Coarse-grained modeling
   HOME

TheInfoList



OR:

Coarse-grained modeling, coarse-grained models, aim at simulating the behaviour of
complex system A complex system is a system composed of many components which may interact with each other. Examples of complex systems are Earth's global climate, organisms, the human brain, infrastructure such as power grid, transportation or communication ...
s using their coarse-grained (simplified) representation. Coarse-grained models are widely used for
molecular modeling Molecular modelling encompasses all methods, theoretical and computational, used to model or mimic the behaviour of molecules. The methods are used in the fields of computational chemistry, drug design, computational biology and materials scien ...
of biomolecules at various
granularity Granularity (also called graininess), the condition of existing in granules or grains, refers to the extent to which a material or system is composed of distinguishable pieces. It can either refer to the extent to which a larger entity is subd ...
levels. A wide range of coarse-grained models have been proposed. They are usually dedicated to computational modeling of specific molecules: proteins, nucleic acids, lipid membranes, carbohydrates or water. In these models, molecules are represented not by individual atoms, but by "pseudo-atoms" approximating groups of atoms, such as whole
amino acid residue Protein structure is the molecular geometry, three-dimensional arrangement of atoms in an amino acid-chain molecule. Proteins are polymers specifically polypeptides formed from sequences of amino acids, the monomers of the polymer. A single ami ...
. By decreasing the degrees of freedom much longer simulation times can be studied at the expense of molecular detail. Coarse-grained models have found practical applications in
molecular dynamics Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the ...
simulations. Another case of interest is the simplification of a given discrete-state system, as very often descriptions of the same system at different levels of detail are possible. An example is given by the chemomechanical dynamics of a molecular machine, such as Kinesin. The coarse-grained modeling originates from work by
Michael Levitt Michael Levitt, ( he, מיכאל לויט; born 9 May 1947) is a South African-born biophysicist and a professor of structural biology at Stanford University, a position he has held since 1987. Levitt received the 2013 Nobel Prize in Chemistr ...
and Ariel Warshel in 1970s. Coarse-grained models are presently often used as components of
multiscale modeling Multiscale modeling or multiscale mathematics is the field of solving problems which have important features at multiple scales of time and/or space. Important problems include multiscale modeling of fluids, solids, polymers, proteins, nucleic ac ...
protocols in combination with reconstruction tools (from coarse-grained to atomistic representation) and atomistic resolution models. Atomistic resolution models alone are presently not efficient enough to handle large system sizes and simulation timescales. Coarse graining and fine graining in statistical mechanics addresses the subject of entropy S, and thus the second law of thermodynamics. One has to realise that the concept of temperature T cannot be attributed to an arbitrarily microscopic particle since this does not radiate thermally like a macroscopic or ``
black body A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. The name "black body" is given because it absorbs all colors of light. A black body ...
´´. However, one can attribute a nonzero entropy S to an object with as few as two states like a ``
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
´´ (and nothing else). The entropies of the two cases are called thermal entropy and von Neumann entropy respectively. They are also distinguished by the terms coarse grained and fine grained respectively. This latter distinction is related to the aspect spelled out above and is elaborated on below. The Liouville theorem (sometimes also called
Liouville equation :''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelf ...
) : \frac(\Delta q\Delta p) = 0 states that a phase space volume \Gamma (spanned by q and p, here in one spatial dimension) remains constant in the course of time, no matter where the point q,p contained in \Delta q\Delta p moves. This is a consideration in classical mechanics. In order to relate this view to macroscopic physics one surrounds each point q,p e.g. with a sphere of some fixed volume - a procedure called coarse graining which lumps together points or states of similar behaviour. The trajectory of this sphere in phase space then covers also other points and hence its volume in phase space grows. The entropy S associated with this consideration, whether zero or not, is called coarse grained entropy or thermal entropy. A large number of such systems, i.e. the one under consideration together with many copies, is called an ensemble. If these systems do not interact with each other or anything else, and each has the same energy E, the ensemble is called a microcanonical ensemble. Each replica system appears with the same probability, and temperature does not enter. Now suppose we define a probability density \rho(q_i,p_i,t) describing the motion of the point q_i,p_i with phase space element \Delta q_i\Delta p_i. In the case of equilibrium or steady motion the equation of continuity implies that the probability density \rho is independent of time t. We take \rho_i=\rho(q_i,p_i) as nonzero only inside the