6 Computational Chemistry and Molecular Modeling
7 ----------------------------------------------
9 |Gromacs| is an engine to perform molecular dynamics simulations and
10 energy minimization. These are two of the many techniques that belong to
11 the realm of computational chemistry and molecular modeling.
12 *Computational chemistry* is just a name to indicate the use of
13 computational techniques in chemistry, ranging from quantum mechanics of
14 molecules to dynamics of large complex molecular aggregates. *Molecular
15 modeling* indicates the general process of describing complex chemical
16 systems in terms of a realistic atomic model, with the goal being to
17 understand and predict macroscopic properties based on detailed
18 knowledge on an atomic scale. Often, molecular modeling is used to
19 design new materials, for which the accurate prediction of physical
20 properties of realistic systems is required.
22 Macroscopic physical properties can be distinguished by
24 #. *static equilibrium properties*, such as the binding constant of an
25 inhibitor to an enzyme, the average potential energy of a system, or the
26 radial distribution function of a liquid, and
28 #. *dynamic or non-equilibrium properties*, such as the viscosity of a liquid,
29 diffusion processes in membranes, the dynamics of phase changes,
30 reaction kinetics, or the dynamics of defects in crystals.
33 technique depends on the question asked and on the feasibility of the
34 method to yield reliable results at the present state of the art.
35 Ideally, the (relativistic) time-dependent Schrödinger equation
36 describes the properties of molecular systems with high accuracy, but
37 anything more complex than the equilibrium state of a few atoms cannot
38 be handled at this *ab initio* level. Thus, approximations are
39 necessary; the higher the complexity of a system and the longer the time
40 span of the processes of interest is, the more severe the required
41 approximations are. At a certain point (reached very much earlier than
42 one would wish), the *ab initio* approach must be augmented or replaced
43 by *empirical* parameterization of the model used. Where simulations
44 based on physical principles of atomic interactions still fail due to
45 the complexity of the system, molecular modeling is based entirely on a
46 similarity analysis of known structural and chemical data. The QSAR
47 methods (Quantitative Structure-Activity Relations) and many
48 homology-based protein structure predictions belong to the latter
51 Macroscopic properties are always ensemble averages over a
52 representative statistical ensemble (either equilibrium or
53 non-equilibrium) of molecular systems. For molecular modeling, this has
54 two important consequences:
56 - The knowledge of a single structure, even if it is the structure of
57 the global energy minimum, is not sufficient. It is necessary to
58 generate a representative ensemble at a given temperature, in order
59 to compute macroscopic properties. But this is not enough to compute
60 thermodynamic equilibrium properties that are based on free energies,
61 such as phase equilibria, binding constants, solubilities, relative
62 stability of molecular conformations, etc. The computation of free
63 energies and thermodynamic potentials requires special extensions of
64 molecular simulation techniques.
66 - While molecular simulations, in principle, provide atomic details of
67 the structures and motions, such details are often not relevant for
68 the macroscopic properties of interest. This opens the way to
69 simplify the description of interactions and average over irrelevant
70 details. The science of statistical mechanics provides the
71 theoretical framework for such simplifications. There is a hierarchy
72 of methods ranging from considering groups of atoms as one unit,
73 describing motion in a reduced number of collective coordinates,
74 averaging over solvent molecules with potentials of mean force
75 combined with stochastic dynamics :ref:`9 <refGunsteren90>`, to
76 *mesoscopic dynamics* describing densities rather than atoms and
77 fluxes as response to thermodynamic gradients rather than velocities
78 or accelerations as response to forces \ :ref:`10 <refFraaije93>`.
80 For the generation of a representative equilibrium ensemble two methods
83 #. *Monte Carlo simulations* and
85 #. *Molecular Dynamics simulations*.
87 For the generation of non-equilibrium
88 ensembles and for the analysis of dynamic events, only the second method
89 is appropriate. While Monte Carlo simulations are more simple than MD
90 (they do not require the computation of forces), they do not yield
91 significantly better statistics than MD in a given amount of computer
92 time. Therefore, MD is the more universal technique. If a starting
93 configuration is very far from equilibrium, the forces may be
94 excessively large and the MD simulation may fail. In those cases, a
95 robust *energy minimization* is required. Another reason to perform an
96 energy minimization is the removal of all kinetic energy from the
97 system: if several “snapshots” from dynamic simulations must be
98 compared, energy minimization reduces the thermal noise in the
99 structures and potential energies so that they can be compared better.
101 Molecular Dynamics Simulations
102 ------------------------------
104 MD simulations solve Newton’s equations of motion for a system of
105 :math:`N` interacting atoms:
107 .. math:: m_i \frac{\partial^2 \mathbf{r}_i}{\partial t^2} = \mathbf{F}_i, \;i=1 \ldots N.
