Suppressing interaction between bonded particles

Abstract

A method for managing flow of particles into an array of pairwise-point-interaction-module includes receiving a first set of particles into a first queue. The first set is a proper subset of a second set of particles that comprises all particles that are to be passed into an array of pairwise-point-interaction-modules during a current time period. Prior to having received all particles from the second set, particles from the first set are allowed to pass from the first queue into the array.

Claims

1. A method comprising causing a simulation machine for molecular-dynamic simulation to manage flow of particles into an array of pairwise-point-interaction-modules, wherein said simulation machine comprises nodes connected to each other by a network, said nodes collectively representing a volume with each node corresponding to a particular portion of said simulation space, said simulation machine further comprising a first queue and an array of pairwise-point-interaction modules, wherein causing said simulation machine for molecular-dynamic simulation to manage flow of said particles into said array comprises receiving a first set of particles into said first queue, said first set of particles being a proper subset of a second set of particles, wherein said second set of particles comprises all particles that are to be passed into said array of pairwise-point-interaction-modules during a current time period, and prior to having received all particles from said second set, allowing said particles from said first set to pass from said first queue into said array.

2. The method of claim 1, further comprising continuing to load particles from said first set into said array as additional particles from said second set are received into said first queue.

3. The method of claim 1, further comprising receiving a third set of particles into a second queue, wherein said third set of particles comprises all particles that are to only be loaded into said array during said current time period, and wherein allowing said particles from said first set to pass from said first queue into said array occurs only after all particles from said third set have been loaded into said array.

4. The method of claim 1, further comprising receiving a third set of particles into second and third queues, wherein said third set of particles comprises all particles that are to be loaded into said array during said current time period, and wherein allowing said particles from said first set to pass from said first queue into said array occurs only after all particles from said third set have been loaded into said array.

5. The method of claim 1, further comprising receiving a fourth set of particles into a third queue, wherein said fourth set of particles is to be both loaded and streamed into said array, wherein streaming of particles from said fourth set commences only after completion of loading of particles from said fourth set.

6. The method of claim 1, wherein said first queue is a logical queue, wherein said method further comprises selecting said first and second set of particles from a plurality of selected physical queues.

7. A non-transitory and tangible computer-readable medium having encoded thereon software that, when executed by a molecular dynamic simulation machine, causes execution of the method of claim 1 by said molecular dynamic simulation machine, wherein said molecular dynamic simulation machine comprises nodes connected to each other by a network, said nodes collectively representing a volume with each node corresponding to a particular portion of said simulation space, a first queue, and an array of pairwise-point-interaction modules.

8. An apparatus comprising a molecular dynamics simulator comprising nodes, an array, and a first queue, wherein said nodes are connected to each other by a network and collectively represent a volume with each node corresponding to a particular portion of said simulation space, wherein said array is an array of pairwise-point-interaction modules, wherein said simulation machine is configured to execute the method of claim 1.

9. The apparatus of claim 8, wherein each of said nodes comprises an application-specific integrated circuit that comprises flex tiles comprising geometry cores and a common memory that is available to all of said geometry cores.

10. The apparatus of claim 9, wherein said flex tiles further comprise a network interface for enabling communication with other components of said node.

11. The apparatus of claim 9, wherein said flex tiles further comprise a dispatch unit to provide hardware support for fine-grained event-driven computation.

12. The apparatus of claim 9, wherein said application-specific integrated circuit further comprises a logic analyzer that captures and stores activity of said node.

13. The apparatus of claim 9, wherein said application-specific integrated circuit further comprises a host interface that provides communication with an external host.

14. The apparatus of claim 9, wherein said application-specific integrated circuit further comprises interaction tiles that receive particles and grid points from said flex tiles via an on-chip mesh network and that enqueue said particles into queues that are stored in a local memory.

15. The apparatus of claim 8, further comprising a particle director, a plurality of queues, and an interaction tile, wherein said first queue being one of said queues in said plurality of queues, wherein said particle director places particles arriving at said interaction tile into different queues.

16. The apparatus of claim 8, wherein said first queue is a first-in-first-out queue.

17. The apparatus of claim 8, wherein said first queue is one of a plurality of queues, each of which is programmed to know how many particles are to arrive during said current time-period.

