Presentation on theme: "Coarse Graining and Mesoscopic Simulations"— Presentation transcript:
1 Coarse Graining and Mesoscopic Simulations C.P. Lowe, University of AmsterdamContents:What are we doing and why?Inverse Monte-CarloLangevin/Brownian dynamicsStochastic rotational dynamicsDissipative particle dynamicsThe Lowe-Andersen thermostatLocal thermostats in MD.Case study: Free energy of confinement of a polymerCase study: The dynamics of mesoscopic bio-filaments(and how to do rigid constraints)
2 What is coarse graining? Answer: grouping things together and treating them as one objectYou are already familiar with the concept.Quantum MDClassical MDn-AlkaneUnited atom model
3 What is mesocopic simulation? Answer: extreme coarse graining to treat things on the mesoscopic scale(The scale ~ 100nm which is huge by atomic standards but where fluctuationsare still relevant)100nmlong polymer
4 What are the benefits of coarse graining? Why stop there? Eg. this lipidPNCOCAll-atom model118 atomsCoarse-grained model10 sitesllcCPU per time step at least 10 times less (or even better)
5 What are the benefits of coarse graining? Why stop there? Eg. this lipidPNCOCAll-atom model118 atomsCoarse-grained model10 sitesllcMaximum time-stepLonger time-steps possible
6 Coarse graining with inverse Monte Carlo Full information(but limited scale)All-atomic modelMD simulationCoarse-graining – simplified modelEffective potentialsfor selected sitesRDFs for selected degrees of freedomReconstruct potentials(inverse Monte Carlo)IncreasescaleEffectivepotentialsProperties on a larger length/time scale
7 Inverse Monte Carlo Model Properties direct inverse Interaction potentialRadial distributionfunctionsEffective potentials for coarse-grained models from "lower level" simulationsEffective potential = potential used to produce certain characteristics of the real systemReconstruct effective potential from experimental RDF
8 Inverse Monte Carlo(A.Lyubartsev and A.Laaksonen, Phys.Rev.A.,52,3730 (1995))Consider Hamiltonian with pair interaction:VaMake “grid approximation”:| | | | | | |Hamiltonian can berewritten as:Rcuta=1,…,MWhere Va=V(Rcuta/M) - potential within a-interval,Sa - number of particle’s pairs with distancebetween them within a-intervalSa is an estimator of RDF:
9 Inverse Monte Carlo b = 1/kT, In the vicinity of an arbitrary point in the space of Hamiltonians one can write:wherewithb = 1/kT,
10 Inverse Monte Carlo Choose trial values Va(0) Direct MC Calculate <Sa> (n) and differencesD<Sa>(n) = <Sa >(n) - Sa*Repeat until convergenceSolve linear equations systemObtain DVa(n)New potential: Va(n+1) =Va(n) +DVa(n)
11 Example: vesicle formation Starting from a square plain piece of membrane, 325x325 Å, 3592 lipids:(courtesy of Alexander Lyubartsev )cut plane
12 The dispersed phase problem Many important problems involve one boring species (usuallya solvent) present in abundance and another intersting speicesthat is a large molecule or molecular structure.ExamplesPolymer solutionsColloidal suspensionsAggregates in solutionThis is very problematic
13 The dispersed phase problem Polymers are long molecules consisting of a large number N (up to many millions) of repeating units. For example polyethyleneConsider a polymer solution at the overlap concentration(where the polymers roughly occupy all space)L~N1/2bwhere b is of the order of the monomer size
14 The dispersed phase problem Volume fraction of momomersVolume fraction of solvent(assuming solvent molecules similar in size to monomers)Number of solvent molecules per polymerSo, if N=106, not unreasonable, we need 109 solvent molecules per polymer
15 Use Stokes-Einstein to estimate Time-scaleConfigurations change on the time-scale it takes the polymer to diffuse a distance of its own size tD. From the diffusion equation root mean squared displacement D as a function of time t isExperimentally: b (polyethylene) = mb(DNA) = mSo for N= lp(polyethylene) = m (1/2m)lp(DNA) = m (50m)Use Stokes-Einstein to estimatek=Boltzmann’s constantT=Temperatureh=shear viscosity of solventkT(room temp.)~ gcm2/s2h(water)~0.01 g/cm sSo
16 The dispersed phase problem Configurations change on the time-scale it takes the polymer to diffuse a distance of its own size tD. From the diffusion equation root mean squared displacement D as a function of time t isCoarse graining requiredExperimentally: b (polyethylene) = mb(DNA) = mSo for N= lp(polyethylene) = m (1/2m)lp(DNA) = m (50m)Use Stokes-Einstein to estimatek=Boltzmann’s constantT=Temperatureh=shear viscosity of solventkT(room temp.)~ gcm2/s2h(water)~0.01 g/cm sSo
17 Hydrodynamic interactions The ultimate solvent coarse grainingThrow it away and reduce the role of the solvent to:1) Just including the thermal effects (i.e thefluctutations that jiggle the polymer around)2) Including the thermal and fluid-like like behaviourof the solvent. This will include the “hyrodynamic interactions”between the monomers.Fv’vHydrodynamic interactions
18 The Langevin Equation (fluctuations only) Solve a Langevin equation for the big phase:Force on particle i-gv is the friction force, here the friction coefficient is related to the monomerdiffusion coeffiient by D=gkTis a random force with the propertyis the sum of all other forcesMany ways to solve this equationForbert HA, Chin SA. Phys Rev E 63, (2001)It is basically a thermostat.