109 The forces are the negative derivatives of a potential function
110 :math:`V(\mathbf{r}_1, \mathbf{r}_2, \ldots, \mathbf{r}_N)`:
112 .. math:: \mathbf{F}_i = - \frac{\partial V}{\partial \mathbf{r}_i}
114 The equations are solved simultaneously in small time steps. The system
115 is followed for some time, taking care that the temperature and pressure
116 remain at the required values, and the coordinates are written to an
117 output file at regular intervals. The coordinates as a function of time
118 represent a *trajectory* of the system. After initial changes, the
119 system will usually reach an *equilibrium state*. By averaging over an
120 equilibrium trajectory, many macroscopic properties can be extracted
121 from the output file.
123 It is useful at this point to consider the limitations of MD
124 simulations. The user should be aware of those limitations and always
125 perform checks on known experimental properties to assess the accuracy
126 of the simulation. We list the approximations below.
128 **The simulations are classical**
130 - Using Newton’s equation of motion automatically implies the use of
131 *classical mechanics* to describe the motion of atoms. This is all
132 right for most atoms at normal temperatures, but there are
133 exceptions. Hydrogen atoms are quite light and the motion of
134 protons is sometimes of essential quantum mechanical character.
135 For example, a proton may *tunnel* through a potential barrier in
136 the course of a transfer over a hydrogen bond. Such processes
137 cannot be properly treated by classical dynamics! Helium liquid at
138 low temperature is another example where classical mechanics
139 breaks down. While helium may not deeply concern us, the high
140 frequency vibrations of covalent bonds should make us worry! The
141 statistical mechanics of a classical harmonic oscillator differs
142 appreciably from that of a real quantum oscillator when the
143 resonance frequency :math:`\nu` approximates or exceeds
144 :math:`k_BT/h`. Now at room temperature the wavenumber
145 :math:`\sigma = 1/\lambda =
146 \nu/c` at which :math:`h
147 \nu = k_BT` is approximately 200 cm\ :math:`^{-1}`. Thus, all
148 frequencies higher than, say, 100 cm\ :math:`^{-1}` may misbehave
149 in classical simulations. This means that practically all bond and
150 bond-angle vibrations are suspect, and even hydrogen-bonded
151 motions as translational or librational H-bond vibrations are
152 beyond the classical limit (see
153 :numref:`Table %s <tab-vibrations>`)
156 .. |H2CX| replace:: H\ :math:`_2`\ CX
157 .. |OHO1| replace:: O-H\ :math:`\cdots`\ O
158 .. |INCM| replace:: :math:`\mathrm{cm}~^{-1}`
163 Typical vibrational frequencies (wavenumbers) in molecules and hydrogen-bonded
164 liquids. Compare :math:`kT/h = 200~\mathrm{cm}^{-1}` at 300 K.
168 +---------------+-------------+------------+
169 | | type of | wavenumber |
170 | type of bond | vibration | |INCM| |
171 +===============+=============+============+
172 | C-H, O-H, N-H | stretch | 3000--3500 |
173 +---------------+-------------+------------+
174 | C=C, C=O | stretch | 1700--2000 |
175 +---------------+-------------+------------+
176 | HOH | bending | 1600 |
177 +---------------+-------------+------------+
178 | C-C | stretch | 1400--1600 |
179 +---------------+-------------+------------+
180 | |H2CX| | sciss, rock | 1000--1500 |
181 +---------------+-------------+------------+
182 | CCC | bending | 800--1000 |
183 +---------------+-------------+------------+
184 | |OHO1| | libration | 400--700 |
185 +---------------+-------------+------------+
186 | |OHO1| | stretch | 50--200 |
187 +---------------+-------------+------------+
191 - Well, apart from real quantum-dynamical simulations, we can do one
194 (a) If we perform MD simulations using harmonic oscillators for
195 bonds, we should make corrections to the total internal energy
196 :math:`U = E_{kin} + E_{pot}` and specific heat :math:`C_V` (and
197 to entropy :math:`S` and free energy :math:`A` or :math:`G` if
198 those are calculated). The corrections to the energy and specific
199 heat of a one-dimensional oscillator with frequency :math:`\nu`
200 are: \ :ref:`11 <refMcQuarrie76>`
204 U^{QM} = U^{cl} +kT \left( {\frac{1}{2}}x - 1 + \frac{x}{e^x-1} \right)
208 C_V^{QM} = C_V^{cl} + k \left( \frac{x^2e^x}{(e^x-1)^2} - 1 \right)
210 where :math:`x=h\nu /kT`. The classical oscillator absorbs too
211 much energy (:math:`kT`), while the high-frequency quantum
212 oscillator is in its ground state at the zero-point energy level
213 of :math:`\frac{1}{2} h\nu`.
