This is an electronic version of a paper published 1981 in Progress in Biophysics and Molecular Biology 37, 1-47. A printing error in eq. (23), omitting differential operators ∂ has been corrected in this version. GENERATION OF BIOLOGICAL PATTERNS AND FORM: SOME PHYSICAL, MATHEMATICAL, AND LOGICAL ASPECTS Alfred Gierer Max Planck Institute für Virusfoschung (now Max Planck Institute for Developmental Biology Spemannstr. 35, D72076 Tübingen SUMMARY While many different mechanisms contribute to the generation of spatial order in the development of an organism, the formation of morphogenetic fields within initially nearuniform cells and tissues which in turn direct cell responses giving rise to pattern and form are of major importance and an essential part of the biological generation cycle. The chemical nature of such fields is not yet known, but it is most likely that they represent spatial concentration patterns of substances (or states of substances) produced by molecular kinetics. Short-range autocatalytic activation in conjunction with longer range inhibition or depletion effects suffice to generate such patterns. These conditions are necessary for the simplest two-factor case and are likely to be fair approximations in multi-component systems in which activation and inhibition may be system parameters, subsuming the action of several agents. The conditions of autocatalysis and lateral inhibition can be shown to be closely related to the analytical results obtained by general stability analysis, although this correspondence is not trivial and not immediately obvious. Very different molecular mechanisms are consistent with the concept, but fairly simple mechanisms known in molecular biology, in suitable i combination, would be sufficient. Gradients, symmetric and periodic patterns in one or several dimensions, stable or pulsing in time can be generated in this way. The pattern-forming systems account for striking self-regulatory features often observed in biological systems, including the reproducible formation of structures from nearuniform initial conditions, polarity, size-regulation and symmetry changes; quantitative aspects of biological systems, such as regenerating hydra and developing insects have been modelled on this basis. The theory is applicable to intercellular as well as intracellular patterns. Generalization to more than two components, with activation and inhibition representing system features, shows that "lateral activationm-mutual activation of two locally exclusive states in adjacent regions, mediated by diffusible substrates-can be subsumed under a generaliza-tion of the condition of lateral inhibition in conjunction with autocatalysis. Chains of induction as they seem to occur in intercalary regeneration can be modelled by further generalizations along these lines. In multicellular tissues, cells may respond to morphogenetic fields by cell determination and differentiation, changes of cell form and interaction, and the regulation of growth and proliferation. Cell determination occurs in development as a sequence of decisions between relatively few alternatives at each stage. There are indications that determination may be combinatorial, with a combination of control circuits turned "on" defining each state of determination. Morphogenetic fields are expected to contribute to control these decisions. In this way, areas may become subdivided into discrete subareas. Combinatorial regulation would imply that there could be an area code in morphogenesis (not necessarily for all tissues, cell layers, and cell types), each area being defined by a different combination of control circuits turned "on" to define the regulatory state of the cells. Real form of an organ or organism can often be traced back to the curvature of cell sheets resulting from evagination or invagination of initially nearly flat sheets in the course of development, the position of evagination being determined by morphogenetic fields. Experimental evidence indicates that often bending moments rather than tangential forces govern the process and that evagination is self-regulating, reversible and sometimes repeatable. This in turn suggests that the process is not time-controlled but rather describable as approach to a stable steady state maintained some time (though not indefinitely) in the course of development and depending on the local value of the morphogenetic field. The form assumed upon evagination may then be describable as minimizing a formal generalized potential. lntracellular mechanisms, e.g. involving intracellular fibres, and contact-mediated cell interaction with other cells, and with medium may contribute to potential in a complex manner. It can be shown that linear relations between potential and (idealized) surface areas of a cell are inconsistent with a freely evaginating stable cell sheet, but simple non-linear features (e.g. due to capping of surface components) are consistent with stability. Local activation of ii cell sheets leads to bending moments, curvature and the generation of form if the sheets are anisotropic in the inside-outside dimension (as many cell sheets obviously are). Shell theory as developed by architects and engineers is useful for calculating form of cell sheets. Elongated structures can be modelled on this basis. It is not claimed that the concept of minimal potential is always applicable-there will be cases requiring explicit treatment of friction, shear and pressures arising from growth; but in the cases where the self-regulatory properties mentioned are experimentally observed it is likely that the concept of minimal potentials is adequate. The essential requirement for non-linear effects of cell form and contact areas for understanding stability, curvature and structure of cell sheets suggests that these parameters may also contribute in a subtle non-linear manner to the regulation of cell proliferation and tissue growth. This notion might have consequences for the understanding of malignant growth as well. The theoretical treatment of morphogenetic fields discussed in this paper indicates that relatively simple molecular kinetics (consistent with conventional molecular biology) suffice to explain basic empirical features of biological pattern formation and regulation. Pattern formation based on autocatalysis and lateral inhibition is formally analogous in some respects to the generation of structures in inorganic physics and economics. However, biological pattern formation has the distinct feature of leading to highly complex re-producible structures, under the control of genes, by the invariance of the pattern formed with respect to many details and fluctuations of initial conditions, and by the defined localization and orientation of patterns, subpatterns, sub-subpatterns etc. in the course of development. Clearly the concepts used in the theories proposed and discussed have to be substantiated and confirmed by biochemical evidence; however, if the mechanism is within the scope of general kinetics on which the theory is based, even a complete list of biochemical structures involved would not in itself lead to an explanation of biological patterns. Rather, the mathematics linking molecular kinetics with developmental regulation con-stitutes, by itself, a necessary aspect for the understanding of spatial order in biological development. iii