Unidimensional life. (Research Paper).

INTRODUCTION

What are the really fundamental differences between inert and living matter? Physics has shown that all material systems are made of simple atoms and thus it opened the prospect to explain all systemic behaviours in terms of atomic dynamics. Sadly, in spite of quite extensive observation and rigorous theorization, this prospect succeeded only for the passive behaviours of physical systems: we still have to understand in the same general terms how living systems come to elaborate active behaviours that ensure survival and development in the midst of inert matter. At least, we know that carbon is the basis of all known life and that it binds to other atoms in preferred geometric configurations; therefore, we may presume that geometry is at stake but we must yet discover the peculiarity that enables carbon to sustain life.

Regarding inert matter, rational mechanics always postulates that physical space is three-dimensional, homogeneous and isotropic; the state of motion of material point-like particles in this space then is completely determined by a pair of conjugate three-component variables, positions and velocities. Accordingly, inert matter formed of such particles tends to fill this space more or less isotropically: for instance, fluids extend indiscriminately into all available parts of their container, and most chemical elements crystallize into compact patterns close to cubic or hexagonal structures. Exceptions only arise for aggregates of extremely asymmetric molecules: for example, fluids made of long molecules exhibit non-isotropic features as in liquid crystals, and solids made of long molecules exhibit privileged directions along which linear and tree-like structures grow spontaneously.

Living matter is the province where such exceptions are the rule: indeed, most organic compounds are formed of linear chains of carbon atoms along which various satellite atoms bind. The genetic material itself consists of very long one-dimensional strings of chemical bases, yet as ingenuous they seem, they carry enough instructions to build a full three-dimensional organism. Those instructions include the primary assembling of proteins that also are long strings of amino acids folding secondarily onto themselves into three-dimensional structures; proteins may thus form long tubular structures determining shapes and motions within cells, or they may form globules graced with chemically active sites that ensure cell metabolism. Protein folding is not encoded explicitly: it occurs spontaneously during synthesis of the strings as thermal agitation tests for their lowest energy states. The procedure is a random search that poses a notorious problem intractable by formal computation (Pattee, 1995).

Unidimensionality is also prevalent at the organismic level in zoology. For example, the earliest-known multicellular organisms were string-like microbes as old as 3.5 billion years (Bradshaw, 1997). Since then, most body plans have consisted of one-dimensional strings of segments specialized in diverse functions, and internal air or fluid circulatory networks have developed as pervading tree-like structures; conversely, a few symmetrical body plans are known, for instance urchins or starfishes, but they characteristically failed to evolve. Certain cells, such as muscle fibres or nerves, also consist of one-dimensional linear segments forming bundles or tree-like structures. Thus, nerves propagate signals along chained paths wired in limbs, spinal cord and neuronal columns of brains; for instance, low-level loops of such neuronal chains organize reflex behaviours devoted to safekeeping. Finally in human beings, the high-level functions of living matter generate mental representations such as music, speech or text; these representations are strictly linear strings of symbols yet they turn out to carry quite complex meanings. Apparently, linear symbolic representations carry meanings for human communication in precisely the same way that linear genetic sequences carry instructions for synthesis of living matter.

Many questions then arise as to why living matter appears to be structurally restricted to unidimensionality even though its functions are definitely multidimensional: is unidimensionality necessary for life to appear and endure? If so, how did unidimensionality appear among exclusively three-dimensional objects such as isolated atoms? Then, in what ways has unidimensionality conditioned functional complexity and evolution of living matter? Finally, has unidimensionality conditioned its capacity to carry complex instructions or meanings, including consciousness and culture, that are the latest but highest achievements of life?

The present paper investigates these challenging questions on the assumption that living organisms formally are one-dimensional dynamic systems, i.e., large ensembles of sharply anisotropic components that are all described by a single pair of one-dimensional dynamic variables. The outcome will emerge that such systems have the exclusive capacity to sustain large-scale internal processes switched on from the outside. Thereby, they become able to adapt and develop in changing environments: unidimensionality indeed endows them with a simple trial-and-error method to solve problems of survival when external conditions become hazardous; moreover, many repeated resolutions of this kind may typically achieve very high levels of complexity. Thus, one-dimensional systems may exhibit elaborate behaviours and improve them continuously, for instance by developing and specializing organs, or ultimately by manufacturing artificial tools. As a result, they consistently refine control over their environment; consciousness and intelligence may then conceivably appear at the highest levels and further lead to social organization and culture. In that way, mathematics and physics together are able to ground a continuous genealogy that connects elementary chemistry to biology and sociology. The new perspective legitimates life and culture as potentialities inherent in inert matter without resorting to extraordinary events or assumptions.

FROM ISOTROPY TO UNIDIMENSIONALITY

The first issue is to explain how three-dimensional physics, which applies to isolated point-like particles, may lead to loss of isotropy in systems formed of many such particles to the point where mere unidimensional descriptions are sufficient. Mathematically, isotropy refers to properties at any position around a centre that are invariant with the direction of the position vector originating from this centre. The physical prototype is the spherically symmetric force field around a point-like electrical charge: its direction is indeed invariantly parallel to the position vector originating at the charge (Figure 1).

[FIGURE 1 OMITTED]

Anisotropy begins to show up when pairs of equal and opposite charges are brought close to each other to form a dipole, usually described by a moment vector oriented from the negative to the positive charge. Around such a dipole, the direction of the composite field varies strongly with respect to the position vector originating at the centre of mass. Indeed, the field is parallel at positions down the dipole moment axis, then it turns perpendicular on a normal axis, and finally it reverses in antiparallel orientation at positions up the moment axis (Figure 2); thus by turning twice as fast as the position vector, the field exhibits distinctive directional variations. The sharpest variations obtain when many such dipoles are clustered in a finite material sample. In this case, they interact, and tend to align with respect to each other so that the composite field depends on the shape of the sample: the more oblong the sample, the more anisotropic the composite field becomes.

[FIGURE 2 OMITTED]

Physical examples include samples composed of ferroelectric or ferromagnetic materials. The simplest is the long cylindrical magnet formed of many atoms carrying magnetic moments (Fivaz, 1982). Mutual interactions between moments coerce them to align within adjacent domains whose total moments have preferential orientations: except in small domains near the ends of the cylinder, total moments are either up or down the axis, but they finally sum to a net value of zero for an isolated magnet. The shape dependence is then described by a demagnetizing coefficient that scales the composite magnetic field with respect to the net moment. Now, according to a general principle in physics, when such a sample is embedded in a finite external field, its net moment density adjusts itself to the value that minimizes the energy in the field. The result in magnetostatics is that the net moment varies linearly in response to the external field; then, both are effectively zero for the isolated sample but proportional in a finite field. The proportionality constant is called magnetic susceptibility and its value depends on the shape of the sample: the linear susceptibility increases with length and the value in …