18. The apparatus of claim 8, wherein said first queue is one of a plurality of queues, each of which is programmed to know how many particles that are expected to arrive during said current time-period have arrived.

19. A method for simulating interactions between pairs of particles using nodes, each of which comprises a module set that comprises one or more pairwise-point-interaction-modules, said method comprising: at each node, carrying out iterations, each of which corresponds to a corresponding time interval from a sequence of time intervals, each of said iterations comprising: at a first node, beginning to receive information about particles in a set of particles, said set of particles consisting of first, second, and third parts that are disjoint from each other, as particles from said first part are received, storing said particles in a first queue, as particles from said second part are received, storing said particles in a second queue, as particles from said third part are received, storing said particles in a third queue, and, before having received all of said information, beginning a simulation of interactions between particles in pairs of particles from said first set, said particles in said pairs of particles consisting of only those that have already been received and excluding those that have yet to be received, wherein said interactions include interactions between particles in said first and second parts and interactions between particles in said second and third parts and exclude interactions between particles in said first and second parts and wherein said simulation includes loading information from all particles in said first and third parts into a module set of said first node, streaming information from particles in at least one of said second and third parts through said module set after having loaded and prior to having received information for all particles in said second part, and evaluating said interactions in said module set.

20. An apparatus comprising a molecular-dynamics simulator for simulating interactions between pairs of particles, wherein said simulator comprises nodes, each of which comprises a plurality of queues and a module set, wherein said module set comprises one or more pairwise-point-interaction-modules, wherein said simulator is configured to carry out iterations, each of which corresponds to a corresponding time interval from a sequence of time intervals, each of said iterations comprising: at a first node, beginning to receive information about particles in a set of particles, said set of particles consisting of first, second, and third parts that are disjoint from each other, as particles from said first part are received, storing said particles in a first queue from said plurality of queues, as particles from said second part are received, storing said particles in a second queue from said plurality of queues, as particles from said third part are received, storing said particles in a third queue from said plurality of queues, and, before having received all of said information, beginning a simulation of interactions between particles in pairs of particles from said first set, said particles in said pairs of particles consisting of only those that have already been received and excluding those that have yet to be received, wherein said interactions include interactions between particles in said first and second parts and interactions between particles in said second and third parts and exclude interactions between particles in said first and second parts and wherein said simulation includes loading information from all particles in said first and third parts into a module set of said first node, streaming information from particles in at least one of said second and third parts through said module set after having loaded and prior to having received information for all particles in said second part, and evaluating said interactions in said module set.

Description

BRIEF DESCRIPTION OF THE FIGURES

(1) FIG. 1 shows some of the nodes in an exemplary molecular dynamics simulation machine, and the architecture of a typical node;

(2) FIG. 1A shows details of a node in FIG. 1;

(3) FIG. 2 shows the architecture of a flex tile from FIG. 1;

(4) FIG. 3 shows the architecture of an interaction tile from FIG. 1;

(5) FIG. 4 shows a structural formula for a molecule;

(6) FIG. 5 shows topological identifiers for a molecule;

(7) FIG. 6 shows how a cyclic graph for a molecule is modified to accommodate tagging of its atoms by topological identifiers;

(8) FIG. 7 shows queues in the high-throughput interaction-subsystem tile of FIG. 3;

(9) FIG. 8 shows a table for use in piecewise approximation of a function; and

(10) FIG. 9 shows a smaller table with accompanying logic to carry out substantially the same piecewise approximation as the table of FIG. 8.

DETAILED DESCRIPTION

(11) Molecular dynamics simulation involves simulating the motion of particles in response to forces. Because many of these forces are short-range forces, most computations involving a particle are restricted to interactions with nearby particles. Thus, computations involving a neighborhood of particles can often be carried out largely independently of computations involving other neighborhoods of particles. This property lends itself to parallel processing.