19 The Andersen thermostat (fluctuations only)Use an Andersen thermostat: A method that satisfies detailed balance(equilibium properties correct)Integrate the equations of motion with a normal velocity VerletalgorithmThen with a probability GDt (G is a “bath” collision probability) setWhere qi is a Gaussian random number with zero mean and unit variance.(i.e. take a new velocity component from the correct Maxwellian)Gives a velocity autocorrelation function C(t)=<v(0)v(t)>Identical to the Langevin equation with g/m=G
20 Andersen vs LangevinQuestion: Should I ever prefer a Langevin thermostat to an Andersenthermostat?Answer: No. Because Andersen satisfies detailed balance you can uselonger time-steps without producing significant errors in the equilibriumproperties(who cares that it is not a stochastic differential equation)
21 (fluctuations and hydrodynamics) Brownian Dynamics(fluctuations and hydrodynamics)Use Brownian/Stokesian dynamicsIntegrates over the inertial time in the Langevin equation and solve the corresponding Smoluchowski equation (a generalized diffusion equation). As such, only particle positions enter.are “random” displacements that satisfyand is the mobility tensorD.L. Ermak and J.A. McCammon, J. Chem. Phys. 969, 1352 (1978)“Computer simulations of liquids”, M.P. Allen and D.J. Tildesley, (O.U. Press, 1987)
22 (fluctuations and hydrodynamics) Brownian Dynamics(fluctuations and hydrodynamics)If the mobility tensor is approximated byThe algorithm is very simple. This corresponds to neglecting hydrodynamicinteractions (HI)Including HI requires the pair terms. A simple approximation based on theOseen tensor (the flow generated by a point force) is.For a more accurate descrition it is much more difficult but doable, see the workof Brady and co-workers.A. J. Banchio and J. F. Brady J. Chem. Phys. 118, (2003)
23 (fluctuations and hydrodynamics) Brownian Dynamics(fluctuations and hydrodynamics)Limitations:Computaionally demanding because of long range nature of mobility tensorDifficult to include boundariesFundamentally only works if inertia can be completely neglectedSo should I just neglect hydrodynamics: NO(hydrodynamics are what make a fluid a fluid)
24 Simple explicit solvent methods An alternative approach: Keep a solvent but make it as simple aspossible (strive for an “ising fluid”).What makes a fluid:Conservation of momentumIsotropyGallilean InvarianceThe right relative time-scaletime it takes momentum to diffuse ltime it takes sound to travel ltime it takes to diffuse l
25 Stochastic Rotational dynamics A. Malevanets and R. Kapral, J. Chem. Phys 110, 8605 (1999).Advectcolliderandom grid shift recoversGallilean invariance
26 Stochastic Rotational dynamics Collide particles in same cellbasically rotates the relative velocity vectorwhere the box centre of mass velocity iswith Ncell the number of particles in a given cell. R is the matrix fora rotation about a random axisAdvantages:TrendyComputationally simpleConserves mometumConserves energyDisadvantagesDoes not conserve angular momentumIntroduces boxesIsotropy?Gallilean invariance jammed in by grid shiftConserves energy (need a thermostat for non-equilibrium simulations)
27 Stochastic Rotational dynamics Equation of state: Ideal gasParametrically: exactly the same as all other ideal gas modelsmust fixnumber of particles per cell (cf r)degree of rotation per collision (cf G)number of cells traversed before velocity is decorrelated (cf L)Time-scalesTransport coefficients: theoretical results accurate in the wrong range of parameters. For realistic parameters, must callibrate.For an analysis seeJ.T. Padding abd A.A. Louis, Phys. Rev. Lett. 93, (2004)
28 Dissipative Particle Dynamics part 1: the methodFirst introduced by Koelman and Hoogerbrugge as an “off-lattice latticegas” method with discrete propagation and collision step.P.J. Hoogerbrugge and J.M.V.A. Koelman, Europhys. Lett. 19, 155 (1992)J.M.V.A. Koelman and P.J. Hoogerbrugge and , Europhys. Lett. 21, 363 (1993)This formulation had no well defined equilibrium state (i.e. correspondedto no known statistical ensemble). This didn’t stop them and others using it though.The formulation usually used now is due to Espanol and Warren.P. Espanol and P.B. Warren, Europhys. Lett. 30, 191, (1995).Particles move according to Newton’s equations of motion:
