F. Piazza Center for Molecular Biophysics and University of Orléans, France. Selected topic in Physical Biology. Lectures 2-3

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1 F. Piazza Center for Molecular Biophysics and University of Orléans, France Selected topic in Physical Biology Lectures 2-3 Elastic network models of proteins. Theory, applications and much more than this

2 The force field The first thing: a force field

3 Structure-energy relation

4 Energy landscapes: local versus global minima 2D representation of a 3nD-dimensional landscape Energy minimization

5 Molecular dynamics simulations: stuck in a local minimum Computational costs at present only allow to explore local minima in general. For small system the ever-growing computational power allows to observe a few barrier-crossing events

6 Molecular dynamics simulations: integrate Newton s equation Nothing more than F = ma

7 Many force fields and full simulation packages are available You can install one of these packages on your local work-station and run your molecular dynamics simulation right away

8 A recent innovation: ANTON A microchip designed and optimized only to run molecular dynamics simulations

9 Coarse-grained models W. G. Noid, Perspective: Coarse-grained models for biomolecular systems The Journal of Chemical Physics 139, (2013) Given the system coordinates, a coordinate mapping, M, determines the configuration, R, of the CG model as a function of the configuration, r, of an underlying atomistic model. The Cartesian coordinates, R I, of site I are typically determined as a linear combination of atomic Cartesian coordinates, r i, with constant, positive coefficients that often correspond to, e.g., the center of mass or geometry for the associated atomic group.

10 A long story, starting in 1975 and recently culminated in a Nobel prize! Abstract A new and very simple representation of protein conformations has been used together with energy minimisation and thermalisation to simulate protein folding. Under certain conditions, the method succeeds in renaturing bovine pancreatic trypsin inhibitor from an open-chain conformation into a folded conformation close to that of the native molecule.

The Go model: the birth of native-centric modeling strategies Independently, in that same year Nobuhiro Go and his collaborators proposed a model an (even simpler) model where the chain of beads is

11 The Go model: the birth of native-centric modeling strategies Independently, in that same year Nobuhiro Go and his collaborators proposed a model an (even simpler) model where the chain of beads is mounted on on a lattice (initially they took a 2D lattice). Each bead would correspond to a residue or even to a secondary structure element of a protein (e.g. an -helix). The protein is restricted to fluctuate about the imposed structure: native-centric

12 Native-centric and off-lattice: the class of Elastic Network Models (ENMs) The story begins in 1996 Monique Tirion shows that the low frequency normal modes (NM) of a protein are not significantly altered when Interatomic interactions are replaced by identical Hookean springs (one-parameter model)!

13 Normal Mode Analysis (NMA) Typical all-atom potential energy V For small enough displacements about the equilibrium position:

14 Normal Mode Analysis (NMA) Displacements The equations of motion reads! (1)! where we have introduced the Hessian matrix! The system of coupled ODEs (1) can be transformed into a set of uncoupled ODEs by the Normal Mode coordinate transformation! Mass-weighted Hessian! Diagonalization! Coordinate transformation: the Normal Modes!

15 Normal Mode Analysis (NMA) Perform the change of variable and use the orthonormality and completeness relations! We get 3N uncoupled ODEs. It is the harmonic oscillator! The solutions are readily determined! So that the bead displacements can be written as!

16 Normal Mode Analysis (NMA) We note that the NM change of variables is such that the total potential energy and kinetic energy quadratic forms be diagonalized simultaneously. As a consequence, one has! 1 2 m i u 2 iα iα iα,jβ where we have introduced the normal mode energies! k = 1 2 Q 2 k + ω 2 kq 2 k u iα K αβ ij u jβ = k k At thermodynamic equilibrium! k = k B T H = 1 2 Ckω 2 k 2 = C k = k 2kB T ω k The amplitude of the k-th mode goes! as the inverse of its frequency. In general,! for proteins modes with frequencies! below cm -1 are responsible! for % of displacements.!

17 Elastic Network Models (ENMs)

18 Elastic Network Models (ENMs): the potential energy V A = 1 2 i>j k ij rij r 0 ij 2 Anisotropic Network Model (ANM) Nearly pure central forces V G = 1 2 = 1 2 i>j k ij r ij r 0 ij 2 k ij u j u i 2 Gaussian Network Model (ANM) Angular forces of the same order as central ones i>j i and j can be atoms (Tirion) or groups of atoms e.g. aggregated particles, e.g. amino acids r ij = r i r j r 0 ij = r 0 i r 0 j r ij r 0 ij = u j u i