(12) To take advantage of this inherent parallelism, a simulation machine 10 for molecular dynamic simulation, as shown in FIG. 1, features a plurality of nodes 12 that are connected by a network. The nodes collectively represent a volume of a simulation space, with each node corresponding to a particular portion of that simulation space. The topological relationship between nodes in the network corresponds to their spatial relationship in the simulation space. This simulation volume is occupied by particles that interact with each other in ways that are to be calculated by the simulation machine 10. In one particular embodiment, the simulation volume is toroidal. However, other volumes are possible, such as prisms.

(13) Because of the inherent parallelism, it is useful to divide the simulation volume into node boxes, each of which is handled by one of the nodes 12. A description of the manner in which calculations are allocated among different nodes can be found in Shaw, “ORTHOGONAL METHOD,” U.S. Pat. No. 7,707,016, issued Apr. 7, 2010, the contents of which are herein incorporated by reference.

(14) Referring to FIG. 1A, each node 12 is implemented as a combination of first hardware elements that are especially designed to perform pairwise interactions and second hardware elements whose function is to provide potentially interacting particles to the first hardware elements. In one embodiment, each node 12 is implemented as an application specific integrated circuit having sixteen flex tiles 14, two interaction tiles 16, a host interface 18, and a logic analyzer 20.

(15) The host interface 18 provides communication with an external host via a PCI link. The logic analyzer 20 is used primarily to capture and store node activity for debugging. Each node 12 also includes communication interfaces 22 for data transmission between neighboring nodes in each of the three local coordinate directions. Within a node 12, data transmission between the components of the node 12 is carried out by an on-chip mesh network.

(16) As shown in FIG. 2, each flex tile 14 has geometry cores 24, each of which has an associated memory 26. In the particular embodiment shown, there are four such geometry cores 24. However, in general other numbers of geometry cores can be used. Each flex tile 14 also includes a network interface 28 for enabling communication with other components of the node 12, and a dispatch unit 30 to provide hardware support for fine-grained event-driven computation. A dispatch unit 30 is described in detail in Grossman, “EVENT-DRIVEN COMPUTATION,” U.S. Patent Publ. 2014-0282576, filed on Sep. 18, 2014, the contents of which are herein incorporated by reference. Also included in a flex tile is a common memory 32 that is available to all four geometry cores 24.

(17) As shown in FIG. 3, each interaction tile 16 features an interaction controller 34, an abbreviated flex tile 44, and a pairwise-point-interaction-module (“PPIM”) array 52. PPIM array 52 is made up of PPIMs 54. Pairwise interaction between particles occurs in the PPIM array 52. In the illustrated embodiment, the PPIM array 52 has thirty-eight PPIMs 54. However, in general, other numbers of PPIMs can be used.

(18) The interaction tile 16, and specifically the interaction controller 34, receives particles and grid points from the flex tiles 14 via the on-chip mesh network. It then enqueues these particles and grid points into queues 36 that are stored in local memory 38. The operation of the interaction controller 34 is controlled by instructions 40 received from a geometry core 42 in the abbreviated flex tile 44.

(19) In addition to a geometry core 42, the abbreviated flex tile 44 has a local memory 46, a dispatch unit 48, and a network interface 50, all of which serve functions similar to those described in connection with FIG. 2.

(20) The simulation machine 10 simulates the evolution of a collection of particles by repeatedly calculating and integrating all inter-particle forces in small time steps. At the beginning of each time step, each flex tile 14 uses the on-chip network to send packets to interaction tiles 16 on the same node and on other nodes. These packets contain information about particles that interact with each other in ways that the interaction tile 16 will ultimately reveal through calculation.

(21) As noted above, each node 12 is responsible for computations concerning particles within its node box. Due to resource limitations, it may be necessary to further divide each node box into sub-boxes. Each sub-box has some particle population. This particle population fluctuates over time as a result of particles moving within the simulation volume in response to forces exerted by other particles. Each interaction tile 16 will receive, from the flex tiles 14, some variable number of particles from multiple sub-boxes. Each interaction tile 16 also receives, from the flex tiles 14, count packets, each of which reports how many particles to expect from each sub-box. These count packets are used by the interaction tile 16 to determine when it has received all particles from all sub-boxes.

(22) To accurately simulate the motion of particles, it has been found necessary to evaluate particle interactions at time intervals that are very close together. In a typical simulation, particle interactions are evaluated every few femtoseconds of simulated time.

