Schlick T. Molecular Modeling and Simulation: An Interdisciplinary Guide

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512 14. Molecular Dynamics: Further Topics

Figure 14.18. Left: Molecular snapshots near open (left column) and closed (right

column) states of pol β for four transition state regions [1031]: (1) Partial thumb closing.

(2) Asp-192 ﬂip. (3) Arg-258 partial rotation. (4) Phe-272 ﬂip. Right: Overall captured

reaction kinetics proﬁle (from TPS) for the conformational transition of pol β (for G:C)

from open (state 1) to closed (state 7) forms showing free energies (in k

T ) associated

with the different transition state regions. The meta-stable basins (in red) along the reaction

coordinate are numbered 1–7.

is used to ﬁnd kinetic “bottleneck regions” by estimating rate constants associated

with reaching subsequent interfaces. Then, FFS concentrates on sampling those

bottlenecks regions only. FFS is appropriate for discretely deﬁned surfaces, like

protein folding on a lattice [23].

Milestoning [354] coarse grains temporal and spatial descriptors of the system

by sequential transitions and then recovers reaction kinetics by integral equations

and global path optimization strategies. Similarly, Markov State Models (MSM)

[919] compute reaction rates from different interfaces by using transition net-

works to describe biomolecular kinetics and thermodynamics. See also [920].

Combinations of MSM with the single-sweep method [828] can be successful

for proteins [957] because phase space is surveyed to map dynamically important

regions, from which free simulations are initiated, and then the transition ma-

trix is constructed by piecing the information together. Clustering techniques for

partitioning space to locate key regions can also be used in this connection [1357].

Milestoning was compared to MSM, FFS, and transition interface sampling in

[1298], where the advantages of milestoning compared to the other methods were

suggested in terms of accuracy, efﬁciency, and parallel-machine implementations.

Various combinations of these methods, such as REMD with TIS (transition

interface sampling) to compute free energy proﬁles and rate constants when the

barriers are high [146], emphasize how different sampling approaches can be

combined in clever ways to suit the problem. Often, coarse-grained models are

14.8. Future Outlook 513

Free Energy (k

Thumb

closing

Thumb

closing

rotation

258

ion

rearrange

ment

Product

192 flip

272 flip

Chemical Reaction

Open

Closed

TS 1

TS 2

TS 3

TS 4

G:C

G:A

192 flip

258 rotation

272 flip

Closed

TS 1

192 flip

TS 2

258 Rotation

TS 3

TS 4

TS 5

Closed

Chemical Reaction

Product

Open

Figure 14.19. Overall captured reaction kinetics proﬁle for pol β’s closing transition fol-

lowed by chemical incorporation of dNTP for G:C and G:A systems [1032]. The barriers

to chemistry (dashed peaks) are derived from experimentally measured kpol values. The

proﬁles were constructed by employing reaction coordinate characterizing order parame-

ters in conjunction with transition path sampling. The potential of mean force along each

reaction coordinate is computed for each conformational event.

used in combination with implicit solvation and Langevin or Brownian dynamics

or MC for added beneﬁts. Still, applications of these methods to biomolecular

complexes at large remain a challenge.

14.8 Future Outlook

14.8.1 Integration Ingenuity

Many Approaches

In this chapter we have discussed various numerical integration techniques for

Newtonian and Langevin/Brownian dynamics and focused on variations that can

increase the timestep and lead to overall computational gains. For very detailed

dynamic pathways, the standard Newtonian approach is necessary, but for certain

systems (like long DNA supercoils or diffusion-controlled mechanisms in enzyme

catalysis), stochastic models are appropriate and computationally advantageous.

The approaches discussed to ameliorate the severe timestep restriction in

biomolecular dynamics involve force splitting or multiple timesteps, various har-

monic approximations, implicit integration, and other separating frameworks.

Each such avenue has encountered different problems along the way, but

these obstacles serve to increase our understanding of the intriguing numeri-

cal, computational, and accuracy issues involved in propagating the motion of

complex nonlinear systems.