19 Elastic Network Models (ENMs): the cutoff issue k ij = kf( r 0 ij ) is the force constant between particles i and j Popular choices for the force constants include: f( r 0 ij ) =c ij 1 for r 0 ij <R c 0 otherwise f( r 0 ij ) exp[ ( r 0 ij /σ) 2 ] f( r 0 ij ) r 0 ij α, α > 0 The sharp cutoff model R c vary between 8 and 16 Ang! Two-parameter models, in the sense that there is 1. One physical force scale gauging homogeneously interparticle force constants 2. One reference length specifying the range of interparticle interactions

20 The Gaussian network model is equivalent to a scalar model The GNM is intrinsically a scalar model!

21 Elastic Network Models (ENMs): the forces. The ANM scheme

22 Elastic Network Models (ENMs): the forces. The GNM ANM scheme

23 Elastic Network Models (ENMs): angular versus central forces Case (a) is meant to illustrate the magnitude of angle-bending forces, case (b) illustrates bond stretching. We want to compare the two kind of forces in the ANM and in the GNM schemes!

24 Elastic Network Models (ENMs): angular versus central forces ANM! GNM! Coupling along the direction of displacement! ANM! GNM! f (a) iy f (b) ix f (a) iy f (b) ix = O[(u/R) 2 ] = O[1]

26 Building an ENM An energy minimization is no longer required. This means that the equilibrium structure is assumed to coincide with the experimentally resolved structure (X-ray Crystallography, NMR). This can be risky! 1 First step: go to a structure repository and download a file containing the atomic coordinates of the macromolecules h"p://pdb.org/pdb/home/home.do The extension of such files is.pdb (for Protein Data Bank) 2 Use a computer program to read the coordinates from the PDB file, which has its own specific format. This will be your r 0 i vectors

27 PDB files contain a wealth of information on the protein x y z Atom number Atom type B factor Amino acid (three- leeer code) Chain number Chain

28 Calculating the mass-weighted Hessian is simple for ENMs! ANM H αβ ij = k ij k ij ˆRα mi m ij ˆRβ ij δ ij j m ˆR α im ˆR β im ˆR α ij r0 i,α r0 j,α r 0 ij Cartesian components ( =x,y,z) of the equilibrium unit vectors of inter-particle bonds GNM V G = 1 2 u T αku α α GNM H αβ ij = 1 2 ut (I 3 K)u = K ijδ αβ mi m j K = u is the 3N-dimensional vector of particle displacement k ij j=i K ij for i = j for i = j

29 Inter-particle correlation: the covariance matrix. The covariance matrix can be computed analytically!! Z u is the partition function! Let us introduce the matrix of eigenvectors of the mass-weighted Hessian matrix! Then we have!

30 Inter-particle correlation: the covariance matrix Let us perform the following change of variables! where!

31 Inter-particle correlation: the covariance matrix where!

32 Inter-particle correlation: the covariance matrix

$In fact, this is not as simple as that, as they are indeed refining parameters used in fitting the X-ray diffraction spectra, measuring line-widths. As such, they also contain (large!$

33 The crystallographic B-factors: definition The crystallographic Debye-Waller factors (so-called isotropic B-factors) are related to atomic fluctuations. In fact, this is not as simple as that, as they are indeed refining parameters used in fitting the X-ray diffraction spectra, measuring line-widths. As such, they also contain (large!) contributions from roto-translations of the protein as a whole in the crystal and static disorder (the atoms of the protein in different crystal cells are not exactly in the very same position) In the GNM the problem is effectively N-dimensional and not 3N-dimensional B i = 8π 3 α = 8πk BT 3m i u 2 iα α (H 1 ) αα ii (H 1 ) αβ ij = m i m j K 1 ij δ αβ = B i =8πk B T K 1 ii

34 Atomic fluctuations in the GNM PnB- Esterase 13 It appears that the protein is almost rigid. With the excepion that the C- terminal region exhibit the largest correlated movement within the whole protein. This could be a linker or entrance mechanism to the acive site. InteresIngly, the residues around the proposed acive site (HIS 399, GLU 308 and SER 187) measured also high slow modes (in contrast of the whole protein)

Atomic fluctuations provide insight into biological function β- lactamase (1BLC) (a) and penicillopepsin (1BXO) (b), illustra7ng the mobility of residues in the first (lowest frequency) GNM mode.