(23) Many interesting events occur on timescales of milliseconds or longer, involve hundreds of thousands of particles, or both. Examples of events of this type include those that arise in biochemical systems in which biological macromolecules interact.

(24) Simulation of systems in which events unfold on such long time scales takes a great deal of real time. This is because the exchange rate between simulation time and real time is presently on the order of a billion to one. Thus, it is necessary to spend microseconds in order to compute the interactions required to advance the simulated time by femtoseconds. While this may seem fast, to place matters in perspective, this means that in order to simulate just one millisecond of real time at this exchange rate, it is necessary for the simulation machine 10 to work for one million seconds, which is a little over eleven days of continuous computation.

(25) The following discussion refers to particles being stored in or streaming through certain hardware. It should be understood that a “particle” means information representing a particle, or a simulated particle, and not actual particles.

(26) Particles arrive at an interaction tile 16 in no particular order. As they arrive, a particle director 74 places them into different first-in-first-out (FIFO)) queues 76, as shown in FIG. 7. In one embodiment, there are 2.sup.7 such queues 76. In one implementation, the particle director 74 is implemented as a programmable look-up table.

(27) Since particles arrive in no particular order, they are also placed in queues 76 in no particular order. However, the order in which queues 76 will be used is known in advance. Additionally, each queue is programmed in advance to know how many sub-boxes worth of particles it is expecting. Because of the count packets being received from the flex tiles 14, each queue can also determine how many particles will arrive for each sub-box. As a result, each queue 76 knows how many particles to expect, and whether or not they have all arrived.

(28) The queues 76 are divided into first queues 78, second queues 80, and third queues 82. Particles that are to be loaded into the PPIM array 52 are placed in a first queue 78. Particles that are to be streamed past the loaded particles are placed in a second queue 80. Particles that are to be both loaded and streamed are placed in a third queue 82.

(29) For example, particles that are only within a first volume of space are placed in the first queue 78, from which they are loaded into the PPIM array 52. Particles that are in only a second volume of space are placed in the second queue 80, from which they are streamed into the PPIM array 52. And, particles that are in the intersection of the first and second volumes of space are placed in the third queue 82, from which they are both loaded and streamed into the PPIM array 52.

(30) In one embodiment, the interaction controller 34 waits until the first and third queues 78, 82 have been filled. As noted above, this information is available because the flex tiles 14 have been sending count packets along with the particles. The interaction controller 34 then loads all the particles from the first and third queues 78, 82 into the PPIMs 54 in a round-robin fashion. Once the particles have been loaded, the PPIM array 52 is ready to accept the streaming particles from the second queue 80 and again from the third queue 82. Control over whether a queue is to begin allowing its particles into the PPIM array 52 or whether it should wait is carried out by software instructions.

(31) Upon detecting that loading is complete, the interaction controller 34 determines whether the second queue 80 is full. The third queue 82 is of course known to be full by this point since loading into the PPIM array 52 would not have completed otherwise. If the second queue 80 is full, the interaction controller 34 unleashes the particles from the second queue 80 and the third queue 82. These particles then begin to stream through all the PPIMs 54 in the PPIM array 52. The PPIMs 54 then begin the process of calculating interactions between the loaded particles from the first and third queues 78, 82 and the streaming particles from the second and third queues 80, 82.

(32) In an alternative method for managing flow of particles into the PPIM array 52, the interaction controller 34 does not bother to wait until the first and third queues 78 have filled. Instead, the interaction controller 34 loads particles from the first and third queues 78, 82 into the PPIMs 54 as soon as it can. This saves time since the PPIM array 52 no longer has to wait until all particles have arrived before it begins to load particles.

(33) Once the particles are all loaded into the PPIM array 52, the interaction controller 34 begins streaming particles into the PPIM array 52 from the second queue 80 and the third queue 82. This occurs even though the second queue 80 may not yet have received all particles designated for streaming.

