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[MATH] . Let us confirm it with fig. The top two figures share the same monitoring period [MATH] , but their maximum velocities are different. The top left diagram corresponding to [MATH] has three branches. This number equals to that of all the integral velocities, two, one and zero, as is depicted in the diagram. The... |
All the end points of the branches as well as the jamming line are on the line [MATH] . This is because the density at the end point is the maximum density [MATH] |
that allows the velocity of the slowest car to be [MATH] . The maximum density [MATH] is determined by [EQUATION] Since all the cars flow at the velocity [MATH] |
when [MATH] , the corresponding flow is given by [MATH] . Thus the relation [MATH] holds. The free line is a branch whose inclination equals to the maximal velocity [MATH] Any other metastable branch and the jamming line branch out from the free line. By observation, the density of the branch point of the branch corres... |
[EQUATION] This observation is explained as follows. Suppose one car, say the car [MATH] runs at the velocity [MATH] and the other [MATH] cars run at the maximum velocity [MATH] At the moment the [MATH] -th car slows down to [MATH] , the headway between the cars [MATH] and [MATH] is [MATH] . Since it takes at least [MA... |
expands up to [MATH] by the time the [MATH] -th car speeds up to [MATH] . If all the cars can obtain the headway [MATH] , slow cars running at the velocity [MATH] disappear in the end. Thus the density at the branch point of the branch corresponding to the velocity [MATH] is given by [MATH] Note that the density at the... |
Concluding remarks Through an extension of the ultradiscretization for the OV model Takahashi2009 we introduced the ds2s–OV ( ) and s2s–OV ( models as ultradiscretizable traffic flow models. The model is a hybrid of the OV Bando1995 |
and the s2s Takayasu1993 models whose ultradiscrete limit gives a generalization of a special case of the uOV model by Takahashi and Matsukidaira Takahashi2009 The phase transition from free to jam phases as well as the existence of multiple metastable states were observed in numerically obtained fundamental diagrams f... |
Detailed studies on the properties of the hybrid models ( ), ), ( 13 ) and ( 14 ) such as exact solutions, comparison with other traffic flow models as well as empirical data remain to be investigated. |
Acknowledgements. The authors are grateful to D. Takahashi, J. Matsukidaira, A. Tomoeda, D. Yanagisawa and R. Nishi for their valuable comments at the spring meeting of JSIAM in March, 2009. |
# Source: arxiv 0908.4426 # Title: Hot Ice Computer # Sections: all # Downloaded: 2026-03-02T08:58:17.529832+00:00 Hot Ice Computer |
Abstract We experimentally demonstrate that supersaturated solution of sodium acetate, commonly called ‘hot ice’, is a massively-parallel unconventional computer. In the hot ice computer data are represented by a spatial configuration of crystallization induction sites and physical obstacles immersed in the experimenta... |
logical gates. Keywords: crystallization, Voronoi diagram, shortest path, unconventional computing, logical gates, physics of computation |
Department of Computer Science, University of the West of England, Bristol, United Kingdom andrew.adamatzky@uwe.ac.uk Introduction |
The last decade has witnessed a rise in the number of unconventional computing devices based on principles and mechanisms of information processing in physical, chemical and biological systems |
. Most results in unconventional computing are theoretical whilst there is a handful of intriguing experimental prototypes. The experimental laboratory research focuses on implementation of computing schemes in novel materials which helps us to grasp the inner nature of physics of computation. The following prototypes,... |
Reaction-diffusion computers are implemented in a spatially extended chemical media. Data and results of computation are represented by spatial distributions of reagent concentrations. Computation is realized by propagating and interacting diffusive or excitation waves |
. Experimental laboratory prototypes of reaction-diffusion computers solve a wide range of geometrical problems, optimization and logical computation |
Physarum computing is based on adaptive foraging behaviour of plasmodium of Physarum polycephalum In Physarum computers data are represented by the distribution of attracting (food) and repelling sources (light). Computation is implemented by the plasmodium which optimizes it feeding pattern under control of attracting... |
