Cellular automaton simulation of pulsed field gel electrophoresis

Electrophoresis 1993, 14,337-343 Cellular automaton simulation of pulsed field gel electrophoresis 337

Mark A. Smith' Yaneer Bar-Yam2

'MIT Laboratory for Computer Science, Cambridge, MA 'Electrical, Computer and Systems Engineering, Boston University, Boston MA

1 Introduction

Cellular automaton simulation of pulsed field gel electrophoresis

We describe simulation techniques well suited to detailed investigation of the microscopic behavior of DNA during electrophoretic separation in the diffusive regime. Long polymers moving diffusively in a medium are simulated using microscopic Monte-Carlo steps. Simulations rely upon a recently introduced two-space abstract polymer that enables fine-grained massively parallel simulation. Tests of the two-space polymer dynamics are reviewed. The scaling with polymer length of the size and relaxation time of isolated polymers are shown to agree with univer- sal scaling relations. The relaxation time is found to be significantly faster than the alternative bond-fluctuation method. Simplicity of implementation enables simulation on cellular automaton machines (CAM) including CAM-6, and a prototype of the new CAM-8, as well as other massively parallel architectures. Preliminary simulations of polymers migrating under an external field through a random medium of obstacles in two dimensions are described. Two sequences of simulations are performed, with different obstacle densities corresponding to pore sizes larger and smaller than the polymer radius of gyration. In the dilute medium polymers are characteristically draped on single obstacles. In the denser medium draping across multiple obstacles results in reduced orientation in the field direction. A demonstration of rapid 90" field direction switching results in polymer motion to- ward the expected intermediate direction.

The separation of DNA chains of varying length using gel- electrophoresis has been greatly enhanced in recent years by the utilization of pulsed fields [ 1,2]. Of special note is the importance of this technique to the Human Genome Proj- ect. Prior to this development, separation was only effective on relatively short chains since longer chains become oriented and exhibit velocities that are independent of chain length. Simulations of the motion of polymers in media have demonstrated the processes of entanglement and disentanglement which lead to the effectiveness of pulsed fields. Extensive large-scale simulations which can probe the effect of realistic models of the gel medium and investigate the diffusion of polymers under variable fields would enhance our understanding and may potentially influence the design of improved gels and field induced separation techniques.

In order to improve our ability to simulate .the complex behavior ofpolymers, we introduced a new approach which is well suited to parallel processing [3,4]. In its most general form, the approach may be used to simulate macromolecules ranging from simple polymers to complex polymers such as proteins. While polymer interactions are nonlocal along the chain, by recognizing that interactions are local in space, it is possible to develop a general domain-decompo- sition approach to parallel processing of polymer dynamics.

The concept of space-oriented dynamics is manifest in the very general category of dynamical models known as cellular automata. We developed two general classes of cellular automaton dynamics which can simulate abstract models

Correspondence: ProfessorY. Bar-Yam, ECS, 44 Cummington St.,Boston University, Boston, MA 02215, USA

Abbreviations: CAM, cellular automaton machine; PFG, pulsed field gel electrophoresis

of high molecular weight polymers, grafted polymers, and block-copolymers in a variety of media including bounda- ries, obstacles and constrictions as well as complex models of gels.The algorithms are easy to implement both on parallel and serial machines. The second class of algorithms, which incorporate a novel two-space concept, are particu- larly efficient for simulation because the long-polymer limit is reached for very few monomers and the polymer relaxation time is short when compared with other models. Thus, we hope to overcome fundamental obstacles to large- scale simulation of polymers.

In this paper we briefly describe the two-space algorithm in Section 2; scaling tests for the radius of gyration and relaxation time of isolated polyers are reviewed in Section 3; a comparison with the bond-fluctuation lattice model is given in Section 4. The central part of this paper presents simulations of polymers diffusing through random media under the influence of external fields as described in Sec- tion 5. The purpose of these preliminary simulations is a demonstration of the capability of describing key aspects of pulsed field gel electrophoresis. Section 6 briefly describes the new cellular automaton machine (CAM-8) architecture, a fine-grained massively parallel architecture expected to dramatically further these and other polymer simulations. A few conclusions are presented in Section 7.