(34) In an alternative practice, it is useful to regard the queues shown in FIG. 7 as logical queues that are assembled by selectively choosing queues from a subset of the set of all physical queues. The particles that are placed in these logical queues are obtained by selecting a subset of the set of physical queues. The choice of which subset to select depends on the neighbors of whatever sub-box in question. This allows the flexibility to re-use physical queues on an as-needed basis.

(35) In the course of being streamed through the PPIMs 54, each streaming particle from the second and third queues 80, 82 encounters each stored particle from the first and third queues 78, 82. At each such encounter, a determination must be made concerning whether or not an interaction should be computed. If the answer is “yes,” the force that results from interaction between the stored particle and the streaming particle is calculated. These forces are accumulated as they are calculated. Once the streaming particle has traversed the PPIM array 52, the total force acting on the particle is sent to an output 56.

(36) A test must therefore be devised to answer the question, “Should an interaction between these two particles be computed?”.

(37) One test for deciding whether or not an interaction should be computed is to ask whether or not the distance between particles is close enough to make computation worthwhile. If two particles are too far apart, no interaction will be calculated.

(38) However, although this test is a good approximation, it is complicated by the fact that sometimes interactions between particles should not be calculated even if the particles are close together. This complication arises in molecular dynamics because atoms can be covalently bonded together to form molecules. In that case, the forces that hold these particles in a bond easily dwarf the inter-particle forces that are being simulated. In known simulation machines, these interactions are still calculated, but are later removed in a correction pipeline.

(39) To remedy this deficiency, and to thereby eliminate the need for a correction pipeline, each particle is associated with a topological identifier that communicates the nature of a topological relationship between that particle and other nearby particles. Without loss of generality, this topological identifier will be discussed in connection with atoms that bond together with other atoms to form molecules, and in particular, biological macromolecules such as proteins and lipids.

(40) Referring to FIG. 4, a molecule 58 can be represented as a graph in which nodes represent atoms 60 and edges represent bonds 62 between atoms 60. The number of edges separating the two atoms 60 in the molecule 58 can be used to define a topological distance between atoms 60. This topological distance can then be used as a basis for suppressing the computation of an interaction between two atoms. The result will be that for two atoms that are next to each other in a molecule, no interaction will be computed. But if two atoms are on opposite ends of a macromolecule, an interaction may be computed.

(41) Referring now to FIG. 5, the topological identifier 64 is a short descriptor that is unique to an atom 60. This uniqueness extends across different molecules.

(42) The topological identifier 64 encodes topological relationships between atoms 60 in a molecule 58. Thus, by comparing the topological identifiers 64 of two atoms 60, it is possible to define a topological distance between the two atoms 60. This enables determination, with high accuracy, of whether an interaction between those atoms 60 should be excluded even if those atoms are otherwise close enough so that interaction would normally be calculated.

(43) Comparing topological identifiers 64 therefore avoids the vast majority of corrections that would normally have been carried out in a correction pipeline, and also avoids wasteful computation. As will be discussed below, there are some special cases where computations will be carried out even if they should not be. However, there are so few of these cases that correction of the calculation can be done by software instead of by having a separate hardware correction pipeline.

(44) The implementation of topological identifiers 64 described herein relies on the fact that many molecules 58 feature a backbone of atoms with primary side-chains branching off the backbone. These primary side-chains can have secondary side-chains. These secondary side-chains can have tertiary side-chains and so an ad infinitum. However, it has been found that most molecules 58 of interest have a backbone with primary side-chains branching off the backbone, and secondary side-chains branching off the primary side-chains. Thus, a practical implementation requires that only primary and secondary side-chains be accounted for.

(45) In the embodiment described herein, each atom 60 is assigned an integer quartet. The members of the quartet identify a backbone, primary and secondary side-chains, and a termination flag. More generally, the topological identifier is an integer tuple with N+2 elements, where N is the number of levels of side chains to be accounted for. In the present embodiment, N=2 because only a primary and secondary side-chain are to be accounted for.

(46) For any atom 60, the first element of the quartet is a backbone identifier that identifies that atom's associated backbone atom. Where an atom 60 is itself the backbone atom, for purposes of assigning the first element, that atom 60 is considered to be associated with itself. All atoms of side-chains that ultimately connect to the same backbone atom would have the same backbone identifier.