514 14. Molecular Dynamics: Further Topics

The most successful integrators that address stability and resonance limitations

of MD integrators due to the high-frequency vibrational modes and the intricate

coupling among the vibrational modes of a biomolecule consist of multiple-

timestep integrators combined with stochastic dynamics. Langevin and Brownian

dynamics (LD and BD), where solvent collisions and thermal ﬂuctuations are in-

corporated in an average sense, have long been used as a way to allow larger

timesteps and hence larger timespans in dynamics simulations. For stochastic

dynamics methods, it is also easier to prove ergodicity.

Certainly, long MD simulations are being reported more regularly now com-

pared to several years ago, made possible by efﬁcient and parallel simulation

packages like NAMD, DESMOND, GROMACS and new computer systems like

Anton, hard-wired for MD simulations [1169,1462]. For example, a 1 μsatomic-

level simulation of the voltage-modulated potassium channel Kv1.2 (120,000

atoms) recently described the opening/closing mechanism involved [136]. Thus,

many problems thought to be intractable several decades ago are now being

solved.

But faster computer technology alone is insufﬁcient to address challenging bio-

molecular problems. Algorithmic developments for enhanced sampling such as

described here, especially those divide and conquer methods suitable for loosely

coupled processor architectures that are more readily available to the average user,

are needed to complement long atomistic simulations.

Resonance

The heightened appreciation of resonance problems [134,821], in particular, con-

trasts with the more systematic error associated with numerical stability that

grows monotonically with the discretization size. Ironically, resonance artifacts

are worse in the modern impulse multiple-timestep methods, formulated to be

symplectic and reversible; the earlier extrapolative variants were abandoned due

to energy drifts. Stochasticity and slow-force averaging, as in the LM method

[594, 595], or stochasticity and extrapolation, as in the LN method [95], can

dampen and/or remove resonance artifacts and allow larger timesteps, but this

requires added forces to standard Newtonian dynamics.

Ultimately, the compromise between the realized speedup and the accuracy

obtained for the governing dynamic model should depend on the applications for

which the dynamic simulations are used (e.g., conﬁgurational sampling versus

detailed dynamics).

14.8.2 Current Challenges

PME Protocols

A problem that remains unsolved in part in connection with MTS methods

is their optimal integration with Ewald and particle-mesh Ewald protocols.

14.8. Future Outlook 515

The problem is due to the presence of fast terms in the reciprocal Ewald com-

ponent; this limits the outer MTS timestep and hence the speedup. For some

discussion and approaches, see [89, 97, 98, 1019, 1025, 1236, 1449]. Improving

such algorithms will allow us to use larger outer timesteps and hence simulate

systems over longer times.

There is also a related parallelization problem in PME implementations: mem-

ory requirements create a bottleneck in typical MD simulations longer than a

microsecond. This is because the contribution of the long-range electrostatic

forces imposes a global data dependency on all the system charges. In PME,

this dependency appears as a convolution between the meshed charge distribution

and a chosen kernel evaluated using 3D Fast Fourier Transforms. These 3D FFT

computations impose communication problems and deﬁne the rate limiting step

for parallel implementations [403]. In practice, parameters such as mesh sizes,

precision, size of real space and k-space have been optimized to delay the commu-

nication bottleneck as possible (e.g., [1462]), but overall errors in long simulations

are far from trivial [1204]. These issues will likely receive more attention in the

future. In addition, these communication requirements and hence limitations on

parallel implementationssuggest that exploring alternatives to PME, like fast mul-

tipole (Chapter 10) or multigrid [1082] methods are needed. These and other

approaches may be necessary as long-time MD is implemented on massively

parallel computer architectures.

Technology’s Role

Not to be downplayed as a factor in the increasing of simulation scope is tech-

nology improvement. The steady increase in computer power, the declining cost

of fast and highly-modular processors, and the rise of efﬁcient parallelization

of MD codes surely are helping to bridge the timescale gap between simulation

range and experimental durations. The triumphant report of Duan and Kollman’s

1 μs trajectory approaching the folding state of a 36-residue villin headpiece,

from a disordered conﬁguration (4 dedicated months of 256 processors on the

Cray T3D/E), is a case in point [338]. However, as discussed in Chapter 1

(see Table 1.2 and Figures 1.3 and 1.4), microsecond simulations are possible

today and simulating milliseconds in the life of a biomolecule is on the hori-

zon [658, 1462]. But many experts consider that certain technical bottlenecks

must be overcome, such as mentioned above, concerning force-ﬁeld accuracy,

long-range force calculations, integration accuracy, and parallelization implemen-

tations. Indeed, designing reliable, efﬁcient, and general software for large-scale

MD applications on multiprocessors remains a challenge.