Extended analog computers represent data as a spatial configuration of point-wise sources of current applied to a conductive medium. Computing operations are determined by physical properties of the conductive medium, (e.g. foam or gel). Results are represented by the distribution of voltages across the medium |
Other less-known prototypes include analog computing schemes using microwaves plane tessellation in gas-discharge semi-conductor systems |
, learning networks of memristors , logical circuits in liquid crystals , and glow discharge systems computing the shortest path |
Most experimental prototypes of unconventional computers either require a tailored hardware interface (analog computers, liquid crystals) and specialized equipment (memristors, gas-discharge systems), or they may have intrinsic limitations on the speed of computation (reaction-diffusion chemical processors). Physarum c... |
We aim, therefore, to provide an example of a novel computing material which is cheap to build, requires minimal resources to operate, implements computational procedures relatively quickly and is capable of solving a wide range of computationally-hard tasks. We show that sodium acetate trihydrate (colloquially called ... |
The paper is structured as follows. Experimental methods are outlined in Sect. . Section presents experimental results on computing the Voronoi diagram. In Sect. we show how to extract direction towards the site of crystallization induction. Computation of one-destination-many-sources paths by crystallization of sodium... |
Methods We prepare a supersaturated clear solution of sodium acetate trihydrate CH COONa [MATH] 3H O, pour the hot solution into Petri dishes and cool the solution down to -5 C. To induce crystallization we briefly immerse aluminum wire (powdered with fine crystals of the sodium acetate) into the solution. |
To compute shortest paths in a space with obstacles we mimic obstacles with blobs of silicon, small Petri dishes fixed to the bottom of experimental container and labyrinths made of Blu-Tack |
Dynamics of crystallization is recorded with FujiPix 6000 digital camera and still high-resolution images are produced by scanning containers with crystallized sodium acetate in HP Deskjet 5100 scanner. Magnified images of the crystalline structure formed are taken using Digital Blue QX-5 USB microscope. |
Software tools for extracting directions of crystal growth from still images and models of cellular-automaton shortest path calculation are coded in Processing |
Voronoi diagram Let [MATH] be a nonempty finite set of planar points. A planar Voronoi diagram of [MATH] is a partition of the plane into such regions, that for any element of [MATH] , a region corresponding to a unique point [MATH] contains all those points of the plane which are closer to [MATH] than to any other nod... |
[MATH] assigned to point [MATH] is called a Voronoi cell of the point [MATH] . The boundary of the Voronoi cell of a point [MATH] is built of segments of bisectors separating pairs of geographically closest points of the given planar set [MATH] . A union of all boundaries of the Voronoi cells determines the planar Voro... |
[MATH] A basic concept of parallel approximation of Voronoi diagram by wave patterns propagating in a spatially extended medium is based on time-to-distance transformation. To compute a bisector separating two given points [MATH] and [MATH] we initiate wave-patterns in the medium’s sites geographically corresponding to... |
Precipitating reaction-diffusion chemical media are proved to be an ideal computing substrate for approximation of the planar Voronoi diagram |
. A Voronoi diagram can be approximated in a two-reagent medium. One reagent [MATH] is saturated in the substrate, drops of another reagent [MATH] are applied to the sites corresponding to planar points to be separated by bisectors. The reagent [MATH] diffuses in the substrate and reacts with reagent [MATH] . Colored p... |
. Thus uncolored loci of the reaction-diffusion medium represent bisectors of the computed diagram. Most reaction-diffusion chemical processors are quite slow in computing a Voronoi diagram at the macro-scale, due to speed limitations caused by diffusion. Crystallization patterns in sodium acetate propagate much quicke... |
We represented data points of set [MATH] by a configuration of pins (pieces of aluminium wire) fixed through the lid of a 9 cm Petri dish (Fig. a). The pins were powdered with fine crystals of sodium acetate. |