2 Two-space algorithm

In the two-space algorithm [ 1,2] monomers of the polymer alternate between two spaces so that odd-numbered monomers are in one space and even-numbered monomers are in the other. The nearest neighbors along the contour of a monomer are located in the other space. Consider the motion of a particular monomer. The two-space construction enables the constraint of maintaining connectivity to be imposed by considering only the positions of monomers in the opposite space. Moreover, the excluded volume con-

0 VCH Vcrlagsgesellschaft mbH, D-6940 Weinheirn, 1993 0173-0835/93/0404-0337 $5.00+.25/0

338 M. A. Smith and Y. Bar-Yam Electrophoresis 1993, 14,337-343

two algorithms on a serial c0mputcr.A comparison of relaxation times reveals that the tito-spacc algorithm is more flexible, reducing the relaxation time for equal lcngih poly- mcrs by a factor of three (Fig.3). Furthermore, the intrinsic monomer update speed o f the bond-fluctuation method is also slower by a factor o f 3. In part. this arises from the ne- cessitv of randonilv selecting a monomcr.The inherent Dar-

Figure I . Schematic illustration of a two-space polymer used in two- space polymer dynamics described in the text. Monomers on the upper plane are shownas filled circles,monomers on the lowerplane are shown as open circles. Monomers are attached (bonded) only to monomers in the other plane. The ‘bonding-neighborhood’ of each monomer is a 3 X 3 region of cells located in the opposite plane. A lightly yhaded region indicates the bonding-neighborhood of the black monomer marked with a white dot. Its two nearest neighbors are located in this bonding-neighborhood. Bonds are indicated by line segments between monomers.

straint is also imposed only through interactions of a monomer with monomers in the opposite space. Even though excluded volume is not explicitly imposed between two monomers in the same space, it arises indirectly through the interactions with the opposite space. Despite the unusual implementation of connectivity and excluded volume, the static and dynamic scaling properties of long chains are pre- served.

The advantages of this algorithm include flexible dynamics (fast relaxation times) as well as effective parallelization. The dynamics is flexible because nearest neighbors which are in opposite spaces can be ‘on top of each other’ so that local expansion and contraction is possible. More interes- tingly, it is possible to move all of the monomers in one space in parallel because both connectivity and excluded volume are implemented through interactions with the other space. The possibility of updating half of the monomers in parallel opens a wide range of parallelization schemes.

3 Scaling tests

The two-space polymer dynamics is an abstract polymer model with an unusual implementation of excluded volume. The unusual local interaction does not affect the asymptotic structural and dynamical behavior, which is in the same universality class as other abstract polymer models. This was confirmed by simulating the scaling of known quantities. In Fig. 2 we plot the radius of gyration of a polymer as a function of its contour length and compare with the exact result in two dimensions, R-LO”. In Fig. 3 we cal- culate the longest relaxation time of the chain and compared with the Rouse model prediction, z-Lz ’.

allelizability of the two-space algorithm enables the monomers to be updated systematically in a space, only the space must be selected at random. The combination of relaxation time and update speed leads to an overall speed up of a factor of 10 of the two-space algorithm from the bond- fluctuation algorithm for polymers of equal numbers of monomers.

5 Electrophoresis demonstration

The polymer dynamics described above can be modified to give the polymers a net drift velocity.This can be done most simply by assigning unequal probabilities of performing steps in different directions. Alternatively, one can, at regu- lar intervals, choose a definite direction of movement (say, to the right) for all the monomers. The bias in motion which is generated by either method corresponds to the effect of an electric field. In physical systems, in the diffusive regime, an electric field causes such a drift on top of the random Brownian motion. This modification of the two-space algorithm can be used to model pulsed field gel electrophoresis (PFG). The gel itself may be implemented directly in the initial conditions of the space as discussed below.