215 (b) We can treat the bonds (and bond angles) as
216 *constraints* in the equations of
217 motion. The rationale behind this is that a quantum oscillator in
218 its ground state resembles a constrained bond more closely than a
219 classical oscillator. A good practical reason for this choice is
220 that the algorithm can use larger time steps when the highest
221 frequencies are removed. In practice the time step can be made
222 four times as large when bonds are constrained than when they are
223 oscillators \ :ref:`12 <refGunsteren77>`. |Gromacs| has this
224 option for the bonds and bond angles. The flexibility of the
225 latter is rather essential to allow for the realistic motion and
226 coverage of configurational space \ :ref:`13 <refGunsteren82>`.
228 **Electrons are in the ground state**
229 In MD we use a *conservative* force field that is a function of
230 the positions of atoms only. This means that the electronic
231 motions are not considered: the electrons are supposed to adjust
232 their dynamics instantly when the atomic positions change (the
234 approximation), and remain in their ground state. This is really
235 all right, almost always. But of course, electron transfer
236 processes and electronically excited states can not be treated.
237 Neither can chemical reactions be treated properly, but there are
238 other reasons to shy away from reactions for the time being.
240 **Force fields are approximate**
243 They are not really a part of the simulation method and their
244 parameters can be modified by the user as the need arises or
245 knowledge improves. But the form of the forces that can be used in
246 a particular program is subject to limitations. The force field
247 that is incorporated in |Gromacs| is described in Chapter 4. In the
248 present version the force field is pair-additive (apart from
249 long-range Coulomb forces), it cannot incorporate
250 polarizabilities, and it does not contain fine-tuning of bonded
251 interactions. This urges the inclusion of some limitations in this
252 list below. For the rest it is quite useful and fairly reliable
253 for biologically-relevant macromolecules in aqueous solution!
255 **The force field is pair-additive**
256 This means that all *non-bonded* forces result from the sum of
257 non-bonded pair interactions. Non pair-additive interactions, the
258 most important example of which is interaction through atomic
259 polarizability, are represented by *effective pair potentials*.
260 Only average non pair-additive contributions are incorporated.
261 This also means that the pair interactions are not pure, *i.e.*,
262 they are not valid for isolated pairs or for situations that
263 differ appreciably from the test systems on which the models were
264 parameterized. In fact, the effective pair potentials are not that
265 bad in practice. But the omission of polarizability also means
266 that electrons in atoms do not provide a dielectric constant as
267 they should. For example, real liquid alkanes have a dielectric
268 constant of slightly more than 2, which reduce the long-range
269 electrostatic interaction between (partial) charges. Thus, the
270 simulations will exaggerate the long-range Coulomb terms. Luckily,
271 the next item compensates this effect a bit.
273 **Long-range interactions are cut off**
274 In this version, |Gromacs| always uses a
276 radius for the Lennard-Jones
277 interactions and sometimes for the Coulomb interactions as well.
278 The “minimum-image convention” used by |Gromacs| requires that only
279 one image of each particle in the periodic boundary conditions is
280 considered for a pair interaction, so the cut-off radius cannot
281 exceed half the box size. That is still pretty big for large
282 systems, and trouble is only expected for systems containing
283 charged particles. But then truly bad things can happen, like
284 accumulation of charges at the cut-off boundary or very wrong
285 energies! For such systems, you should consider using one of the
286 implemented long-range electrostatic algorithms, such as
287 particle-mesh Ewald \ :ref:`14 <refDarden93>`,
288 :ref:`15 <refEssmann95>`.
290 **Boundary conditions are unnatural**
291 Since system size is small (even 10,000 particles is small), a
292 cluster of particles will have a lot of unwanted boundary with its
293 environment (vacuum). We must avoid this condition if we wish to
294 simulate a bulk system. As such, we use periodic boundary
295 conditions to avoid real phase boundaries. Since liquids are not
296 crystals, something unnatural remains. This item is mentioned last
297 because it is the least of the evils. For large systems, the
298 errors are small, but for small systems with a lot of internal
299 spatial correlation, the periodic boundaries may enhance internal
300 correlation. In that case, beware of, and test, the influence of
301 system size. This is especially important when using lattice sums
302 for long-range electrostatics, since these are known to sometimes
303 introduce extra ordering.
305 Energy Minimization and Search Methods
306 --------------------------------------
308 As mentioned in sec. :ref:`Compchem`, in many cases energy minimization
309 is required. |Gromacs| provides a number of methods for local energy
310 minimization, as detailed in sec. :ref:`EM`.