Sampling Issues

The scope of dynamics simulations is certainly being enhanced when used in com-

bination with various methods to analyze biomolecular motion, sample the large

conformation space, and obtain relevant information on reaction mechanisms,

pathways, and rates. Because coarse-grained models are vital for addressing

516 14. Molecular Dynamics: Further Topics

problems related to large macromolecular complexes or ensemble properties of

smaller systems, Monte Carlo methods deserve further consideration and devel-

opment in general. In addition, exciting new approaches for rigorous frameworks

for general multiscale models are on the horizon.

While harmonic approximation methods like PCA, NMA, ED and elastic net-

works continue to add valuable insights into biomolecular ﬂexibility and function,

they are also participating in more applications with the growth of network models

for molecular machines that help dissect and distill complex functional motions.

Simple potential modiﬁcations to MD like TMD can provide conformational

insights, and REMD approaches can enhance sampling, but success has been

rather empirical rather than rigorous. Because the actual kinetics of the systems

are altered, caution is warranted in biological interpretations. However, enhanced

sampling protocols can be useful in speciﬁc contexts, as recently demonstrated by

using accelerated MD [504] to reproduce residual dipolar coupling measurements

concerning the slow modes from NMR data on a domain of the protein GB3 and

pinpoint slow motions [831].

REMD has recently received some scrutiny concerning effective ensemble

exchange protocols, general computational efﬁciency, and ergodicity questions.

This has led to variants with stochastic elements — motivated by REMD’s ori-

gin in Monte Carlo parallel tempering methods — that can improve sampling

(e.g., [264]). Because REMD applications require concurrent processors (one

per replica), the technique is not always practical, especially for large atom-

istic systems. The distributed replica variant [1060] that performs stochastic

moves of independent replicas instead of pairwise exchanges of replicas offers

an alternative.

TPS, Markov state models, milestoning and approaches that aim to compute

reaction rates have been more rigorously grounded in theory, and successful bio-

molecular applications have been reported. Still, their application to biomolecules

in general remains far from routine. In this goal, various combinations of meth-

ods that deduce mechanisms and compute reaction rates by divide and conquer

approaches will undoubtedly be effective on today’s readily available distributed

computing resources of cluster networks, especially by enhancing them with

coarse-grained models and Monte Carlo elements.

Signiﬁcantly, all these approaches for enhanced sampling can be combined for

cumulative and signiﬁcant computational advantages.

Tip of the Iceberg

These long-time and greater-sampling approaches represent only the tip of the

iceberg of those possible. Undoubtedly, motivated by fundamental biological

problems like protein folding, the zealous computational biologists and the sci-

entists they enlist in these quests will continue to combine algorithmic ingenuity

with state-of-the-art computing to overcome, or circumvent, one of the fun-

damental challenges in molecular dynamics simulations. All these advances

are collectively opening the way to exciting applications of a rich variety of

14.8. Future Outlook 517

biomolecular systems regarding large-scale conformational changes and func-

tional dynamics on millisecond and longer timescales that are helping close the

gap between experimental and theoretical timeframes.

Ultimately, the most successful applications reﬂect a combination of novel and

rigorous mathematical ideas with physical intuition. And such applications can be

successful not only by reproducing experimental data concerning slow motions

but also in helping suggest mechanistic and energetic details. Clearly, the gap

between experimental and theoretical timeframes is steadily narrowing, leading to

renewed interest in MD-based modeling by theorists and experimentalists alike.