29 Dissipative Particle Dynamics So what are the forces?They are three fold and are each pairwise additiveThe “conservative” force:rcrijwhere;Is a “repulsion” parameterIs an interaction cut-off range parameter
30 Dissipative Particle Dynamics What is the Conservative force?Simple: a repulsive potential with the formU(r)aijrcIt is “soft” in that, compared to molecular dynamicsit does not diverge to infinity at any point (there isno hard core repulsion.rcThe “dissipative” forceComponent of relativevelocity along line of centres
31 Dissipative Particle Dynamics What is the Dissipative force?A friction force that dissipates relative momentum(hence kinetic energy)A friction force that transports momentumbetween particles?wdrcThe random force:
32 Fluctuation Dissipation 1To have the correct canonical distribution function(constant NVT) the dissipative (cools the system)and random (heats the system) forces are related:wdrcThe weight functions are relatedFor historical (convenient?) reasonswd is given the same form as theconservative forceAs are the amplitudes
33 DPD as Soft Particles and a Thermostat Without the random and dissipative force, this would simply be molecular dynamics with a soft repulsive potential.With the dissipative and random forces the system has a canonical distribution, so they act as a thermostat.These two parts of the method are quite separate but the thermostat has a number of nice features.LocalConserves MomentumGallilean Invariant
34 Integrating the equations of motion How to solve the DPD equations of motion is itself something of an issue.The nice property of molecular dynamics type algorithms (e.g. satisfyingdetailed balance) are lost because of the velocity dependent dissipative force.This is particularly true in the parametrically correct regimeWhy is this important?Any of these algorithms are okay if the time-step is small enoughThe longer a time-step you can use, the less computational time yoursimulations needHow long a time step can I use?Beware to check more than that the temperature is correctThe radial distribution function is a more sensitive test. The temperaturecan be okay while other equilibrium properties are severely inaccurate.L-J.Chen, Z-Y Lu, H-J ian, Z-Li, and C-C Sun, J. Chem. Phys. 122, (2005)
35 P. Espanol and P.B. Warren, Europhys. Lett. 30, 191, (1995). Integrating the equations of motionEuler-type algorithmP. Espanol and P.B. Warren, Europhys. Lett. 30, 191, (1995).And note that, because we are solving a stochastic differential equation(Applies for all the following except the LA thermostat)
36 R.D. Groot and P.B. Warren, J. Chem. Phys. 107, 4423, (1997). Integrating the equations of motionModified velocity Verlet algorithmR.D. Groot and P.B. Warren, J. Chem. Phys. 107, 4423, (1997).Here l is an adjustable parameter in the range 0-1Still widely usedActually equivalent to the Euler-like scheme
37 Integrating the equations of motion Self-consistent algorithm:I Pagonabarraga, M.H.J. Hagen and D. Frenkel, Europhys. Lett. 42, 377, (1998).Updating of velocities is performed iterativelySatisfies detailed balance (longer time-steps possible)Computationally more demanding
38 Which method should I use? 1) It depends on the conservative force (interaction potential). The timestep must always be small enough such that the conservative equationsof motion adequately conserve total energy. To check this, run thesimulation without the thermostat and check total energy.2) If this limits the time-step the methods that satisfy detailed balancelose their advantage.3) If not, use the self-consistent or LAT methods. Never Euler or modifiedVerlet.4) There are some much better methods that still do not strictlysatisfy detailed balance (based on more sophisticated Langevin-typealgorithms).W.K. den Otter and J.H.R. Clarke, Europhys. Lett. 53, 426 (2001).T. Shardlowe, SIAM J. Sci. Comput. (USA) 24, 1267 (2003).5) For a review seeP. Nikunen, M. Karttunen and I. Vattulainen, Comp. Phys. Comm. 153, 407 (2003).