(47) The second element of the quartet is a primary side-chain identifier that identifies the primary side-chain associated with the atom 60. The primary side-chain has an atom 60 that is bonded directly to the backbone. All atoms that are in the same side-chain, or are in side-chains connected to that same side-chain would have the same primary side-chain identifier.

(48) The third element of the quartet is a secondary side-chain identifier that defines the second level side-chain associated with the atom.

(49) Finally, the fourth element is a terminal flag that identifies whether or not the atom is a terminal atom. A terminal atom is one that is bound to the rest of the molecule by only one covalent bond. As used herein, the term “covalent bond” is independent of the number of electronic orbitals participating in the bond, and therefore includes double bonds and triple bonds.

(50) It should be apparent that the above scheme is recursive in nature and can be extended to any number of side-chains by simply adding suitable elements between the terminal flag and the backbone identifier.

(51) In the illustrated embodiment, the topological identifier is an integer quartet (n, m, k, t). Backbone atoms are identified as (n, 0, 0, 0). Atoms in a primary side-chain off the nth backbone atom are identified as (n, m, 0, 0) where m is an integer greater than or equal to 1 that represents the distance along the chain between that atom and the backbone atom. Atoms in a secondary side-chain are identified as (n, m, k, 0) where k is an integer greater than or equal to 1 that represents the distance between that atom and the atom at which the secondary side-chain intersects the primary side-chain, (n, m, 0, 0). A terminal atom, which only has a single neighbor, has its terminal flag set to 1. Thus, a terminal atom that has, as its neighbor, atom (n, m, k, 0) will have as its topological identifier 64 the integer quartet (n, m, k, 1).

(52) FIG. 5 shows a graph corresponding to portion of a macromolecule with atoms having the relevant topological identifiers as assigned by the foregoing rules. The illustrated graph shows five backbone atoms numbered 4 through 8. Consistent with the foregoing rules, these are numbered (n, 0, 0, 0) for n=4 to 8.

(53) The atom attached to backbone atom 4 is a terminal atom because it has only one neighbor. Consistent with the rules, its quartet is the same as its neighbor's quartet, i.e. backbone atom 8's quartet, except its terminal flag is set to 1. Terminal atoms can also be found attached to backbone atoms 6 and 7 with corresponding quartets built according to the same rule.

(54) The assignment of topological identifiers to side-chains can be seen by inspecting the identifiers of atoms that are in the side-chain off of backbone atom 5. As shown, each quartet for all atoms that ultimately connect to backbone atom 5 will have n=5. All atoms in the same primary side-chain have the same value of m, while all atoms in the same secondary side-chain have the same value of k.

(55) The topological distance between two atoms having integer quartets (n.sub.1, m.sub.1, k.sub.1, t.sub.1) and (n.sub.2, m.sub.2, k.sub.2, t.sub.2) resolves into three cases.

(56) In the first case, the atoms are attached to different backbone atoms and therefore have different backbone identifiers. This means that n.sub.1≠n.sub.2. In such a case, the distance is obtained by taking the magnitude of the difference between the backbone identifiers and adding it to the sum of all the remaining elements of the two integer quartets:
|n.sub.1−n.sub.2|+m.sub.1+m.sub.2+k.sub.1+k.sub.2+t.sub.1+t.sub.2

(57) In the second case, the two backbone identifiers are the same, but the two atoms are on different primary side-chains. Thus, n.sub.1=n.sub.2 but m.sub.1≠m.sub.2. In that case, the distance is identical to that for the first case, but instead of adding together the primary side-chain identifiers, one evaluates the magnitude of their difference:
|n.sub.1−n.sub.2|+|m.sub.1−m.sub.2|+k.sub.1+k.sub.2+t.sub.1+t.sub.2

(58) In the third case, the two atoms are on the said primary side-chain but they are on different secondary side-chains. This means that n.sub.1=n.sub.2 and m.sub.1=m.sub.2, but k.sub.1≠k.sub.2. In that case, the distance is computed the same way as the second case, but instead of adding together the secondary side-chain identifiers, one evaluates the magnitude of their difference:
|n.sub.1−n.sub.2|+m.sub.1−m.sub.2|+|k.sub.1−k.sub.2|+t.sub.1+t.sub.2

(59) The foregoing method for assigning a topological identifier 64 assumes that the graph of a molecule 58 is an acyclic graph with at most one side-chain emanating from any atom. While this is true for most molecules of interest, there are exceptions. In such cases, a small number of edges are removed from the molecules graph until this condition is met.