phase space volume V_. One then defines the entropy S by the relation : S=-\Sigma_i\rho_i\ln\rho_i, \;\; where \;\;\Sigma_i\rho_i=1. Then,by maximisation for a given energy E, i.e. linking \delta S=0 with \delta of the other sum equal to zero via a Lagrange multiplier \lambda, one obtains (as in the case of a lattice of spins or with a bit at each lattice point) : V_= e^=\frac \;\;\; and \;\;\; S=\ln V_, the volume of \Gamma being proportional to the exponential of S. This is again a consideration in classical mechanics. In quantum mechanics the phase space becomes a space of states, and the probability density \rho an operator with a subspace of states \Gamma of dimension or number of states N_ specified by a projection operator P_. Then the entropy S is (obtained as above) : S= -Tr\rho\ln\rho = \ln N_, and is described as fine grained or von Neumann entropy. If N_=1, the entropy vanishes and the system is said to be in a pure state. Here the exponential of S is proportional to the number of states. The microcanonical ensemble is again a large number of noninteracting copies of the given system and S, energy E etc. become ensemble averages. Now consider interaction of a given system with another one - or in ensemble terminology - the given system and the large number of replicas all immersed in a big one called a heat bath characterised by \rho. Since the systems interact only via the heat bath, the individual systems of the ensemble can have different energies E_i, E_j, ... depending on which energy state E_i, E_j, ... they are in. This interaction is described as entanglement and the ensemble as canonical ensemble (the macrocanonical ensemble permits also exchange of particles). The interaction of the ensemble elements via the heat bath leads to temperature T, as we now show. Considering two elements with energies E_i,E_j, the probability of finding these in the heat bath is proportional to \rho(E_i)\rho(E_j), and this is proportional to \rho(E_i+E_j) if we consider the binary system as a system in the same heat bath defined by the function \rho. It follows that \rho(E)\propto e^ (the only way to satisfy the proportionality), where \mu is a constant. Normalisation then implies :\rho(E_i) = \frac, \Sigma_i \rho(E_i) =1. Then in terms of ensemble averages : =-, and \mu\equiv\frac, \;k_B=1, or by comparison with the second law of thermodynamics. is now the entanglement entropy or fine grained von Neumann entropy. This is zero if the system is in a pure state, and is nonzero when in a mixed (entangled) state. Above we considered a system immersed in another huge one called heat bath with the possibility of allowing heat exchange between them. Frequently one considers a different situation, i.e. two systems A and B with a small hole in the partition between them. Suppose B is originally empty but A contains an explosive device which fills A instantaneously with photons. Originally A and B have energies E_A and E_B respectively, and there is no interaction. Hence originally both are in pure quantum states and have zero fine grained entropies. Immediately after explosion A is filled with photons, the energy still being E_A and that of B also E_B (no photon has yet escaped). Since A is filled with photons, these obey a Planck distribution law and hence the coarse grained thermal entropy of A is nonzero (recall: lots of configurations of the photons in A, lots of states with one maximal), although the fine grained quantum mechanical entropy is still zero (same energy state), as also that of B. Now allow photons to leak slowly (i.e. with no disturbance of the equilibrium) from A to B. With fewer photons in A, its coarse grained entropy diminishes but that of B increases. This entanglement of A and B implies they are now quantum mechanically in mixed states, and so their fine grained entropies are no longer zero. Finally when all photons are in B, the coarse grained entropy of A as well as its fine grained entropy vanish and A is again in a pure state but with new energy. On the other hand B now has an increased thermal entropy, but since the entanglement is over it is quantum mechanically again in a pure state, its ground state, and that has zero fine grained von Neumann entropy. Consider B: In the course of the entanglement with A its fine grained or entanglement entropy started and ended in pure states (thus with zero entropies). Its coarse grained entropy, however, rose from zero to its final nonzero value. Roughly half way through the procedure the entanglement entropy of B reaches a maximum and then decreases to zero at the end. The classical coarse grained thermal entropy of the second law of thermodynamics is not the same as the (mostly smaller) quantum mechanical fine grained entropy. The difference is called
information Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random ...
. As may be deduced from the foregoing arguments, this difference is roughly zero before the entanglement entropy (which is the same for A and B) attains its maximum. An example of coarse graining is provided by
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.


Software packages

* Large-scale Atomic/Molecular Massively Parallel Simulator (
LAMMPS Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) is a molecular dynamics program from Sandia National Laboratories. LAMMPS makes use of Message Passing Interface (MPI) for parallel communication and is free and open-source soft ...
) * Extensible Simulation Package for Research on Soft Matte
ESPResSo
(external link)


References

{{Reflist Molecular modelling Biomolecules