To start computation we place the lid on the dish with the supersaturated solution. The pins become immersed into the solution. They induce crystallization. Patterns of crystallization propagate – as classical target waves – from the sites of crystallization induction. |
A crystallization wave stops when it encounters another wave of crystallization (Fig. bc). Boundaries between stationary patterns of crystallization represent edges of planar Voronoi diagram [MATH] . The experimental boundaries perfectly correspond to bisectors of the diagram computed by the classical Fortune’s sweepli... |
(Fig. d). Detecting directions towards sites of crystallization induction Crystals growing from from nucleation sites bear distinctive elongated shapes expanding towards their proximal ends (Fig. a). Not only a crystal’s overall shape but also the orientation of saw-tooth edges indicate the direction of the crystal’s g... |
The direction of crystal growth can be detected by conventional image processing techniques, e.g. edge detection procedures (Fig. ), or by a complementary method of detecting directional uniformity of image domains as discussed below. |
We detect local directions of crystal grows in the following way. A fine grid of nodes is applied onto given image and for every node [MATH] of the grid we calculate a set [MATH] of vectors of length [MATH] |
originating at [MATH] and directed at angles [MATH] . For each vector [MATH] we calculate a sum of standard deviations [MATH] of RGB colors of the image pixels coinciding with [MATH] . A vector [MATH] with minimal sum of color deviations indicates a local direction of crystallization propagation. A configuration of loc... |
The vector configuration represents sinks determined by points from set [MATH] . If we place a mobile agent at any site but bisectors of the vector configuration Fig. d the agent will be attracted to the closest data point [MATH] . This leads us to another problem solvable with sodium acetate — computation of a collisi... |
Computation of a collision-free path Previously we have designed a cellular-automaton algorithm for constructing one-destination-many-sources shortest path. The automaton is a regular array of finite state machines (cells) which takes discrete states and update their states simultaneously and in discrete time depending... |
To store the computed path we supply every cell with a pointer. Pointers in all cells are set to nil initially. When a cell becomes excited its pointer orients towards the direction from where the excitation wave came. An example is shown in Fig. a. |
Obstacles are imitated by always-resting, or non-excitable, cells. Examples of the pointer configuration and shortest collision-free paths are shown in Fig. b. A cell assigned to be a destination (in the north-west corner of the array Fig. b) is excited. Waves of excitation propagate from the site of initial stimulatio... |
As we demonstrated in Sect. longitudinal crystals growing in sodium acetate are physical analogs of pointers indicating from where — locally – the wave of crystallization came from. That is crystallization of sodium acetate is the physical implementation of the original cellular-automaton algorithm |
for computation of shortest collision-free path. Let us consider experimental realization. We represent obstacles by impenetrable barriers and induce crystallization at a single site of the medium. For demonstration purposes we have tested various types of objects to represent obstacles, including 35 mm Petri dish glue... |
From the images (Fig. ) we calculate a configuration of local vectors (Fig. ). A vector at each point indicates the direction from where the wave of crystallization came from. The vectors represent an attracting field, which ‘pulls’ virtual objects, placed in the field, towards the destination (site of initial disturba... |
To check whether vector configurations extracted from scanned images of crystallized sodium acetate provide sufficient representation of many-sources-one-destination shortest paths we employed pixelbots |
. A pixelbot is a virtual pixel-sized automaton. The pixelbot moves along the direction of nearest non-zero vector. The pixelbot imitates a tiny (2-5 mm) in diameter robot which travels along sodium acetate crystals or in the longitudinal grooves between the crystals. If there are no non-zero vectors in the pixelbot’s ... |
Examples of test-run trajectories of pixelbots traveling in the vector configurations are superimposed on scanned images of crystallized sodium acetate in Fig. . Pixelbots easily reach the destination in the case of simple obstacles (Fig. ab). They spent more time wandering at random in arenas with complex obstacles (F... |