(60) FIG. 6 shows an example of a molecule 68 having a cyclic graph. The aforementioned method for assigning topological identifiers does not work when the molecule's graph is cyclic or contains cyclic portions. To accommodate this case, it is necessary to remove two edges 70, 72 from the graph to open up the cyclic portions, thus yielding a corresponding acyclic molecule 74. A correction is later carried out in software to address this irregularity.

(61) An atom's topological identifiers and its position location are bundled with its position as it makes its way through the PPIM array 52. As a result, the encoding must be as compact as possible.

(62) As a practical matter, in molecules of interest, most side-chains are short. For most lipids and proteins, three bits is sufficient to encode the primary side-chain identifier, and one bit is enough to encode the secondary side-chain identifier. One more bit is needed to encode the terminal flag. Thus, the remaining bits can be used to encode the backbone identifier.

(63) In some cases, a chemical system is too large for all backbone identifiers to be encoded because there are not enough bits allocated to carry out the encoding. In other cases, side-chains cannot be encoded in the bits available. Both of these cases, like the case in which the molecule has rings, must also be corrected in software.

(64) During the course of evaluating quantities, such as inter-particle forces, it is often necessary to carry out computations that involve evaluating a function of an argument. Evaluating a function, particularly a transcendental function, is a time-consuming task. To speed up this task, it is known to simply look up the value of the function for a particular argument in a look-up table. However, a table that provides low approximation error would be prohibitively large. Another approach is to divide the domain of the function into parts and to approximate the desired function using a parametric form, such as a cubic form. In such cases, a table provides a mapping from a domain region to coefficients of a cubic polynomial. A non-uniform partition of the domain can be used to provide a finer partition in those parts of the domain in which the cubic polynomial does not match the function well, for example, in those parts of the domain in which the function to be approximated is changing fast. Conversely, coarser partitions can be used where the match is good.

(65) In some cases, the function's value changes so rapidly for certain regions of its domain that even using the cubic polynomial would require that the look-up table of cubic coefficients have prohibitively many entries in order to adequately model the function in those portions of its domain.

(66) FIG. 8 illustrates a look-up table 84 for a function that is piecewise approximated by a cubic polynomial. To obtain the value of the function u(x.sub.o) for a particular value of x.sub.o, one looks up the entry 86 corresponding to that portion of the domain that includes x.sub.o. This entry 86 yields the coefficients of the cubic polynomial, which can then be evaluated using suitable combinatorial logic 88 to yield a value of the function u(x.sub.o).

(67) Naturally, since the approximation is a piecewise one, the coefficients of the polynomial change throughout the approximated function's domain. In fact, this is precisely why there have to be multiple entries in the table 84. As is apparent from FIG. 8, for certain regions of its domain, the approximated function changes so rapidly that the entries in the table are only valid for very small such regions. Of course, one can easily solve this problem by having more table entries, each of which covers a smaller part of the function's domain. But it is desirable to avoid this because such tables consume large amounts of die-space.

(68) In an alternative embodiment, shown in FIG. 9, the table 84 has fewer entries, even for those parts of the approximated function's domain in which the approximated function changes rapidly. To obtain an approximated value of the function u(x.sub.o) for a particular value of x.sub.o, one looks up the entry 86 corresponding to that portion of the domain that includes x.sub.o. This entry 86 yields the coefficients of a cubic polynomial. These coefficients can then be used by suitable combinatorial logic 88 to yield a first approximation of the value of the function at the argument x.sub.o, namely ũ(x.sub.o). To improve this first approximation, the argument x.sub.o is also provided to a look-up table 89 that outputs one or more parameters based on the argument. The one or more parameters are then provided to growth/decay logic 90. The growth/decay logic 90 implements functions that are easily executed in logic and that can yield a second approximation. The output of the growth/delay logic 90 is a value of a function ƒ(x.sub.o). This value and the first approximation ũ(x.sub.o) are provided to a multiplier 92, the output of which is the second approximation u(x.sub.o)=ũ(x.sub.o) ƒ(x.sub.o). In a preferred embodiment, ƒ(x.sub.o) is x.sub.o.sup.k.Math.exp(−x.sub.o.sup.m), where k and m are the parameters obtained from the look-up table 89 for a particular value of x.sub.o.

