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John Pearson, Gary Doolen, Brosl Hasslacher, Bill Reynolds, Los Alamos National Laboratory; Werner Horsthemke, Southern Methodist University; Kyoung Lee, Bill McCormick, Harry Swinney, University of Texas at Austin; Silvina Ponce-Dawson, Universidad de Buenos Aires

In 1952, Alan Turing [1] suggested a possible connection between patterns in biological systems and patterns that could, in principle, form spontaneously in chemical reaction- diffusion systems. These patterns are spatial variations in the concentration fields of the reacting chemicals. Turing's analysis stimulated considerable theoretical research on pattern formation in mathematical models, but Turing-type patterns were not observed in controlled laboratory experiments until 1990[2-3]. Subsequently there has been a renewed interest in chemical pattern formation and in the relationship of chemical patterns to the remarkably similar patterns observed in diverse physical as well as biological systems.

A recent numerical study of a simple chemical model revealed spot patterns that undergo a continuous process of birth through replication and death through over-crowding[4]. The self-replicating spot patterns were first discovered during a successful attempt to simulate the labyrinthine patterns experimentally observed in the ferrocyanide- iodate-sulfite reaction[5]. The numerical study was performed on 16,000 nodes of a CM-200 and the study involved several hundred simulations of a simple reaction-diffusion model. Although the individual runs could have been performed on a workstation, it would have been impossible to map out the different dynamical behaviors observed at different regions of parameter space in a timely manner without the speed of the Thinking Machines Connection Machine.

In spite of obvious differences between the experimental chemical kinetics and the kinetics employed in the numerical study, there has been considerable interplay between the numerics and the experiments. The numerical study reproduced the labyrinthine patterns and discovered the self-replicating spots. The self-replicating spots were then found experimentally, where it was discovered that a single spot in isolation grew into an annulus that then fractured to produce more spots[6]. This behavior was subsequently observed in the numerical model.

A theoretical analysis[7] based on the numerical model provides an intuitive explanation of this behavior. This explanation is given in terms of a rapidly diffusing fuel and a slowly diffusing fire. If the fuel becomes sufficiently hot, the autocatalytic process of burning begins. If a small spot is ignited, a local depletion of the fuel occurs. Therefore, gradients are created that induce a lateral flux of fuel into the spot. The fate of an ignited spot depends on the lateral diffusive flux of fuel into it. If this flux is small enough, the system can sustain a steady, localized region of high temperature. As the fuel flux increases, the spot grows. Once the burning spot is sufficiently large, the diffusive flux of fuel into its interior becomes inadequate to maintain the center in the burning state. The center burns out leaving two burning spots on either side. This is the mechanism by which spots replicate. Mathematically, the analysis indicates that the spots can be viewed as points moving on a two-sheeted Riemann surface.

The numerical model is a variant of a skeletal model of glycolysis first introduced by Selkov. Glycolysis is the process by which the cell breaks down glucose into ATP, the cell's energy currency. The observation that the glycolytic pathway contains the positive feedback loops required for chemical pattern formation raises the possibility that such patterns occur inside the cell. Whether such chemical patterns actually occur in the cell can only be answered by experiment, but lattice gas simulations of the simple chemical models[8, 9] indicate that chemical patterns are robust against a number of density fluctuations down to distance scales of a few tenths of a micron. (The lattice gas method was originally developed for parallel hydrodynamic simulations[10]. The group uses a variant that was developed for the purpose of studying fluctuations in reaction-diffusion systems[11].) Below some distance on the order of a tenth of a micron, fluctuations dominate, thus preventing the formation of chemical patterns. Since a typical eukaryotic cell diameter is on the order of ten microns, in principle, chemical patterns can exist inside the cell. What biological function would such patterns perform?

Prior to cell division, all eukaryotic cells must segregate the replicated chromosomes into two congruent sets so that one complete set can go to each daughter cell[12]. This segregation process is known as mitosis. During mitosis, a bipolar structure known as the mitotic spindle forms. The mitotic spindle consists of two organizing centers called centrosomes from which microtubule fibers emanate. The replicated chromosomes migrate to the spindle equator and are then separated by forces exerted by the microtubule on the chromosomes. These forces are generated by the action of the ATP engines dynein and kinesin, which convert chemical potential energy to work. The mitotic spindle forms not only in cells with artificial chromosomes, but also in cells from which the entire nucleus has been completely removed[13]. Moreover, microtubules are observed to organize into a coherent network in cell fragments containing no centrosomes or chromosomes. Thus it seems much more likely that the formation of the ubiquitous spindle is governed by simple physical processes such as chemical reactions and diffusion than by complex genetic mechanisms.


[1] Turing, A.M., Phil. Trans. R. Soc. Lond. B 327, 37-72 (1952).

[2] Castets, V., Dulos, E., Boissonade, J., and De Kepper, P., Phys. Rev. Lett., 64, 2953-2965 (1990).

[3] Ouyang, Q. and Swinney, H.L., Nature, 352, 610-612 (1991).

[4] Pearson, J.E., Science, 261, 189-192 (1993).

[5] Lee, K.J., McCormick, M.D., Ouyang, Q., Swinney, H.L., Science, 261, 192-194 (1993).

[6] [5] Lee, K.J., McCormick, M.D., Pearson, J.E., and Swinney, H.L., to appear in Nature, May 1994.

[7] Reynolds, W.E., Pearson, J.E., and Dawson, S.P., Phys. Rev. Lett., 72, 2797, (1994).

[8] Hasslacher, B. Kapral, R., and Lawniczak, A., Chaos, 3, 7, (1993).

[9] Dawson, S.P., Hasslacher, B. and Pearson, J.E., "Pattern Formation and Lattice Gas Automata," (Proceedings of the NATO Workshop, Waterloo, Canada, June 7-12, 1993), A. Lawniczak and R. Kapral, eds. Fields Institute Communications, (Amer. Math. Soc.)

[10]Frisch, U., Hasslacher, B., and Pomeau, Y., Phys. Rev. Lett., 56, 1505, (1986).

[11]Dab, D., Lawniczak, A., Boon, J-P, and Kapral, R., Phys. Rev. Lett., 64, 2462, (1990).

[12] Alberts, B., Bray, D., Lewis, J., Raff, M., Roberts, K., Watson, J., Molecular Biology of the Cell, 3rd edition, Garland Publishing Inc. (New York), ISBN 0-8153-1619-4, (1994).

[13] Brachet, Jean, "Molecular Cytology," V. 1, The Cell Cycle, pgs. 296 and 338, (Academic Press, 1985).

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