Similarity and Diversity in Chemical

Design

Chapter 15 Notation

YMBOL DEFINITION

Matrices

A Dataset matrix (n × m)

Rank k approximation to A

covariance matrix (m × m), elements c



projection matrix

SVD factor of A (n × n), contains left singular values

SVD factor of A (m × m), contains right singular values;

also eigenvector matrix of C

low-rank approximation to eigenvector matrix (m × k)

SVD factor (n × m), contains singular values

low-rank approximation to Σ

Vectors

left singular value

right singular value

vector of compound i (components Xi

,Xi

, ···Xi

)

scaled version of X

Yi projection of X

; also principal component of C

Scalars & Functions

intercompound distance ij in the projected

representation

f, E

target optimization functions

lower bounds on intercompound distance ij

upper bounds on intercompound distance ij

number of dataset descriptors

number of dataset components

T. Schlick, Molecular Modeling and Simulation: An Interdisciplinary Guide, 519

Interdisciplinary Applied Mathematics 21, DOI 10.1007/978-1-4419-6351-2

15,

 Springer Science+Business Media, LLC 2010

520 15. Similarity and Diversity in Chemical Design

Chapter 15 Notation Table (continued)

YMBOL DEFINITION

N number of variables

total number of distance segments satisfying a given

deviation from target

α, β

scaling factors

Euclidean distance (with upper/lower bounds u, l)

eigenvalues

mean value

weights used in target optimization function

singular values

Every sentence I utter must be understood not as an afﬁrmation but

as a question.

Niels Bohr (1885–1962).

15.1 Introduction to Drug Design

Following a simple introduction to drug discovery research, this chapter presents

some mathematical formulations and approaches to problems involved in chem-

ical database analysis that might interest mathematical/physical scientists. With

continued advances in structure determination, genomics, and high-throughput

screening and related (more focused) techniques, in silico drug design is play-

ing an important role as never before. Thus, traditional structure-directed library

design methods in combination with newer approaches like fragment-based drug

design [496,1447], virtual screening [453,1179], and system-scale approaches to

drug design [236,278,649] will form important areas of research.

For a historical perspective of drug discovery, see [7, 159, 335, 507, 589, 727,

772], for example, and for specialized treatments in drug design modeling consult

the texts by Leach [709] and Cohen [254].

15.1.1 Chemical Libraries

The ﬁeld of combinatorial chemistry was recognized by Science in 1997 as one

of nine “discoveries that transform our ideas about the natural world and also

offer potential beneﬁts to society”. Indeed, the systematic assembly of chemical

building blocks to form potential biologically-active compounds and their rapid

testing for bioactivity has experienced a rapid growth in both experimental and

theoretical approaches (e.g., [640,692,1241]); see the editorial overview on com-

binatorial chemistry [207] and the associated group of articles. Two combinatorial

chemistry journals were launched in 1997, with new journals since then, and a

Gordon Research conference on Combinatorial Chemistry was created. The num-

ber of new-drug candidates reaching the clinical-trial stage is greater than ever.

15.1. Introduction to Drug Design 521

Indeed, it was stated in 1999: “Recent advances in solid-phase synthesis, infor-

matics, and high-throughputscreening suggest combinatorial chemistry is coming

of age” [151].

Accelerated (automated and parallel) synthesis techniques combined with

screening by molecular modeling and database analysis are the tools of combi-

natorial chemists. These tools can be applied to propose candidate molecules that

resemble antibiotics, to ﬁnd novel catalysts for certain reactions, to design in-

hibitors for the HIV protease, or to construct molecular sieves for the chemical

industries based on zeolites. Thus, combinatorial technology is used to develop

not only new drugs but also new materials, such as for electronic devices. Indeed,

as electronic instruments become smaller, thin insulating materials for integrated

circuit technology are needed. For example, the design of a new thin-ﬁlm insu-

lator at Bell Labs of Lucent Technologies [333] combined an optimal mixture of

the metals zirconium (Zr), tin (Sn), and titanium (Ti) with oxygen.

As such experimental synthesis techniques are becoming cheaper and faster,

huge chemical databases are becoming available for computer-aided [159]and

structure-based [41, 453, 1179, 1447] drug design; the development of reliable

computational tools for the study of these database compounds is thus becoming

more important than ever. The term cheminformatics (chemical informatics,also

called chemoinformatics), has been coined to describe this emerging discipline

that aims at transforming such data into information, and that information into

knowledge useful for faster identiﬁcation and optimization of lead drugs.