312 The potential energy function of a (macro)molecular system is a very
313 complex landscape (or *hypersurface*) in a large number of dimensions.
314 It has one deepest point, the *global minimum* and a very large number
315 of *local minima*, where all derivatives of the potential energy
316 function with respect to the coordinates are zero and all second
317 derivatives are non-negative. The matrix of second derivatives, which is
318 called the *Hessian matrix*, has non-negative eigenvalues; only the
319 collective coordinates that correspond to translation and rotation (for
320 an isolated molecule) have zero eigenvalues. In between the local minima
321 there are *saddle points*, where the Hessian matrix has only one
322 negative eigenvalue. These points are the mountain passes through which
323 the system can migrate from one local minimum to another.
325 Knowledge of all local minima, including the global one, and of all
326 saddle points would enable us to describe the relevant structures and
327 conformations and their free energies, as well as the dynamics of
328 structural transitions. Unfortunately, the dimensionality of the
329 configurational space and the number of local minima is so high that it
330 is impossible to sample the space at a sufficient number of points to
331 obtain a complete survey. In particular, no minimization method exists
332 that guarantees the determination of the global minimum in any practical
333 amount of time. Impractical methods exist, some much faster than
334 others \ :ref:`16 <refGeman84>`. However, given a starting configuration,
335 it is possible to find the *nearest local minimum*. “Nearest” in this
336 context does not always imply “nearest” in a geometrical sense (*i.e.*,
337 the least sum of square coordinate differences), but means the minimum
338 that can be reached by systematically moving down the steepest local
339 gradient. Finding this nearest local minimum is all that |Gromacs| can do
340 for you, sorry! If you want to find other minima and hope to discover
341 the global minimum in the process, the best advice is to experiment with
342 temperature-coupled MD: run your system at a high temperature for a
343 while and then quench it slowly down to the required temperature; do
344 this repeatedly! If something as a melting or glass transition
345 temperature exists, it is wise to stay for some time slightly below that
346 temperature and cool down slowly according to some clever scheme, a
347 process called *simulated annealing*. Since no physical truth is
348 required, you can use your imagination to speed up this process. One
349 trick that often works is to make hydrogen atoms heavier (mass 10 or
350 so): although that will slow down the otherwise very rapid motions of
351 hydrogen atoms, it will hardly influence the slower motions in the
352 system, while enabling you to increase the time step by a factor of 3 or
353 4. You can also modify the potential energy function during the search
354 procedure, *e.g.* by removing barriers (remove dihedral angle functions
355 or replace repulsive potentials by *soft-core*
356 potentials \ :ref:`17 <refNilges88>`), but always take care to restore the correct
357 functions slowly. The best search method that allows rather drastic
358 structural changes is to allow excursions into four-dimensional
359 space \ :ref:`18 <refSchaik93>`, but this requires some extra programming
360 beyond the standard capabilities of |Gromacs|.
362 Three possible energy minimization methods are:
364 - Those that require only function evaluations. Examples are the
365 simplex method and its variants. A step is made on the basis of the
366 results of previous evaluations. If derivative information is
367 available, such methods are inferior to those that use this
370 - Those that use derivative information. Since the partial derivatives
371 of the potential energy with respect to all coordinates are known in
372 MD programs (these are equal to minus the forces) this class of
373 methods is very suitable as modification of MD programs.
375 - Those that use second derivative information as well. These methods
376 are superior in their convergence properties near the minimum: a
377 quadratic potential function is minimized in one step! The problem is
378 that for :math:`N` particles a :math:`3N\times 3N` matrix must be
379 computed, stored, and inverted. Apart from the extra programming to
380 obtain second derivatives, for most systems of interest this is
381 beyond the available capacity. There are intermediate methods that
382 build up the Hessian matrix on the fly, but they also suffer from
383 excessive storage requirements. So |Gromacs| will shy away from this
386 The *steepest descent* method, available in |Gromacs|, is of the second
387 class. It simply takes a step in the direction of the negative gradient
388 (hence in the direction of the force), without any consideration of the
389 history built up in previous steps. The step size is adjusted such that
390 the search is fast, but the motion is always downhill. This is a simple
391 and sturdy, but somewhat stupid, method: its convergence can be quite
392 slow, especially in the vicinity of the local minimum! The
393 faster-converging *conjugate gradient method* (see *e.g.*
394 :ref:`19 <refZimmerman91>`) uses gradient information from previous steps. In general,
395 steepest descents will bring you close to the nearest local minimum very
396 quickly, while conjugate gradients brings you *very* close to the local
397 minimum, but performs worse far away from the minimum. |Gromacs| also
398 supports the L-BFGS minimizer, which is mostly comparable to *conjugate
399 gradient method*, but in some cases converges faster.