilities to 1 xixi AB U(x) in A in B Entropy  S(A) = k  (P i ln P i ), where the sum is over points i in A Free Energy  G(A) = U(A) - TS(A) Entropy and Free Energy are only defined for distinctly diff. “states” -- e.g. A (“unfolded”)and B (“folded”)  State B has a lower U and its minimum is more probable than State A  However, state A has a broader minimum that can be occupied in more ways Relative Prob  P(A)/P(B) = exp(-G(A)/kT) exp (G(B)/kT)

25 25 (c) M Gerstein, 2006, Yale, gersteinlab.org Number Density =Number of atoms per unit volume averaged over simulation divided by the number you expect to have in the same volume of an ideal “gas” Spatially average over all directions gives 1D RDF = [ Avg. Num. Neighbors at r ] [Expected Num. Neighbors at r ] “at r” means contained in a thin shell of thickness dr and radius r.

26 26 (c) M Gerstein, 2006, Yale, gersteinlab.org Number Density (cont) Advantages: Intuitive, Relates to scattering expts D/A: Not applicable to real proteins  1D RDF not structural  2D proj. only useful with "toy" systems Number densities measure spatial correlations, not packing  Low value does not imply cavities  Complicated by asymmetric molecules  How things pack and fit is property of instantaneous structure - not average

27 27 (c) M Gerstein, 2006, Yale, gersteinlab.org Measurement of Dynamic Quantities I The time-course of a relevant variable is characterized by (1) Amplitude (or magnitude), usually characterized by an RMS value R = sqrt[ ) 2 > ] R = sqrt[ + 2 > ] R = sqrt[ - 2 ] similar to SD fluctuation Relevant variables include bond length, solvent molecule position, H-bond angle, torsion angle Illustration from M Levitt, Stanford University

28 28 (c) M Gerstein, 2006, Yale, gersteinlab.org Measurement of Dynamic Quantities II The time-course of a relevant variable is characterized by (2) Rate or time-constant  Time Correlation function  C A (t) = = [ averaging over all s ]  Correlation usually exponentially decays with time t  decay constant is given by the integral of C(t) from t=0 to t=infinity Relevant variables include bond length, solvent molecule position, H-bond angle, torsion angle Illustration from M Levitt, Stanford University

29 29 (c) M Gerstein, 2006, Yale, gersteinlab.org D & RMS Diffusion constant  Measures average rate of increase in variance of position of the particles  Suitable for liquids, not really for proteins RMS more suitable to proteins  di = Difference in position of protein atom at t from the initial position, after structures have been optimally rotated translated to minimize RMS(t)  Solution of optimal rotation has been solved a number of ways (Kabsch, SVD)

30 30 (c) M Gerstein, 2006, Yale, gersteinlab.org Other Things to Calculate Fraction of Native Contacts Percent Helix Radius of Gyration Illustration and Caption from Duan & Kollman (1998) Caption: Time evolution of (A) fractional native helical content, (B) fractional native contacts, (C) R and the main chain rmsd from the native structure, and (D) SFE of the protein. The helical content and the native contacts are plotted on a logarithmic time scale. The helical content was measured by the main chain - angle (60° ± 30°, 40° ± 30°). The native contacts were measured as the number of neighboring residues present in 80% of the last 50 ns of the native simulation. Residues are taken to be in contact if any of the atom pairs are closer than 2.8 Å, excluding residues i and i+1, which always have the contacts through main chain atoms. The SFE was calculated as described by Eisenberg and McLachlan (31) using their parameters (0.0163, , , , and , in kcal mol Å2, for the surface areas of nonpolar, polar, sulfur, charged oxygen, and charged nitrogen, respectively). The straight line represents the SFE of the native structure.

