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In fact, this large variability of parameters makes the choice of biological agents for network computing difficult, as many other, less studied parameters, e. The size of an SSP network is determined by i its unit cell size, designed for a specific computing agent; ii the cardinality of the problem, i.

The SSP unit cell size is determined by the geometrical parameters of the computing agents, e. The SSP cardinality determines the number of computing agents required to solve the problem, including some additional number to offset possible errors. Consequently, for a given compactness of the series, the size and the number of computing agents needed determine the area of the SSP computing system. In principle, a larger combinatorial problem requires, by necessity, a larger number of computing agents. However, network-based computing, as described before for SSP [ 19 , 63 ] presents specific advantages, and disadvantages, regarding its scalability when compared to other massively parallel bio-computing approaches, e.

DNA computing [ 9 ]. In contrast, in network-based computation of SSP, the exploration of the 2 C computation paths is distributed in time and space, by recycling of agents. Consequently, network-based SSP calculation will use considerably less mass of agents, but at the expense of a much larger computation time. Presently, network-based computing of SSP assumes [ 20 ] that the agents do not perform any function other than visiting junctions, and thus calculating various paths in the SSP-encoding network. In principle, as discussed further, the agents could perform additional functions, e.

However, this higher technological complexity of the agents, while valuable in accelerating the overall calculation, will not decrease the number of agents required to solve the problem, which is determined by the SSP cardinality.

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The SSP specifications ii and iii mentioned above determine together the total sum of the set. The compactness of the series also determines the type of complexity of the SSP network. Subset sum complexity classes I and II explained in terms of split and join junctions. In Complexity Class I there is only one possible route to every legal exit, and consequently, there are only split and pass junctions active. The series in the set is strongly expanding with the cardinality.

For this case, the exponential series is shown, displayed in two forms: i with descending numbers binary tree and ii with ascending numbers and crossing traffic lines at pass junctions, but still with the same number of routes and exits in compliance with the commutative property of addition.

Conversely, in Complexity Class II there are exits that can be reached through multiple routes and, hence, there are also join junctions active. The series in the Complexity Class II sets can be very compact.

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For instance, the most compact series possible is Pascal's triangle. Tellingly, the set for Pascal's triangle has cardinality 7, compared to cardinality 3 for the binary tree, but occupies the same area. The fundamental difference between the two complexity classes, i. In the combinatorial run mode, the traffic density is falling orders of magnitude for all series, and the bottleneck risk of traffic jam is located at the starting point of the network.

Conversely, in the multiplication run mode, beyond a threshold cardinality value, the traffic density is rising again, orders of magnitude for most series, resulting in a traffic jam further down the network. Note that only the exponential series would show constant traffic density in case of the multiplication run mode.

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Solving SSP by means of network computing requires that the sequence of coordinates each and every agent passes through, or, at the very least, the sequence of the junctions it passes by, is fully recorded. Consequently, the tracks of all the agents should be captured, preferably, in one optical field-of-view FoV , and at a resolution allowing the identification of individual agents.

Alternatively, if the overall computing area is too large to be visualized in one FoV, the optical recording needs to visit several sectors covering the overall movement, but at a frequency high enough to avoid confusion regarding the positioning or identity of the agents. Three traffic scenarios should be considered, discussed in order of decreasing tracking complexity:. Note that channel widths and heights of less than two times the agent widths would prevent overtaking, but the risk of clogging is too large, therefore larger channel widths and heights e.

Obviously, the first scenario cannot be tracked error free, as optical tracking is performed in the x — y plane only; if one agent crawls over others, temporarily obscuring them, the tracking information becomes unreliable afterwards. The second scenario would need a pixel size smaller than the agent width and the agent length in order to preserve reliable traffic information when agents pass each other e.

The horizontal black dashed lines delimit the sizes of 4-, 6- and 8-inch silicon wafers—the standards in semiconductor industry. The vertical blue bars indicate the agent width for molecular motor-driven cytoskeletal filaments, i. The triangular work windows are shown for various microscopy techniques. Because of the competition between resolution and the FoV [ 64 ], the whole imaging of the computing area requires the employment of the maximum useable pixel size MUPS that can still resolve individual agents, i.

The black crossed arrows indicate the intersection of the largest attainable FoV as a square root with the minimum attainable pixel size for various optical imaging technologies, i. To fully exploit the frame size available, the MUPS value should be as close as possible to the resolution limit. The E. For E. The third scenario is described in detail in the electronic supplementary material, SI-2, and the nomogram in electronic supplementary material, figure S1.

It follows that a network with cardinality 5 for E.