A version of the polymer rule was implemented on a special-purpose computer known as a CAM-6 which was developed at the MIT Laboratory for Computer Science. This machine has a 256 X 256 lattice of cells which is updated at 60 Hz in such a way that it can be displayed on a television screen to give a real-time animation of the dynamics. This architecture is programmed by a rule that specifies a se- quence of steps detailing how each cell is updated in paral-

100

10

Rg

1

4 Comparison with bond-fluctuation method 0.1 ! I 1 10 100 lo00

L One of the simplest algorithms for effective simulation of abstract polymers is the bond-fluctuation method 151 which

ymer motion on a single lattice. The bond-fluctuation

F1gure2. Simulations of the radius of gyration of a polymer X,, as a function of the polymer length L (one less than the number of monomers), testing the two-space algorithm. An asymptotic fit indicated by the dashed line is R,-0.8 LO7” consistent with the exact exponent 0.75. Agreement with the asymptotic values are reached for rcmarkably small polymers of length L = 2.

bond lengths to vary so as to more

method is not especially well-suited fOrparakliZatiOn. It is nevertheless of interest to compare the efficiency of the

E/eclrophoresis 1993, 14, 337-343 Cellular automaton siinularioii of pulsed field gel electrophoresis 339

lel with all othercells. Forthe sake of eficiencyon this particular device, the choice of space and the directions of the moves are made in a deterministic cycle, and whetherto ac- tually accept a legal move is made randomly rather than the other way around. Such a Monte Carlo scheme will also sample the configuration space uniformly though the detailed behavior of the dynamics on short time scales is different. Note that because of the deterministic choice of direction in this CAM-6 implementation, the underlying rule satisfies ‘semi-detailed balance’ rather than detailed balance. This may influence some of the observed dynamical phenomena. Future studies of PFG will make use of slightly modified rules satisfying detailed balance such as one recently implemented on CAM-8 (see below) and other parallel architectures [6].

The CAM-6 polymer rule was subsequently modified to add a systematic drift. This was done by eliminating the possibility of moving left once out of every four times that such a choice would normally by possible. In the actual implementation a full cycle contains sixteen steps. Half of the steps correspond to attempts to move left or right. For one quarter of these, only moves to the right are accepted. Hence, on two out of the sixteen steps (one for each space), the direction of any move is to the right. Thus each and every monomer is biased to move to the right four times for every three times it moves to the 1eft.The simplest model of a gel matrix is a fixed set of obstacles. In two dimensions these obstacles can be thought of as polymer strands in a three-dimensional space that intersect the two-dimensional plane. In the CAM-6 implementation described above, such immovable obstructions can be made out of monomers arranged in a block, and therefore no further modifications to the dynamic rule are necessary. It turns out that a 2 X 2 block of even monomers superimposed on a similar block of odd monomers (eight monomers total) forms a tightly bound block that cannot move. Moreover, there is a one-cell excluded volume around each block, so that no other monomers can enter this 4 X 4 volume. The 4

0.1 4 1 10 100

L Figure 3. Simulations of the relaxation Lime r of a polymer as a function of polymer length L. testing the two-space algorithm (A). Simulations were performed on an RISC station. The asymptotic fit indicated by the dashed line was obtained as T-0.12 LZs’ consistent with the Rouse model exponent of 2.5. ?he small prefactor indicates the efficiency of the two-space algorithm. Polymers of length 30-50 already approach the asymptotic behavior. For comparison the bond-fluctuation algorithm is shown (A) with a relaxation time 3 times larger. The single step speed is also 3 times slower (see text).

X 4 excluded volume of each block reduccs the pore size ac- cordingly. The biased dynamics together with an initial con- dition containing both polymers and blocks forms the PFG model discussed in this paper.