(69) It is known to use the PPIM array 52 in connection with charge spreading and force interpolation using GSE methods described in Shan et al., “GRID BASED COMPUTATION FOR MULTIPLE BODY SIMULATION,” U.S. Pat. No. 7,526,415, issued Apr. 28, 2009, the contents of which are herein incorporated by reference. The mathematical basis for the methods applied in Shan are set forth in Shan, et al., “GAUSSIAN SPLIT EWALD: A FAST EWALD MESH METHOD FOR MOLECULAR SIMULATION, J. Chem. Phy. 122 054101 (2005), the contents of which are herein incorporated by reference. Details on how particles and grid locations are distributed in the PPIM array 52 are described in Shaw et al., “PARALLEL COMPUTER ARCHITECTURE FOR COMPUTATION OF PARTICLE INTERACTIONS,” U.S. Patent Publ. 2013/0091341, published Apr. 11, 2013, the contents of which are herein incorporated by reference.

(70) In an improvement of the method described therein, in which all grid locations are passed into the PPIM array 52, an alternative method exploits the fact that grid locations are not randomly located but actually have some spatial regularity. By exploiting this regularity, it becomes possible to pass selected locations into the array and derive the grid locations from those selected locations.

(71) For example, let X be a set of m grid locations (x.sub.1, x.sub.2 . . . x.sub.m) where each x.sub.i is a position vector having a dimensionality that is appropriate to the simulation space. According to the prior art method, to pass the points grid locations into X, one would pass all m points into the PPIM array 52.

(72) In an improved method, there exists a set Y of n locations (y.sub.1, y.sub.2 . . . y.sub.n) where n<m. There also exists a rule R such that X=R(Y). Thus, rather than pass X into the PPIM array 52 one only as to pass Y and implement the rule R in the PPIM array 52 to reconstruct X. This amounts to a compression technique, with the extent of compression being dependent on the ratio of m to n.

(73) For example, if two vectors in the set X were (x, y, z.sub.1) and (x, y, z.sub.2), then one could simply pass a set Y that included the vector (x, y, (z.sub.1+z.sub.2)/2). Then, if one knew the grid spacing, one could derive the original two vectors from the set X. Alternatively, the set Y could just equal every other point from set X, in which case one could reconstruct the original set X by adding the appropriate grid spacing to the appropriate coordinates in the vectors in Y.

(74) In one method, the set Y of locations is loaded into the PPIM array 52 and atoms are streamed through the PPIM array 52. This procedure is used to carry out charge spreading. In another practice, the roles are reversed, and atoms are loaded into the PPIM array 52 while the set Y of locations is streamed through the PPIM array 52. This procedure is used to carry out force interpolation.

(75) In one embodiment of the simulation machine 10, the particles that arrive at the interaction tile 16 for processing are always associated with a current time step. However, an alternative embodiment introduces a phase bit associated with data packets carrying particle data into the interaction tile 16. The phase bit's value is associated with a particular time step. This provides a way to distinguish between particles in two different time steps. As a result, it is possible for the interaction tile 16 to receive data packets associated with two time steps.

(76) In an embodiment that accommodates a phase bit, the interaction controller 34 maintains queues 36 associated with each value of phase bit. Upon receiving a particle, the phase bit of the particle is inspected and the particle is placed in a queue that is appropriate to the phase bit. The phase bit thus permits the interaction tile 16 to receive data associated with different time steps, and thereby eliminates the need to synchronize the interaction tiles 16 and the flex tiles 14. In operation, only queues corresponding to the current time step are loaded into the PPIM 54.

(77) The invention is described in further detail in the attached appendix, the content of which is hereby incorporated by reference in its entirety.