Implementation of and and or logical gates We implement logical gates in a geometrically constrained, T-shaped, containers with sodium acetate. Shoulders of a T-shaped container represent inputs [MATH] and [MATH] and vertical stem — output [MATH] . When we induce crystallization in any one or both shoulders the whole s... |
If we induce crystallization in both shoulders of the container crystallization waves collide at a junction and form a boundary in result of the collision (Fig. ). If crystallization is induced only in one of the shoulders no boundary is formed in [MATH] -output. This hints to the following interpretation. Presence/abs... |
True False . Presence/absence of boundary (bisector of crystallization induction sites in [MATH] and [MATH] represents [MATH] Truth False . In such interpretation sodium acetate in [MATH] -shaped container implements |
and gate. Discussion We experimentally demonstrated that crystals growing in supersaturated solution of sodium acetate (‘hot ice’) compute planar Voronoi diagram, approximate set of collision-free shortest paths in a space with obstacles and implement Boolean conjunction and disjunction in a geometrically constrained m... |
Every micro-volume of the sodium acetate solution can input external stimuli/data, participate in crystal growth and represent a boundary between collided waves of crystal growth. Data are inputted in the sodium acetate solution in parallel, e.g. we use a set of pins to input planar point set to compute a Voronoi diagr... |
With regards to Voronoi diagram computation the hot ice computer is a highly accelerated version of precipitating reaction-diffusion chemical Voronoi processors |
We demonstrate in experiments that boundaries between crystal patterns perfectly represent edges of Voronoi diagram constructed on sites of crystallization induction. We did not show however how exactly the edges can be extracted from still images of the crystallized sodium acetate. This may be a topic of further studi... |
In laboratory experiments we proved that hot ice computer approximates a set of many-source-one-destination collision-free shortest paths in a space with obstacles. The shortest path problem is a typical benchmark task for unconventional computing devices. There experimental evidences that shortest path can be computed... |
, in precipitating chemical medium interfaced with cellular automaton , and in plasmodium of Physarum polycephalum These implementations, although very appealing, are however far from perfect. Some of them require substantial assistance of external hardware/software in the detection of the ‘intersection’ of waves initi... |
or extraction of a shortest path , others just do not produce a directed path at all . Hot ice computer physically represents a many-source-one-destination set of shortest paths, where destination is a site of crystallization induction. If there would be a mobile robot compatible in size with width of crystals, such a ... |
We demonstrated that two logical gates – and and or – are implemented in crystallization of sodium acetate solution. This construction complements experimental realization of xor gate in precipitating reaction-diffusion chemical medium |
. One drawback of our approach is that different properties of the medium represent values of output Boolean variables: liquid and solid phases in or gate, and presence/absence of a bisector in and gate. Thus, in principle, and and or gates cannot be cascaded. One can consider three valued variables, where liquid phase... |
[EQUATION] Whether such operation is useful for implementation of logical circuits or not will be studied elsewhere. Finally we will say a few words about re-usability. In |
we provide comparative analysis of existing non-linear medium computers in terms of re-usability. All excitable processors, including Belousov-Zhabotinsky medium, are re-usable chemical computers, because when development of excitation patterns is finished (and a dissipative structure of excitations representing result... |
Precipitating chemical processors, e.g. , have memory. They represent results of computation in the spatial distribution of precipitate which, when dried, can stay unchanged for years. However, the precipitating chemical processors cannot be re-used. They are not re-usable but “disposable”. |
