A DYNAMIC MODEL FOR LIPIDIC VESICLES WITH AUTOCATALYTIC REACTIONS

The purpose of this work is to contribute to understand the basic conditions required for the appearance of an elementary metabolism in protocells. For this, we develop and analyze here a simple mathematical model that considers that, in some appropriate aquatic precinct (e.g., in a deep-ocean hydrothermal vent or a terrestrial geothermal field), lipid vesicles form spontaneously from a rich pool of lipids present in the media. During their formation, the vesicles can randomly capture molecules of a certain compound (“c”), which cannot permeate across the vesicle membrane. The precinct also possesses another compound (“a”) that can freely diffuse into the vesicle interior through its membrane. For simplicity, we assume that all the vesicles are similar, in both size and shape. Like other previous authors, we assume that an autocatalytic reaction inside the vesicles may have constituted the primordial seed for the further development of progressively more complex metabolic organizations. In our case, we consider that, in the presence of c, the entering a rapidly reacts with c to produce more c according to the scheme a+c→2c+r, where r is another non-permeant product. This reaction only takes place if c acts as a catalyst. Accumulation of c and r inside a vesicle increases its osmotic pressure. We assume that the vesicle breakdown only occurs by osmotic lysis. The main dependent variables considered are the average concentration of c in the vesicles and the number of vesicles that have captured at least one molecule of c (N_V). Among other results, we found that, for ranges of values of the parameters, N_v may reach asymptotically stable steady states. This result ultimately suggests that the model introduced here may contain some elementary characteristics of the possible origin of metabolic pathways inside protovesicles.