39 Alternatively, change the method The complications arise because the stochastic differential equationis difficult to solve without violating detailed balance (see Langevinvs Andersen thermostats)In the same spirit let us modify the Andersen scheme such thatBath collisions exchange relative momentum between pair of particlesby taking a new relative velocity from the Maxwellian distribution forrelative velocitiesImpose the new relative velocity in such a way that linear and angularmomentum is conserved.Following the same arguments as Andersen, detail balance is satisfiedLeads to the Lowe-Andersen thermostat
40 The Lowe-Andersen thermostat Lowe-Andersen thermostat (LAT):C.P.Lowe, Europhys. Lett. 47, 145, (1999).“Bath” collisionHere G is a bath collision frequency (plays a similar role to g/m in DPD)Bath collisions are processed for all pairs with rij<rcThe current value of the velocity is always used in the bath collision (hencethe lack of an explicit time on the R.H.S.)The quantity x is a random number uniformly distributed in the range 0-1The quantity mij is the reduced mass for particles i and j, mij=mi mj/(mi+mj)
41 The Lowe-Andersen vs DPD (as a thermostat)Conserve linear momentum (BOTH)Conserve angular momentum (BOTH)Gallilean invariant (BOTH)Local (BOTH)Simple integration scheme satisfies detailed balance (LA YES, DPD NO)
42 The Lowe-Andersen vs DPD (as a thermostat)Disadvantage?: It does not use weight functions wd and wr (or alternativelyyou could say it uses a hat shaped weight functions)But, no-one has ever shown these are useful or what form they should best take. The form wr=(1-rij/rc) is only used for convenience (work for someone?)They could be introduced using a distance dependent collision probabilityIn the limit of small time-steps LAT and DPD are actually equivalent!E.A.J.F. Peters, Europhys. Lett. 66, 311 (2004).Word of warning: in the LAT, bath collisions must be processed in a random orderIs the DPD thermostat ever better than the Lowe-Andersen thermostat?In simple terms: you can take a longer time-step with LA than with DPD withoutscrewing things up.and there are no disadvantages so….NO
43 Can I use these thermostats in normal MD? Yes and in fact they have a number of advantages:1) Because they are Gallilean invariant they do not see translationalmotion as an increase in temperature. Nose-Hoover (which is not Gallileaninvariant does)T. Soddemann , B. Dünweg and K. Kremer, Phys. Rev. E 68, (2003)2) Because they preserve hydrodynamic behavior, even in equilibriumthey disturb the dynamics of the system much less than methods that do not(the Andersen thermostat for example)
44 Can I use these thermostats in normal MD? Example: a well know disadvantage of the Andersen thermostat is that athigh thermostating rates diffusion in the system is suppressed (leading toinefficient sampling of phase space)Lowe-AndersenAndersenwhereas for the Lowe-Andersen thermostat it is not.E. A. Koopman and C.P. Lowe, J. Chem. Phys. 124, (2006)
45 Can I use these thermostats in normal MD? 3) Where is heat actually dissipated? At the boundaries of the system.Because these thermostats are local (whereas Nose-Hoover is global)one can enforce local heat dissipation.Carbon nanotubemodelled by “frozen”carbon structureHeat exchange of diffusants with the nanotube modelled by localthermostating during diffusant-microtubule interactionsS. Jakobtorweihen,M. G. Verbeek, C. P. Lowe, F. J. Keil, and B. Smit Phys. Rev. Lett. 95, (2005)
46 DPD SummaryThe dissipative and random forces combine to act as a thermostat(Fullfilling the same function as Nose-Hoover or Andersen thermostatsin MD)As a thermostat it has a number of advantages over some commonlyused MD thermostatsThe conservative force corresponds to a simple soft repulsive harmonicpotential between particles, but in principle it could be anything(The DPD thermostat can also be used in MDT. Soddemann, B. Dunweg and K. Kremer, Phys. Rev. E68, (2003) )The equations of motion are awkward to integrate accurately withlarge time-steps. Chose your algorithm and test it with care.The Lowe-Andersen thermostat has the same features as the DPDthermostat but is computationally more efficient as it allows longertime-steps.