Hot ice computer is an experimental example of re-usable non-linear medium processor with non-volatile memory. Patterns of crystallization, which represent results of computation, stay almost intact for weeks and months. As soon as they are heated above 58 C the patterns are transformed into liquid phase, thus being re... |
Acknowledgment I am indebted to Sebastian Adamatzky for pestering me into playing with the hot ice, and for participating in all experiments, and to Jeff Jones for editing text and participating in discussions. |
# Source: arxiv 0909.2631 # Title: Specificity and Completion Time Distributions of Biochemical Processes # Sections: all # Downloaded: 2026-03-03T05:14:50.206828+00:00 |
Specificity and Completion Time Distributions of Biochemical Processes March 15, 2024 Abstract In order to produce specific complex structures from a large set of similar biochemical building blocks, many biochemical systems require high sensitivity to small molecular differences. The first and most common model used t... |
pacs: 05.10.Gg,05.20.Dd,82.39.Rt Introduction The strong bias toward the correct assembly of particular molecular constructs, or specificity, plays a key role in myriad biochemical processes such as DNA assembly, cell signaling, protein folding, and others. A common model accounting for the almost error free completion... |
Yan et al. ( 1999 ); Sancar et al. ( 2004 ); Goulian et al. ( 1968 Similar proofreading ideas appear in other contexts such as protein translation |
Hopfield ( 1974 ); Blanchard et al. ( 2004 , molecular transport Jovanovic-Talisman et al. ( 2008 receptor-initiated signaling Mckeitan ( 1995 ); Rabinowitz et al. ( 1996 ); Rosette et al. ( 2001 ); Liu et al. ( 2001 ); Goldstein et al. ( 2004 ); Faeder et al. ( 2003 RNA transcription Springgate and Loeb ( 1975 , and o... |
Various aspects of the kinetic proofreading concept have already been studied. Hopfield Hopfield ( 1974 and Ninio Ninio ( 1975 demonstrated the possible increases in specificity due to single step proofreading. Later explorations of similar proofreading models considered the multi-step proofreading process as a “black ... |
In addition to process specificity, the time required to reach this specificity also plays an important role in biochemical processes. A proofreading strategy must be efficient as well as specific. In different contexts Zilman et al. ; D’Orsogna and Chou ( 2005 ); Redner ( 2001 ); Bel and Barkai ( 2005 2006 it was show... |
In this article, we investigate the temporal behavior of different kinetic proofreading (KPR) schemes. We derive the chemical master equation (CME– van Kampen ( 2001 ) and its transform into the Laplace domain, which provides analytical expressions for the directional and non-directional completion time distribution. I... |
II The Model Here we consider the general model of kinetic proofreading (KPR), which can be represented by the Markov chain in Fig. . The initiation state is represented by the star in the center of the chain, and is denoted by [MATH] . Depending upon the system, the state [MATH] may have different meanings; in protein... |
or [MATH] , or back to the origin with rate [MATH] or [MATH] . The two branches of the chain have [MATH] or [MATH] nodes correspondingly, the last of which, [MATH] or [MATH] is an absorbing point (representing the formation of the relevant final product). The chemical master equation (CME) describing the dynamics of th... |
[EQUATION] For any given specific case, this CME may be solved using various methods, such as various projection approaches Munsky and Khammash ( 2006 ); Burrage et al. ( 2006 ); Munsky and Khammash ( 2007 ); Peles et al. ( 2006 ); Munsky and Khammash ( 2008a , or simulated using stochastic simulations Gillespie ( 1976... |
[EQUATION] where we are using lowercase variables to represent quantities in the time domain and uppercase variables to represent the corresponding quantities in the Laplace domain. Upon application of the Laplace transform, the probabilities are now described by the following algebraic master equation |
[EQUATION] For the above equation we have already imposed the initial condition [MATH] where [MATH] is the Kronecker delta. In other words, [MATH] and [MATH] for all [MATH] The general solution of these equations is explicitly written as |
[EQUATION] The space independent parameters [MATH] and [MATH] are obtained from the solution of the quadratic equations [EQUATION] |
which come from the expressions for [MATH] at the interior points of the two branches. The boundary conditions are satisfied by proper choice of the coefficients [MATH] and [MATH] The boundary condition at [MATH] (see Eq. 17 ) is expressed as: |