35 Atomic fluctuations provide insight into biological function β- lactamase (1BLC) (a) and penicillopepsin (1BXO) (b), illustra7ng the mobility of residues in the first (lowest frequency) GNM mode. The color code is blue- red- yellow- green in the order of increasing mobility. Both enzymes contain an inhibitor (shown in space filling, gray) bound near the most constrained (lowest mobility) region. (c) and (d) Corresponding square fluctua7on profiles and posi7ons of cataly7c and inhibitor- binding residues. Residues directly involved in cataly1c func1on at ac1ve sites are shown by the green open circles, inhibitor- binding residues are shown by the red squares and residues serving both cataly1c and inhibitor- binding func1ons are marked by the orange diamond. Cataly1c residues tend to lie in the s1ffest por1ons of the protein structures ElasIc network models for understanding biomolecular machinery: from enzymes to supramolecular assemblies Chakra Chennubhotla, A J Rader, Lee- Wei Yang and Ivet Bahar Phys. Biol. 2 (2005) S173- S180

36 Catalytic Residues Coincide or communicate with Global Hinge Regions Fluctuation profiles in the global mode (k = 1) and position of catalytic and inhibitor binding residues illustrated for six enzymes. Residues involved in catalytic function are marked with an open circle, inhibitors binding sites are marked with a closed square, and residues serving both catalytic and inhibitor binding functions are marked with a closed circle. Arrows indicate the hinge sites. Lee-Wei Yang and Ivet Bahar, Coupling between Catalytic Site and Collective Dynamics: A Requirement for Mechanochemical Activity of Enzymes, Structure, 13, , June, 2005

37 With care: B factors do not only contain internal motions!

38 Conformational changes in proteins Many proteins exist in open (apo) and closed (holo, liganded) form. The biological function is closely related to the conformational change brought about by the apo-holo transition Open (top and middle) and closed (bottom) forms of lysine-arginine-ornithine (LAO) binding protein as shown usually (top) and modeled as an ENM coarse-grained at the C level with a cutoff R c = 8 Ang. F. Tama and Y.-H. Sanejouand, Conformational changes in proteins arising from normal mode calculations, Prot. Eng., 14, 1-6 (2001). One can ask the following question: How many NMs (a complete orthonormal basis in E 3N ) will be required to describe (reconstruct) the conformational change (a given percentage of it)? Will I require a number order N or order 1?

39 Conformational changes in proteins reconstructed through NMs: the overlap coefficients Let us consider our NMs as normalized If I use the whole basis of NMs, the (normalized) conformational change is reconstructed exactly ˆR iα Rholo iα Rapo 3N 6 iα R holo R apo = k=1 I k a k iα I k iα ˆR iα a k iα Overlap coefficients

40 The surprising answer There exists a single mode m for which I m is of the order !!

41 Normal modes of the open conformations perform better than those of the closed conformations In the open conformations domains are better separated, hence better defined. Therefore a coarse-grained description of the large-scale dynamics works better.

42 This is usually true for collective motions (low-frequency NMs) but There exist conformational changes that are captured by more localized modes I m = 0.3

43 I k iα ˆR iα a k iα Overlap coefficients Collectivity index The more collective the conformational change, the better the one-mode overlap c k = 1 3N iα (ak iα ak )( R iα R) σ(a k )σ( R) Correlation coefficients η = 1 3N exp jβ ˆR jβ 2 log ˆR jβ 2

44 Case study: predicting active sites in enzymes We have seen catalytic sites seem to have a tendency to lie at hinge-like regions in enzyme structures. This indication comes from the fact that these sites surprisingly often coincide with nodes of low-frequency normal modes.! Is it possible to devise specific indicators that help identify and predict active sites?! Simple tools from network theory: the connectivity graph! C i = j c ij Connectivity. How many neighbors at each node! CC i = j ij 1 Closeness centrality Inverse of the sum of the shortest paths from a given node to all other nodes.! χ i = k S hf ξ k i 2 Spectral stiffness Contribution of a reduced subset of! high-frequency NMs to the local fluctuation of a given node!

45 Predicting active sites in enzymes: high-pass filter Filtering procedure! We compute the indicator patterns but apply a high-pass filter, so as to only retain a reduced number of peaks! Arginin Glycineaminotransferase. PDB code 1JDW!

46 Predicting active sites in enzymes: the cutoff lensing idea Study the patterns of our structural indicators as functions of the cutoff used to build the connectivity graph. For the sake of the argument, we can also push it to values that may be thought of unphysical (excessively connected structures)! R c = 10 Å R c = 20 Å Cutoff lensing effect Some irrelevant peaks disappear and an additional peak appears flagging an active site.! Arginin Kinase. PDB code 1BG0!

47 Predicting active sites in enzymes: varying the cutoff The three indicators behave differently when the cutoff is let increase. In particular the connectivity pattern becomes less interesting at high cutoff as the structures become more and more connected.! We need to study the number of peaks as a function of the cutoff! Arginin Glycineaminotransferase. PDB code 1JDW!

48 Analysis of the enzyme database: the catalytic site atlas (CSA) n Fraction of catalytic residues within sites from! the nearest peak versus cutoff, as computed over the ensemble of enzymes from the CSA! Average peak fraction (number of peaks! divided by number of residues) computed over the CSA!

