An Algorithmic Study of Manufacturing Paperclips and Other Folded Structures

We study algorithmic aspects of bending wires and sheet metal into a specified structure. Problems of this type are closely related to the question of deciding whether a simple non-self-intersecting wire structure (a carpenter's ruler) can be straightened, a problem that was open for several years and has only recently been solved in the affirmative. If we impose some of the constraints that are imposed by the manufacturing process, we obtain quite different results. In particular, we study the variant of the carpenter's ruler problem in which there is a restriction that only one joint can be modified at a time. For a linkage that does not self-intersect or self-touch, the recent results of Connelly et al. and Streinu imply that it can always be straightened, modifying one joint at a time. However, we show that for a linkage with even a single vertex degeneracy, it becomes NP-hard to decide if it can be straightened while altering only one joint at a time. If we add the restriction that each joint can be altered at most once, we show that the problem is NP-complete even without vertex degeneracies. In the special case, arising in wire forming manufacturing, that each joint can be altered at most once, and must be done sequentially from one or both ends of the linkage, we give an efficient algorithm to determine if a linkage can be straightened.Comment: 28 pages, 14 figures, Latex, to appear in Computational Geometry - Theory and Application

Fundamental Limits to Position Determination by Concentration Gradients

Position determination in biological systems is often achieved through protein concentration gradients. Measuring the local concentration of such a protein with a spatially-varying distribution allows the measurement of position within the system. In order for these systems to work effectively, position determination must be robust to noise. Here, we calculate fundamental limits to the precision of position determination by concentration gradients due to unavoidable biochemical noise perturbing the gradients. We focus on gradient proteins with first order reaction kinetics. Systems of this type have been experimentally characterised in both developmental and cell biology settings. For a single gradient we show that, through time-averaging, great precision can potentially be achieved even with very low protein copy numbers. As a second example, we investigate the ability of a system with oppositely directed gradients to find its centre. With this mechanism, positional precision close to the centre improves more slowly with increasing averaging time, and so longer averaging times or higher copy numbers are required for high precision. For both single and double gradients, we demonstrate the existence of optimal length scales for the gradients, where precision is maximized, as well as analyzing how precision depends on the size of the concentration measuring apparatus. Our results provide fundamental constraints on the positional precision supplied by concentration gradients in various contexts, including both in developmental biology and also within a single cell.Comment: 24 pages, 2 figure

Towards synthetic biological approaches to resource utilization on space missions.

This paper demonstrates the significant utility of deploying non-traditional biological techniques to harness available volatiles and waste resources on manned missions to explore the Moon and Mars. Compared with anticipated non-biological approaches, it is determined that for 916 day Martian missions: 205 days of high-quality methane and oxygen Mars bioproduction with Methanobacterium thermoautotrophicum can reduce the mass of a Martian fuel-manufacture plant by 56%; 496 days of biomass generation with Arthrospira platensis and Arthrospira maxima on Mars can decrease the shipped wet-food mixed-menu mass for a Mars stay and a one-way voyage by 38%; 202 days of Mars polyhydroxybutyrate synthesis with Cupriavidus necator can lower the shipped mass to three-dimensional print a 120 m(3) six-person habitat by 85% and a few days of acetaminophen production with engineered Synechocystis sp. PCC 6803 can completely replenish expired or irradiated stocks of the pharmaceutical, thereby providing independence from unmanned resupply spacecraft that take up to 210 days to arrive. Analogous outcomes are included for lunar missions. Because of the benign assumptions involved, the results provide a glimpse of the intriguing potential of 'space synthetic biology', and help focus related efforts for immediate, near-term impact

A Standard Parts List for Biological Circuitry

One of the hallmarks of biochemical circuits found in nature is analog, asymmetric, asynchronous design. That is, there is little standardization of parts, e.g. all the promoters have different strengths and kinetics, transcription factors are designed to have different effects at different loci, and each enzymatic reaction has its own idiosyncratic mechanism and rates. In addition, all of the heterogeneous circuit elements are executing their functions concurrently and asynchronously. Biological circuits are seemingly designed to deal with the fluctuating delays, different time-scales and energy requirements associated with each component process of the overall network. These factors also make design of novel biochemical circuitry from existent parts difficult to achieve. Without standardization, the qualitative design methods used in other engineering fields are simply inapplicable. The de facto design methodology for biological circuitry is natural selection. Rational design of biological systems by humans has remained restricted to rather small or hit-or-miss efforts and has often relied on the ability to "select" for biochemical parts that fulfill some criteria. In practice however biological-designers are rare, and solutions are usually realized through an expensive stepwise trial and error approach or through mutation and selection. Furthermore, these otherwise practical approaches are limited in terms of the problems they can solve. We believe that implementation of designed biological circuitry is limited by issues of practice