[EQUATION] The boundary condition at [MATH] is written as: [EQUATION] and the boundary condition at [MATH] is [EQUATION] Using the definitions of [MATH] (see Eq. 21 ) we can rewrite Eq. ( 23 ) as |
[EQUATION] Similarly using the definitions of [MATH] we rewrite Eq. ( 24 ) as [EQUATION] Finally, using Eqs. ( 25 26 ) one can simplify Eq. ( 22 |
[EQUATION] Note that in deriving Eqs.( 25 26 27 ) we assumed that the parameters [MATH] are different than zero. In order to study the temporal behavior of the kinetic proofreading model, we compute (i) the probability that the system will reach the correct terminus point and (ii) the distribution of time until the sys... |
[EQUATION] According to Eqs. ( 28 ) and ( 20 ) the Laplace transform of the first passage time PDF is given by [EQUATION] This expression now contains a wealth of information about the moments of the escape time distributions. For example, the probability of reaching the correct absorbing site, [MATH] , is found by eva... |
at [MATH] . Furthermore, the [MATH] moment of the arbitrary completion time is [EQUATION] and the [MATH] normalized moment of the escape time to the correct site [MATH] is: |
[EQUATION] III Results and Discussion The non-normalized Laplace transforms of the two branches, [MATH] and [MATH] provide a complete description of the completion process and in particular, we analyze two important quantities: (1) the probability that the process completes via one branch or the other and (2) the distr... |
[EQUATION] Where the expressions are not sufficiently compact, particularly for the higher moments of the completion time distributions, we will use numerical examples to illustrate their dependence on parameters. For these numerical examples, we fix the length of each branch to involve [MATH] steps. To explore the eff... |
III.1 “Correct” and “Wrong” Completion Probabilities In a kinetic proofreading process, the biochemical process must somehow give preference to completing in the correct way, i.e. adding the correct amino acid to the growing protein chain or initiating intracellular signaling when the correct ligand is bound to the rec... |
[EQUATION] For example, one can use these expressions to derive expressions for the directional completion probabilities for the directed kinetic proofreading (dKPR) scheme ( [MATH] and [MATH] ) which are |
[EQUATION] where we have used the notation [MATH] Fig. A shows the probability of completing in the first direction as a function of the kinetic proofreading rates [MATH] in the case of equal forward rates ( [MATH] ). From the figure, it is apparent that a large amount of specificity is achievable for the properly chos... |
The objective of kinetic proofreading is to provide large amplification in directional specificity despite small changes in the parameters [MATH] or [MATH] . To compare how well the dKPR and AM processes achieve this objective we have drawn red dashed lines in each plot corresponding to [MATH] or [MATH] i.e., there is ... |
[MATH] and [MATH] . As the backward and proofreading rates increase, the specificity also increases for both process, as can be seen by how the dashed lines cross the contour levels. The first observation to note is that both the dKPR and the AM process can attain 90% specificity with twenty percent difference in rates... |
Figs. A-B show the completion probabilities for a case where the forward rates are different from one branch to the next. While many qualitative trends of this case are similar to the previous case with equal forward rates, the analysis becomes a little more complicated. First, the fact of different rates already provi... |
III.2 Average Completion Times In the perspective of kinetic proofreading, in addition to forming the correct product, a process must complete this construction in a timely manner. For example, the AM and dKPR schemes may make the same amplification of specificity, but one may be able to do so faster than the other. Wh... |
[EQUATION] Similarly, we find the mean “wrong” completion time [EQUATION] The average arbitrary completion time (without specifying correct or wrong completion) is |
[EQUATION] Figs. C-D show contour plots for the average completion times of the dKPR and AM processes for ranges comparable to the specificity plots in Figs. A-B and [MATH] From these plots, we can observe that as the backward or proofreading rates increase, the amount of time required to complete the process increases... |
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