47 Dissipative particle dynamics part 2: why this form for the conservative force?In principle the conservative force can be anything you like, what are the reasons for this choice? Some common statements;“It is the effective interaction between blobs of fluid”No it isn’t, at least not unless you are very careful about what you mean by effective.“A soft potential that allows longer time-steps”Maybe, but relative to what?Factually: it is not a Lennard-Jones (or molecular-like) potential.It is the simplest soft potential with a force that vanishes at some distancercAs with any soft potential it has a simple equation of state in the fluidregime and at high densities.
48 The equation of state of a DPD Fluid For a single component fluid with pairwise additive spherically symmetric interparticle potentails the pressure P in terms of the radialdistribution function g(r) iswhere r is the density. For a soft potential withrange rc at high densities, r>>3/(4prc3) g(r)~1 sog(r) real fluidWhere a is a constant. For DPD a=.101aijrc4g(r) DPD fluidNote though that :If r is too high or kT too low the DPD fluid will freezemaking the method useless.And a/kT is not the the true second Virial coefficientso this does not hold at low densitiesEoS
49 Mapping a DPD Fluid to a real fluid R.D. Groot and P.B. Warren, J. Chem. Phys. 107, 4423, (1997).Match the dimensionless compressibility k for a DPD fluid to that of real fluidFor a (high density) DPD fluid, from the equation of stateFor water k-1~16 so in DPD aij=75kT/rrc4Once the density is fixed, this fixes the repulsion parameter.You can use a similar procedure to map the dimensionless compressibility of other fluids.
50 What’s right and what’s wrong By setting the dimensionless compressibility correctly we will get thecorrect thermodynamic driving forces FThfor small pressure gradients(the chemical potential gradient is also correct)P0PxFThTechnically, we reproduce the structure factor at long wavelengths correctly.But, other things are completely wrong, eg the compressibility factorP/rkTAnd this assumes on DPD particle is one water molecule. If it representsn water molecules the r(real)=nr(model) so aij must be naij(n=1). Thatis the repulsion parameter is scaled with n and if n>1 the fluid freezes.R.D. Groot and K.L. Rabone, Biophys. J. 81, 725 (2001).
51 DPD for a given equation of state I. Paganabarraga and D. Frenkel, J. Chem. Phys 155, 5015 (2001)The basic idea is to “input” an equation of state.To do so a local “density” is definedwhere r is a weight function that vanishes for rij>rcThe conservative force is the the derivative of the free-energy(as a function of r) w.r.t. the particle positionsWhere is the excess free energy per partilce as calculated from the EoSIs the density really the density?Is it a free energy or a potential energy?
52 DPD for a given equation of state Eg a van der Waals fluidwhere A and B are parameters (related to the critical propertiesof the fluid). Simplest EoS that gives a gas liquid transition.
53 Free energy of a spherically confined polymer Case StudyFree energy of a spherically confined polymerEffective potentials: potential between blobs is the (theoretical) effecivepotential between long polymersWorks up to a point:For higher degreesof confinementyou need more blobsIn Mesocopic modelling we have always thrown information out. The onusis on the user to justify what is in what is out and whether it matters.