The initialization of the system proceeds in three stages. First, a given number (100 in the low density case and 1000 in the high density case, Fig. 4 and Fig. 5, respectively) of 2 X 2 blocks comprising the matrix are placed at random, one at a time. Placements within a prespecified distance from any other block are rejected and another placement is found for the block. This causes an anticlustering of the blocks which makes them more evenly distributed than they would be without the distance constraint. For ease of visualization, block placement near the edges of the space were also rejected; however, the simulation treats the space as periodic. Second, the polymers are randomly placed in a similar fashion to the blocks by using a straight line, vertical polymer configuration. In this case the polymers must be kept at least one cell away from the blocks and other polymers. Finally, the polymers are allowed to relax into an equilibrium configuration using a fast, nonlocal Monte Carlo dynamics. This so-called ‘reptation dynamics’ moves monomers from either end of a polymer to the other end, subject to the constraints of the model. The algorithm pro- duces an equilibrium ensemble and is used here only to es- tablish the starting configuration for dynamic simulations.

We performed trial simulations on CAM-4 of the dynamics, and the results are illustrated for two gel densities in Figs. 4 and 5. In both cases, the space is a 256 X 256 lattice with periodic boundary conditions, and the system is initialized with 128 polymers having 30 monomers each. The density of polymers is thus artificially high in order to see many kinds of interactions in a fairly small area. The average radius of gyration of these polymers, when isolated, is 5.6 in lattice units, resulting in a characteristic diameter of 11.2. Each cycle of sixteen steps updating the whole 256 X 256 lattice takes 0.8 seconds to run on CAM-6. Simulations on CAM-8 have been timed to run roughly an order of magni- tude faster.

Figure 4 shows what happens when the average distance between blocks is about the same as the relaxed length of the polymers, For these low density gel simulations there are 100 blocks in the whole space with a minimum allowed separation of 16 (in lattice units), giving a typical separation of 24 and a pore size of 20 (reduced by the obstacle excluded volume). This is larger than the characteristic polymer diameter. In Fig. 4a the polymers have equilibrated, and their shapes are largely unaffected by the matrix. The effects of the periodic boundary conditions are apparent as some of the polymers can be seen wrapping around to the opposite side of the space. Figure 4b shows the configuration of polymers after 100 cycles of the dynamics (1600 steps of the polymer rule). The polymers begin to feel the effect of the matrix, though many are still just drifting. After 200 cycles (Fig. 4c) of the dynamics (3200 steps of the polymer rule), most of the polymers are draped over the obstacles. A few polymers have succeeded in freeing themselves from ini- tially encountered obstacles and continue to drift. A few polymers have piled up to form larger aggregates. At the high density of polymers we have elected to simulate for this demonstration, after 10 000 cycles (Fig. 4d) of the dy-

340 M. A. Smith and Y. Bar-Yam Electrophoresis 1993, 14, 337-343

Figure 4. Demonstration of gel electrophoresis simulations using the two-space lattice polymer dynamics algorithm wilh a drift. Each of the 128 irregular segments represents a polymer chain containing 30 monomers. The 2 X 2 squares are immovable obstacles which represent a gel matrix. A drift in the motion of polymers is imposed to represent a field applied to the right. The characteristic diameter of a polymer is smaller than the gel pore size. (a) Initial state for the simulations with the polymers in equilibrium. (b) A short time later (100 polymer update cycles). The polymers have drifted a distance approximatelyequal to the mean obstacle separation, and they have begun to run into the obstacles. (c) After 200 cycles the polymers are typically draped across the obstacles. The freeing of a polymerby diffusion results in continued migration. (d) After 10 000 cycles,an approximate steady stale is achieved with significant polymer clusters at this extremely high polymer density. (e) A demonstration of alternating field directions involving fields down and to the right switched five times (see Section 5) . Beginning from the configuration of (b) the direction of the field was switched after every 20 cycles. This figure may be compared with (c) which was simulated for the same amount of time.