31 31 (c) M Gerstein, 2006, Yale, gersteinlab.org Timescales Values from McCammon & Harvey (1987) and Eisenberg & Kauzmann

32 32 (c) M Gerstein, 2006, Yale, gersteinlab.org Simplified Solvent Figures from Smit et al. (1990) Smit et al. (1990) Surfactant simulation Three types of particles, o, w and s  s consists of w-w-o-o-o-o  s has additional springs all particles interact through L-J potential  o-w interaction truncated so purely repulsive Above sufficient to give rise to the formation of micelles, membranes, &c

33 33 (c) M Gerstein, 2006, Yale, gersteinlab.org Normal Modes

34 34 (c) M Gerstein, 2006, Yale, gersteinlab.org NMA formalism and implementation: Hinsen (1998): Minimize pot. energy Then diagonalize the 2nd derivative of the potential energy Simplified potential Solve:

35 35 (c) M Gerstein, 2006, Yale, gersteinlab.org Normal Mode Analysis Examine vibrational motions  For a system of N particles: Total number of displacement = 3N Total number of vibrational modes = 3N-6 Mode frequency indicates the type of motion:  Low frequency: Collective motion  High frequency: Localized motion

36 36 (c) M Gerstein, 2006, Yale, gersteinlab.org What is “NMA”? +

37 37 (c) M Gerstein, 2006, Yale, gersteinlab.org

38 38 (c) M Gerstein, 2006, Yale, gersteinlab.org Normal (Natural) Modes for MOLA

39 39 (c) M Gerstein, 2006, Yale, gersteinlab.org Adiabatic mapping Interpolate then minimize  Gives apx. energy (H) landscape through a barrier  can sort of estimate transition rate rate = (kT/h) exp (- dG/kT)  Used for ring flips, hinge motions

40 40 (c) M Gerstein, 2006, Yale, gersteinlab.org Molecular Dynamics Give each atoms a velocity.  If no forces, new position of atom (at t + dt) would be determined only by velocity x(t+dt) = x(t) + v dt Forces change the velocity, complicating things immensely  F = dp/dt = m dv/dt

41 41 (c) M Gerstein, 2006, Yale, gersteinlab.org Molecular Dynamics (cont) On computer make very small steps so force is nearly constant and velocity change can be calculated (uniform a) [Avg. v over  t] = (v +  v/2) Trivial to update positions: Step must be very small  t ~ 1fs (atom moves 1/500 of its diameter)  This is why you need fast computers Actual integration schemes slightly more complicated  Verlet (explicit half-step)  Beeman, Gear (higher order terms than acceleration)

42 42 (c) M Gerstein, 2006, Yale, gersteinlab.org MC vs/+ MD MD usually used for proteins. Difficult to make moves with complicated chain. MC often used for liquids. Can be made into a very efficient sampler. Hybrid approaches (Brownian dynamics) Simulated Annealing. Heat simulation up to high T then gradually cool and minimize to find global minimum.

43 43 (c) M Gerstein, 2006, Yale, gersteinlab.org Practical Aspects of Simulation

44 44 (c) M Gerstein, 2006, Yale, gersteinlab.org Typical Systems: DNA + Water

45 45 (c) M Gerstein, 2006, Yale, gersteinlab.org Typical Systems: Protein + Water

46 46 (c) M Gerstein, 2006, Yale, gersteinlab.org Practical Aspects: simulation cycle I Divide atoms into types (e.g. alpha carbon except for Gly, carbonyl oxygen) Initially  Associate each atom with a mass and a point charge  Give each atom an initial velocity Calculate Potential Calculating non-bonded interactions take up all the time  Electrostatics hardest since longest ranged  Neighbor lists Illustration Credit: McCammon & Harvey (1987)

47 47 (c) M Gerstein, 2006, Yale, gersteinlab.org Practical Aspects: simulation cycle II Update Positions with MD equations, then recalculate potential and continue Momentum conservation Energy Conserved in NVE ensemble Hydrophobic interaction naturally arises from water behavior Illustration Credit: McCammon & Harvey (1987)

48 48 (c) M Gerstein, 2006, Yale, gersteinlab.org Periodic Boundary Conditions Make simulation system seem larger than it is Ewald Summation for electrostatics (Fourier transform)

49 49 (c) M Gerstein, 2006, Yale, gersteinlab.org End of class M8 [2006,11.27] Start of class M9 [2006,11.29]


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