54 Case Study: The dynamics of biofilaments M. C. Lagomarsino, I. Pagoabarraga and C.P. LowePhys. Rev. Lett. 94, (2004).M. C. Lagomarsino, F.Capuani and C.P. Lowe, J. Theor. Biol.224, 205 (2003)Nature uses a lot of mesoscopic filaments for structure and transportExample, microtubules (which act as tracks for molecular motors)Globular protienWe want to simulate the dynamics of these things in solutions.Without coarse graining – forget it
55 Case Study: The dynamics of biofilaments Effective potential: The bending energy is that of an ideal elastic filament
56 Mesoscopic model Ff Ft Fb Fx Fb - bending force (from the bending energy for afilament with stiffness k that we described earlier)Ft - Tension force (satisfies constraint of no relativedisplacement along the line of the links)Ff - Fluid force (from the model discussed earlier,with F the sum of all non hydrodynamic forces)Fx - External forceSolve equations of motion using a Langevin Equation!!
57 Imposing rigid constraints Note that this is a quite generic problem:MD (fixing bond lengths to integrate out fast vibrational degrees of freedom)roboticscomputer animationStep 1) Write the new positions in terms of the unconstrained positions (nc)and the as yet unknown constraint forcesStep 2) Constraint forces are equal and opposite, directed along currentconnector vector (conserve linear and angular momentum). Write in terms ofscalar multipliers
58 Imposing rigid constraints soStep 3) Linearize (violations of the constraints are small)Or in matrix form…
59 Imposing rigid constraints Matrix inversion is an order n3 process so solve iteratively using SHAKESimply satisfy successive linearized constraints even though satisfying oneviolates othersOr
60 Imposing rigid constraints For a linear chain matrix is tri-diagonal with elementsAnd inverting a tri-diagonal matrix in order n operations is trivial(for rings and branches the matrix can be diagonalized in ordern operation to make the problem equivalent to the linear chain)
61 Imposing rigid constraints this method is termed “Matrix inverted linearized constrains – SHAKE”A. Bailey and C.P. Lowe, J. Comp. Phys. (in press 2008)Milc SHAKESHAKE
62 The fluid forceA simple model, a chain of rigidly connectedpoint particles with a friction coefficient gVfF
63 Why might this not give a complete picture? A simple model, a chain of rigidly connected point particles witha friction coefficient g subject to an external force FVfFFf = -g (v-vf)Vfv
64 Stokesian dynamics without the the fluctuations The Oseen tensor gives the solution to the fluid flow equations (ona small scale) for a point force acting on a fluid. This givesthe velocity of the fluid due to the force on another bead asThese equations are linear so solutions just addStokesian dynamics without the the fluctuations
65 Approximate the solution as an integral. For a uniform perpendicular force.s = the distance along a rod of unit lengthb = is the bead separation
66 Approximate the solution as an integral. For a uniform perpendicular force.s = the distance along a rod of unit lengthb = is the bead separationIf the velocity is uniform the friction is higher at the end than in the middleConstructing the mesoscopic model gives us a theory
67 What happens with uniform force acting downwards? Sed = B= FL2/k = ratio of bending to hydrodynamic forcesIf the filament is long enough, the bending modulus small enough orthe force high enough, the filament bends significantly.
71 Movie timeE. Koopman presentsB = 1, filament aligned at 450
72 So a torque acts on the fibre to rotate it towards the prependicular Why?Aligned more parallel,lower friction forceFFA component of the force perpendicularto the force bends it and moves it left.Aligned more perpendicular,higher friction forceSo a torque acts on the fibre to rotate it towards the prependicular
73 How long does it take to reorientate? From this we can;work out what conditions are necessaryin the real world to see the effectwork out when the approximation ofneglecting diffusion and dipole orientationis sensible.
74 Is this practically relevant? For a microtubule the bending modulus is knownand we estimate, B ~ 1 requires F~1 pN for a 10 micronmicrotubule. This is reasonable on the micrometer scale.For sedimentation (external force is gravity) , no.Gravity is not strong enough. You’d need a ultracentrifugeMicrotubules are barely charged and the charge is known, we estimatean electric field of 100 V/m for B ~ 200 (L=30 microns). So itshould be doable.
75 But this only orientate the filament in a plane (perpendicular to the force direction) What if we apply a force in a direction that rotates?Circularly polarised electric fieldElectric field as a function of time
76 Dimensional Analysis 30m Microtubule in water, Field= 100 V/m Frequecy 1 HzMovie time~real timeCircularly polarised electric field
77 Dimensional Analysis 30m Microtubule in water, Field= 100 V/m Frequecy 1 HzMovie time~real timeCircularly polarised electric field