Electrophoresis 1993, 14. 337-343 Cellular automaton simulation of pulsed field gel electrophoresis 341

Figure 5. Demonstration o f gel electrophoresis simulations using the two-space lattice polymer dynamics algorithm with a drift. This simulation repeats the simulation of Fig. 4 but with a higher-densitygel consist- ing ofatotal of 1000 blocks.Forthis density the characteristic diameter of a polymer is larger than the gel pore size. (a) Initial state for the simulations with the polymers in equilibrium. (b) After 100 polymer update cycles, most of the polymers are entangled in the gel matrix. Note that polymers are draped across more than one obstacle. (c) After 1340 cycles. (d) After 100 000 cycles, the polymers have become matted into a few regions. (e) A demonstration of alternating field directions involving fields down and to the right switched five times (see Section 5). Beginning from the configuration of (b) the direction of the field was switched after every 20 cycles.

342 M. A. Smith and Y. Bar-Yam Elrctruphurerir 1993, 14, 337-343

namics (160 000 steps), a characteristic configuration is reached where the strands preferentially pile up to form semicrystalline logjams on individual pegs. It is important to recognize that this configuration could only be achieved by significant numbers of polymers successfully freeing themselves from barriers around which they were tempora- rily draped. Longer runs reveal that the polymers can continue to slide off, but that they are less likely to do so for larger piles. In Fig. 4e we show an initial demonstration of the effect of alternating field directions. Resuming from the configuration of Fig. 4b we alternated bias direction between right and down (900 between field directions). A total of 5 field direction changes were made. In each field direction 20 consecutive cycles (complete polymer updates) were performed for a total of 100 additional cycles. At this high frequency of field switching the primary effect is to cause polymer drift at a 45" intermediate angle. Comparing Fig. 4e to Fig. 4c one may suggest some changes in the effect of the obstacles due to the alternation offield direction. Detailed studies must be performed to investigate this further. The total elapsed time of the simulation on CAM-6 to Fig. 4e is 160s. Further studies will be performed on CAM-8.

In Fig. 5 we show the results of simulations for the case of polymers larger than the typical distance between obstructions. The same length and number of polymers was used as in Fig. 4; however, a larger number of obstacles were placed, resulting in a higher density gel. In this instance, there are 1000 blocks in the space with a minimum allowed separation of 5, giving a typical separation of 8 (for a typical pore size of 4). This is smaller than the characteristic polymer diameter. The minimum separation barely allows polymers to fit between blocks situated horizontally or verti- cally, though under some circumstances there may not be enough room between diagonally situated blocks. Figure 5a shows an initial configuration for this dense gel. After 100 cycles (1600 steps),virtuallyall ofthe polymers have encountered an obstruction (Fig. 5b). At this gel density, polymers encounter obstacles much more frequently. Indeed, the characteristic of polymers draped over obstacles changes. Each polymer is draped over more than one obstacle and therefore polymers are not fully aligned with the field direction. This modified draping may enable the polymers to diffuse off the barriers more easily so that the mobility should not scale simply with obstacle density. Similar to the case ofFig. 4 the extended simulation of polymer motion results in polymer clusters. Fig. 5c shows the configuration reached after 1340 cycles (21 500 steps). Some cluster- ing is visible. After 100 000 cycles (Fig. 5d) of the rule (1.6 million steps), the polymers entangle with the matrix to form extended mats.The simulation resulting in Fig. 5d ran for 22.22 h on CAM-6. Fig. 5e shows the effect of alternating field directions in a manner similar to the simulation of Fig. 4e. The higher gel density simulation began from the configuration of Fig. 5b and was treated with the same pro- cedure as 4e. As in Fig. 4e, the primary effect is the change of direction of polymer motion to a 45" intermediate angle. The effect of multiple rather than single barrier draping con- tinues to play an important role in the dynamics.