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When the area to be imaged and monitored in time exceeds the FoV of the imaging system, a powerful option to enlarge the effective FoV is image stitching of cyclic sampled frames. The loss of information can be minimized through faster switching speeds, which in turn are limited by the mechanical capabilities of the microscope stage. In the electronic supplementary material SI-3, the possibilities and limits of image stitching for our SSP calculation networks are modelled. In the case of high density traffic, agent speed and body length determine the sample frequency, and in the case of low density traffic, agent speed and junction distance are decisive.

In electronic supplementary material, table S1, it is shown that in a typical setting used to monitor E. For traffic scenario ii , at a resolution of 0. One corollary of the above analysis is that, for E. Even if that were technically feasible, the data storage needed would be very large. Moreover, from a fabrication point of view, such large chips are very vulnerable to fatal errors by dust particles in the lithographic steps. It appears that the scaling of networks, in particular for solving SSP, is the most problematic, albeit technological and not fundamental, aspect of network computing with biological agents.

Indeed, the chip area, which grows with the size of the problem, requires FoVs which are not presently available. Alternatively, to limit the explosion of the chip area with the size of the problem would require smaller agents, which in turn would require a higher resolution, but this would further raise problems for the achievable FoV.

Ultimately, a technology that allows the agents to report their own travel history at the exits would not need optical recording of the total network. In the first instance, the time to solve an SSP depends on the mode of operation of the computing agents, the extent of the series, i. More compact series will result in a smaller computing area and consequently a shorter computing time. For a given series and given cardinality, the track length is the same for all run modes.

As expected, the highest computing times are observed for the sequential run mode, and the lowest are observed for the multiplication run mode. The difference in run time between the sequential and the combinatorial run modes is small for compact series, but quite large for expanding series. The multiplication run modes for the various series are all following the same straight line because, effectively, only one agent starts and the off-spring that takes the longest track is monitored, but all are assumed to run with the same average speed.

A network with cardinality 30 would only fit on a standard wafer for the prime number and the Pascal series. Also indicated, by blue arrows, are time frames. Only the multiplication run mode would allow a cardinality 30 network to be run in a reasonable time.

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  • While electronic computers perform computations in a serial manner, they are many orders of magnitude faster per operation than it is reasonable to expect from network computing with biological agents. Consequently, the immediate scaling question is to what extent an ideal set of agent parameters, i. In order to have a comparison between the ideal performance of network computing with biological agents, and electronic computers, two sets of simulations have been performed. At this junction an important distinction must be made when comparing the performance of electronic computers with any other alternative computing devices, including the one recently proposed for solving SSP [ 20 ].

    Pisinger's [ 66 ], which can solve SSP very quickly if run by electronic computers.

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    However, the alternative computation approaches, including DNA, quantum, and networks-based computing, to name a few, propose in the first instance computing devices with associated operational procedures , rather new algorithms, which indeed might be required to be developed to capitalize on the potential benefits offered by the new computing hardware.

    Consequently, and taking into consideration the tentative or early stage of development of the new computing devices, any meaningful comparison of the computing power of electronic computers and any new paradigmatic computing device must use comparative algorithmic procedures, rather than the most advanced ones, which by virtue of decades long history of microelectronics have been solely and specifically created and optimized for sequential electronic computers.

    On this background, a computer program was designed to solve the SSP by brute force i. The program is described in detail in the electronic supplementary material, SI Intel's , , and single core Pentium and a present-day MacBook chip. As opposed to all electronic chips, which perform computation in a sequential run mode, the simulated computation by biological agents is performed in the combinatorial run mode, for cytoskeletal filaments and the chosen bacterial agents, and in the multiplication run mode for the latter, assuming multiplication rates reported in the literature; the following doubling times have been used: M.

    Comparison of the computing performance of the electronic computers bottom-right half and biological computers top-left half solving the prime numbers subset problem. Even a cursory inspection of the computing performance comparison of the electronic and network-based computers reveals several evident trends.

    Moreover, this performance gap remains constant, or increases slightly, throughout the range of cardinalities tested.

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    In contrast, and aside from other physical limitations assessed in the previous sections, e. Indeed, the speed of the faster biological agent, i. Intel's While some improvement can be achieved, in principle, using faster biological agents operating in the pure combinatorial run mode, the computing performance of electronic computers will remain unmatched for the foreseeable future. However, in itself, the scale of combinatorial or complex problems of practical importance translates into large amounts of energy used, if the computation is performed by sequential electronic computers.

    For instance, solving large complex problems, even if not necessarily combinatorial in nature, would require scaling up HPC to exascale computing, i. Consequently, and aside from the difficulty of solving large combinatorial problems, it appears that electronic computers are also unsustainable energy-wise. The most energy-efficient systems are, expectedly, molecular computers , of which the most well-known is DNA computing [ 9 ], followed by numerous variations [ 74 ].

    Energy efficiency of various computing systems. It typically does not include the parallel job launch and teardown, which is required to run for at least one minute. Consequently, no energy consumption is reported for environmental, e.