49 Reliability of the stiffness indicator The reliability is defined as the fraction of predicted catalytic sites (within namino acids along the sequence) divided by the fraction of stiffness peaks (number of peaks per amino acid).! Average number of peaks in the reduced! stiffness patterns per catalytic site!! The optimal cutoff corresponds to nearly! one peak per catalytic site.!! Extreme predictive precision!

50 Predicting active sites in enzymes: size matters! n fraction of catalytic sites within sites from the nearest peak of the reduced stiffness! patterns computed over three different size classes in the CSA database.!

The best predictions can be obtained by combining the 3 indicators at optimal cutoff in a sequential way The connectivity profiles should be examined first. These are the ones with the largest number!

51 The best predictions can be obtained by combining the 3 indicators at optimal cutoff in a sequential way The connectivity profiles should be examined first. These are the ones with the largest number! of peaks, often coalescing to highlight extended regions. The search should be subsequently! narrowed down with the corresponding closeness profile, typically featuring more localized peaks,! albeit many of them likely to be orphan ones. The prediction should then be refined through the! reduced stiffness patterns, the ones with the least number of peaks.!!"##$%&'()*+ R c = 20!"!!"##$%&'()*+ R c = 20!"! #$!%! ++!%! '(!%! )*!%! #'!%! '#!%! ))!%! )-!%! #&!%! )+!%! ''!%! &'!%! ''!%! ',!%!!/".$#$..+ R c = 28!"!!n!= 1!,&-#$..+ R c = 22!"!!/".$#$..+ R c = 28!"!!n!= 2!,&-#$..+ R c = 22!"!

52 Scientific Reports 5, Article number: (2015) doi: srep14874!

53 Normal modes strictly refer to very low temperature The thermal overlap coefficients

54 The thermal overlap coefficients

55 Aggregated spectral weight over a few modes is a good temperature-insensitive indicator There is redistribution of spectral weight at the working temperature. CAUTION IN USING T=0 normal modes!

56 The overall shape matters for spectral reconstructions of conformational fluctuations at non-zero temperature The less globular, the less cooperative, the worse 1G2F!

57 Proteins live immersed in a solvent. There is friction! (1) (2)

58 (3) (4) (5) (3)

59 Langevin modes is the block covariance matrix (6) (7) (8)

60 The block covariance matrix can be shown to obey the following solution (this is a straightforward consequence of imposing a Gaussian ansatz for the Fokker-Planck equation) (9) Eq. (9) has the following solution (see e.g. book by Risken, The Fokker-Planck equation ) (10) (11) (12)

61 (12) (12) (13) (10) (14) (11) (15) (15)

62 Intramolecular energy flux in proteins or within other complex three-dimensional molecular structures. General question!! 2-electrode setup: inject energy! at some site and monitor energy! outflux at a different site.!! 1. What are the energy transduction pathways?! 2. Are specific site pairs characterized by low impedance?!!!!

63 A typical non-equilibrium setup! Hot thermostat T 1! Generalized impedance! i j i j Cold thermostat T 2!

64 Energy flux We need first to introduce a measure of local energy flow in protein structures.! This comes naturally if we take the time derivative of the local energies!! Taking the time derivative!

65 Let us introduce the current from i to j, from i to j), then! (positive if energy flows from which leads to! Note that at equilibrium the following relations hold! Hence the total incoming and outgoing energy current is zero at each site!

In the harmonic approximation the expression for the energy current can be simplified further. One has! Hot thermostat T 1! i j Cold thermostat T 2!

66 In the harmonic approximation the expression for the energy current can be simplified further. One has! Hot thermostat T 1! i j Cold thermostat T 2! This expression can be used to measure the energy current between two given sites in a nonequilibrium setting such as the one we are interested into!

67 J i j ss = 1 2 αβ K αβ ij [ u iαu jβ ss u jβ u iα ss ] where (ss = steady-state) is an average computed with respect to the non-equilibrium steady-state measure! β 1 i Γ ij The imposed temperature field!

68 Practically, the sequence of operations is the following: 1. β 1 i Γ ij The imposed temperature field! 2. Compute the non-equilibrium covariance matrix! 3. Isolate the 3x3 blocks in the off-diagonal block that correspond to the two electrode beads and compute the current! J i j ss = 1 2 αβ K αβ ij [ u iαu jβ ss u jβ u iα ss ]

69 Questions!!?!

Protein Simulations in Confined Environments

Critical Review Lecture Protein Simulations in Confined Environments Murat Cetinkaya 1, Jorge Sofo 2, Melik C. Demirel 1 1. 2. College of Engineering, Pennsylvania State University, University Park, 16802,