The extension of these simulations to investigate a wide variety of key questions in PFG is presently being considered. The tracing of individual polymer motion may be easily per-

formed to investigate mobility and the time evolution of other polymer properties. Moreover, many other modifications to the basic drift rule are also possible to model vari- ous experimental conditions or to increase the physical fi- delity-such as through more accurate gel models. A potential problem with this rule as it stands (and in lattice models in general) is that tension is not transmitted along the chain as readily as it would be in a real polymer. For long enough polymers the diffusive regime where tension is not relevant is expected to be valid. However, the relevant regime for PFG is not well known. Furthermore, sliding motion may be unphysically limited. In the case of the high density polymer simulation described above this may be seen in the semi-crystalline pileups. It is hard to accurately assess the relevance of local interactions to the physical conditions where inter-monomer interactions and structure might also be expected to play a role. The dominant dependence of DNA fragment mobility on mass rather than composi- tion suggests that the local interactions should not play an essential role. However, if tension and sliding motion do become relevant to the polymer dynamics, a continuum version of this algorithm, in which both are included, may be easily constructed. A continuum implementation of the two-space algorithm would retain the inherent parallelism.

6 The new CAM-8

An important development for large-scale PFG simulations (as well as other cellular dynamical models) is the recent construction of CAM-8 modules [7]. This indefinitely extensible modular architecture enables both two- and three-dimensional space simulations. It is well suited for both visual observation of the simulations through direct video output and also well suited for measuring the avera- ges of physically meaningful quantities. As an attachment to a SPARC station host it is a versatile and powerful machine for simulating spatially extended physical systems. The two-space polymer rule has recently been implemented on a prototype machine.

7 Conclusions

The simplicity of the two-space polymer algorithm enables direct adaptation for simulation of pulsed-field-gel-electrophoresis. Preliminary simulations suggest the potential for further detailed investigations of microscopic quantities. Direct visual real-time representation and observation of simulations enables development of an intuitive understanding of central microscopic quantities that affect the macroscopic phenomena. The preliminary simulations were used to illustrate the difference between gels with pore sizes larger and smaller than the polymer radius of gyration. Results showed significant differences in polymer- gel interactions due to simultaneous interaction of a single polymer with multiple obstacles in the smaller pore size ex- ample.

Received February 27, 1993

Electrophoresis 1993, 14, 337-343

8 References

[l] Schwartz, D. C. and Cantor, C. R., Cell 1984, 37, 67-75. [2] Smith, C. L., Current Opinion BiotechnoL 1991,2, 86-91. [3 Bar-Yam, Y., Rabin, Y. and Smith, M. A., Macromolecules Reprints

[4 Smith,M. A,, Bar-Yam,Y.,Rabin,Y., Bennett, C. H.,Margolus,N. and Toffoli, T. in: Sirota, E. B., Weitz, D., Witten, T. and Isrealachvili, J. (Eds.), Complex Fluids, Materials Research Society Symposium Pro- ceedings, Pittsburgh 1992 Vol. 248 p. 483; J . Conzputational Polymer Sci. 1992, 2, 165-171.

1992,25,2985-2986.

Cellular automaton simulation of pulsed field gel electrophoresis 343

[S] Carmesin, I. and Kremer, K., Macromolecules 1988,21,2819-2823. [6] Ostrovsky, B., Smith, M. A., Biafore, M., Bar-Yam,Y., Rabin,Y., Mar-

golus,N. and Toffoli,T.,in: Sincovec,R.F.,Keyes,D. E.,Leuze,M.R., Petzold, L. R. and Reed, D. A. (Eds.), PTOCeedingS of the sixth SIAM Conference on Parallel Processing for Scientrfic Computing, Society for Industrial and Applied Mathematics, Philadelphia 1992, Vol. I, pp.

[7] Margolus,N. and Toffoli,T., in: Doolen, G. D., Frisch, E., Hasslacher, B.,Orzag,N.and Wolfram, S. (Eds.), Lattice Gas Methods ofPartialD$- .ferential Equations, Addison-Wesley Longman Publishing Group, New York 1989, pp. 219-249.

193-202.

Documents

Cellular automaton simulation of pulsed field gel electrophoresis