A "partitioned leaping" approach for multiscale modeling of chemical reaction dynamics

We present a novel multiscale simulation approach for modeling stochasticity in chemical reaction networks. The approach seamlessly integrates exact-stochastic and "leaping" methodologies into a single "partitioned leaping" algorithmic framework. The technique correctly accounts for stochastic noise at significantly reduced computational cost, requires the definition of only three model-independent parameters and is particularly well-suited for simulating systems containing widely disparate species populations. We present the theoretical foundations of partitioned leaping, discuss various options for its practical implementation and demonstrate the utility of the method via illustrative examples.Comment: v4: 12 pages, 5 figures, final accepted version. Error found and fixed in Appendi

A mass conserved reaction-diffusion system captures properties of cell polarity

Various molecules exclusively accumulate at the front or back of migrating eukaryotic cells in response to a shallow gradient of extracellular signals. Directional sensing and signal amplification highlight the essential properties in the migrating cells, known as cell polarity. In addition to these, such properties of cell polarity involve unique determination of migrating direction (uniqueness of axis) and localized gradient sensing at the front edge (localization of sensitivity), both of which may be required for smooth migration. Here we provide the mass conservation system based on the reaction-diffusion system with two components, where the mass of the two components is always conserved. Using two models belonging to this mass conservation system, we demonstrate through both numerical simulation and analytical approximations that the spatial pattern with a single peak (uniqueness of axis) can be generally observed and that the existent peak senses a gradient of parameters at the peak position, which guides the movement of the peak. We extended this system with multiple components, and we developed a multiple-component model in which cross-talk between members of the Rho family of small GTPases is involved. This model also exhibits the essential properties of the two models with two components. Thus, the mass conservation system shows properties similar to those of cell polarity, such as uniqueness of axis and localization of sensitivity, in addition to directional sensing and signal amplification.Comment: PDF onl

Stochastic Gene Expression in a Lentiviral Positive Feedback Loop: HIV-1 Tat Fluctuations Drive Phenotypic Diversity

Stochastic gene expression has been implicated in a variety of cellular processes, including cell differentiation and disease. In this issue of Cell, Weinberger et al. (2005) take an integrated computational-experimental approach to study the Tat transactivation feedback loop in HIV-1 and show that fluctuations in a key regulator, Tat, can result in a phenotypic bifurcation. This phenomenon is observed in an isogenic population where individual cells display two distinct expression states corresponding to latent and productive infection by HIV-1. These findings demonstrate the importance of stochastic gene expression in molecular "decision-making."Comment: Supplemental data available as q-bio.MN/060800

Algorithms for Rapidly Dispersing Robot Swarms in Unknown Environments

We develop and analyze algorithms for dispersing a swarm of primitive robots in an unknown environment, R. The primary objective is to minimize the makespan, that is, the time to fill the entire region. An environment is composed of pixels that form a connected subset of the integer grid. There is at most one robot per pixel and robots move horizontally or vertically at unit speed. Robots enter R by means of k>=1 door pixels Robots are primitive finite automata, only having local communication, local sensors, and a constant-sized memory. We first give algorithms for the single-door case (i.e., k=1), analyzing the algorithms both theoretically and experimentally. We prove that our algorithms have optimal makespan 2A-1, where A is the area of R. We next give an algorithm for the multi-door case (k>1), based on a wall-following version of the leader-follower strategy. We prove that our strategy is O(log(k+1))-competitive, and that this bound is tight for our strategy and other related strategies.Comment: 17 pages, 4 figures, Latex, to appear in Workshop on Algorithmic Foundations of Robotics, 200

Fast, cheap and somewhat in control

Efforts to manipulate living organisms have raised the question of whether engineering principles of hierarchy, abstraction and design can be applied to biological systems. Here, we consider the practical challenges to controlling living organisms that must be surmounted, or at least managed, if synthetic biology and cellular bioengineering are to be productive