Timed automata

Figure 1,8, for example, plots the probability that a cell has value 1 at time t4-l - labeled Pt+i - versus the probability that a cell had value 1 at time t -labeled Pt - for a particular four dimensional cellular automaton rule. The rule itself is unimportant, as there are many rules that display essentially the same kind of behavior. The point is that while the behavior of this rule is locally featureless - its space-time diagram would look like noise on a television screen - the global density of cells with value 1 jumps around in quasi-periodic fashion. We emphasize that this quasi-periodicity is a global property of the system, and that no evidence for this kind of behavior is apparent in the local dynamics. 

Given an automaton M that starts in state CTi, and any finite string s A, a, s) will represent the final output state that J<4 will enter after having processed s, one symbol at a time, from left to right. J<4 is said to accept the word s if ai,s) E the word s is rejected if and only if it is not accepted. Finally, we may define the language C JA) accepted by M as the set of all words s A that are accepted by Ad. A language C is called regular if there is a finite automaton Ad that accepts it. 

Finite automata such as these are the simplest kind of computational model, and are not very powerful. For example, no finite automaton can accept the set of all palindromes over some specified alphabet. They certainly do not wield, in abstract terms, the full computational power of a conventional computer. For that we need a suitable generalization of the these primitive computational models. Despite the literally hundreds of computing models that have been proposed at one time or another since the beginning of computer science, it has been found that each has been essentially equivalent to just one of four fundamental models finite automata, pushdown automata, linear bounded automata and Turing machines. 

In LGCA models, time and space are discrete this means that the model system is defined on a lattice and the state of the automaton is only defined at regular points in time with separation St. The distance between nearest-neighbor sites in the lattice is denoted by 5/. At discrete times, particles with mass m are situated at the lattice sites with b possible velocities ch where i e 1, 2,. .., b. The set c can be chosen in many different ways, although they are restricted by the constraint that... 

The state of the automaton at time t can be completely determined by the boolean variable n, r,t), which is equal to 1(0) if a particle is present (absent) on site r with velocity c,. From this it follows that the local microscopic density p and flow velocity u at site r are given by... 

Fig. 10.4 The timed automata models of the resources Ri, R2, R3 and the place automaton Pi.

Chovan et al.30 described a system that integrates different components of bioanalysis including automatic in vitro incubation, automatic method development (mainly SRM transitions for LC/MS/ MS analysis), and a generic LC method for sample analysis to minimize human intervention and streamline information flow. Automaton software (Applied Biosystems) was used for automatic MS method development. Flow injection was used instead of a HPLC column to decrease run time to 0.8 min per injection. Two injections were performed. The first was performed to locate the precursor ion and optimal declustering potential (DP). The second injection was performed to locate the product ion and optimal collision energy (CE). 

Sample integrations similar to pharmaceutical approaches were already examined in 1997 [39]. Here, a chip-like microsystem was integrated into a laboratory automaton that was equipped with a miniaturized micro-titer plate. Microstructures were introduced later [40] for catalytic gas-phase reactions. The authors also demonstrated [41] the rapid screening of reaction conditions on a chip-like reactor for two immiscible liquids on a silicon wafer (Fig. 4.8). Process conditions, like residence time and temperature profile, were adjustable. A third reactant could be added to enable a two-step reaction as well as a heat transfer fluid which was used as a mean to quench the products. 

Fig. 10.2 Waves through cell cycle phases in absence (a, b) or presence (c, d) of entrainment by the circadian clock. The variability of durations for all cell cycle phases is equal to 0% (left column) or 15% (right column). The curves, generated by numerical simulations of the cell cycle automaton model, show the proportions of cells in Cl, S, G2 or M phase as a function of time, for days 10-13. The time step used for simulations is equal to 1 min. The duration of the cell cycle before or in the absence of entrainment is 22 h. The successive phases of the cell cycle have the following mean durations G1 (9 h),...

In simulating the cell cycle automaton response to 5-FU, we impose a circadian profile of the anticancer medication similar to that used in clinical oncology [30, 31] 5-FU is delivered in a semi-sinusoidal manner from 10p.m. to 10a.m., with a peak at 4 a.m. (Fig. 10.3b). During the remaining hours of the day and night, the drug concentration is set to zero. For comparison, we consider similar drug delivery patterns shifted in time, with peak delivery either at 10 a.m., 4 p.m., or 10 p.m. 

To clarify the reason why different circadian schedules of 5-FU delivery have distinct cytotoxic effects, we used the cell cycle automaton model to determine the time evolution of the fraction of cells in S phase in response to different patterns of circadian drug administration, for a cell cycle variability of 15%. The results, shown in Fig. 10.5, correspond to the case considered in Fig. 10.4, namely, entrainment of a 22-h cell cycle by the circadian clock. The data for Fig. 10.5a clearly indicate why the circadian schedule with a peak at 4 a.m. is the least toxic. The reason is that the fraction of cells in S phase is then precisely in antiphase with the circadian profile of 5-FU. Since 5-FU only affects cells in the S phase, the circadian delivery of the anticancer drug in this case kills but a negligible amount of cells. 

The cases of peak delivery at 10 a.m. (Fig. 10.5b) or 10 p.m. (Fig. 10.5d) are intermediate between the two preceding cases. Overlap between the peak of 5-FU and the peak of cells in S phase is only partial, but it is still greater in the case of the peak at 10 a.m., so that this pattern is the second most toxic, followed by the circadian delivery centered around 10 p.m. The comparison of the four panels Fig. 10.5a-d explains the results of Fig. 10.4a on the marked differences in cytotoxic effects of the four 5-FU circadian delivery schedules. The use of the cell cycle automaton helps clarify the dynamic bases that underlie the distinctive effects of the peak time in the circadian pattern of anticancer drug delivery. 

General class of algorithmic methods that involve a stochastic element, i.e., that let the computer make random decisions (Binder Landau, 2000). An important subclass, the so-called Markov Chain Monte Carlo (MCMC) methods can be understood as acting on Markov chains. A Markov chain is a stochastic finite automaton that consists of states and transitions between states (Feller, 1968). At each time we consider ourselves as resident in a certain state of the Markov chain. At discrete time steps we leave this state and move to another state of the chain. This transition is taken with a certain probability characteristic for the given Markov chain. The probability depends only on the two states involved in the transition. Often one can only move to a state in a small set of states from a given state. This set of states is called the neighborhood of the state from which the move originates. 

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