15.1.2 Early Drug Development Work

Before the 1970s, proposals for new drug candidates came mostly from labora-

tory syntheses or extractions from Nature. A notable example of the latter is Carl

Djerassi’s use of locally grown yams near his laboratory in Mexico City to syn-

thesize cortisone; a year later, this led to his creation of the ﬁrst steroid effective

as a birth control pill [323]. Synthetic technology has certainly risen, but natu-

ral products have been and remain vital as pharmaceuticals (see [666, 1006]and

Box 15.1 for a historical perspective).

A pioneer in the systematic development of therapeutic substances is James W.

Black, who won the Nobel Prize in Physiology or Medicine in 1988 for his re-

search on drugs beginning in 1964, including histamine H

-receptor antagonists.

Black’s team at Smith Kline & French in England synthesized and tested sys-

tematically compounds to block histamine, a natural component produced in the

stomach that stimulates secretion of gastric juices. Their work led to development

of a classic ‘rationally-designed’ drug in 1972 known as Tagamet (cimetidine).

This drug effectively inhibits gastric-acid production and has revolutionized the

treatment of peptic ulcers.

Later, the term rational drug design was introduced as our understanding of

biochemical processes increased, as computer technology improved, and as the

ﬁeld of molecular modeling gained wider acceptance. ‘Rational drug design’

refers to the systematic study of correlations between compound composition and

its bioactive properties.

522 15. Similarity and Diversity in Chemical Design

Box 15.1: Natural Pharmaceuticals

Though burdened by political, environmental, and economic issues, pharmaceutical in-

dustries have long explored unusual venues for disease remedies, many in remote parts of

the world and involving indigenous cures. Micro-organisms and fungi, in particular, are

globally available and can be reproduced readily. For example, among the world’s 25 top-

selling drugs in 1997, seven were derived from natural sources. Some notable examples of

products derived from Nature are listed below.

• A fungus found on a Japanese golf course is being used by Merck to make the

cholesterol lowering drug mevacor, one of the 25 top-sellers of 1997.

• A fungus found on a Norwegian mountain is the basis for another 1997 top-seller,

the transplant drug Cyclosporin, made by Novartis.

• A fungus from a Paciﬁc yew tree is also the source of the anticancer agent paclitaxel

(taxol).

• The rosy periwinkle of Madagascar is the source of Eli Lilly’s two cancer drugs

vincristine and vinblastine, which have helped ﬁght testicular cancer and childhood

leukemia since the 1960s.

• A microbe discovered in a Yellowstone hot spring is the source of a heat-resistant

enzyme now key in DNA ampliﬁcation processes.

• Ocean salmon is a source for osteoporosis drugs (Calcimar and Miacalcin), and

coral extracts are used for bone replacement.

• The versatile polymer chitosan, extracted from crab and shrimp shells, is a well

known fat-binding weight-loss aid, in addition to its usage in paper additives, pool

cleaners, cosmetics, and hair gels.

• The Artemisiam annua plant (also known as sweet wormwood), which grows in

China, Vietnam, and some parts of the United States, provides the raw material for

a malaria drug, artemisinin.

• Frog-skin secretions serve as models for development of painkillers with fewer side

effects than morphine. This chemical secret, long exploited by Amazon rain forest

tribesmen, is now being pursued with frogs from Ecuador by Abbott Labs.

• Marine organisms from the Philippines are being investigated as sources of

chemicals toxic to cancer cells.

• The venomous lizard termed Gila monster inhabiting Phoenix, Arizona, may pro-

vide a powerful peptide, exenden, for treating diabetes, because it stimulates insulin

secretion and aids digestion in lizards that gorge thrice-yearly.

• A compound isolated from a ﬂowering plant in a Malaysian rainforest, calanolide A,

is a promising drug candidate for AIDS therapy, in the class of non-nucleoside

reverse transcriptase inhibitors.

• A protein from a West African berry was identiﬁed by University of Wisconsin

scientists as 2000 times sweeter than sugar; sweeteners are being developed from

this source to make possible sweeter food products by gene insertion.