Gause's principle and the effect of resource partitioning on the dynamical coexistence of replicating templates

Abstract

Models of competitive template replication, although basic for replicator dynamics and primordial evolution, have not yet taken different sequences explicitly into account, neither have they analyzed the effect of resource partitioning (feeding on different resources) on coexistence. Here we show by analytical and numerical calculations that Gause's principle of competitive exclusion holds for template replicators if resources (nucleotides) affect growth linearly and coexistence is at fixed point attractors. Cases of complementary or homologous pairing between building blocks with parallel or antiparallel strands show no deviation from the rule that the nucleotide compositions of stably coexisting species must be different and there cannot be more coexisting replicator species than nucleotide types. Besides this overlooked mechanism of template coexistence we show also that interesting sequence effects prevail as parts of sequences that are copied earlier affect coexistence more strongly due to the higher concentration of the corresponding replication intermediates. Template and copy always count as one species due their constraint of strict stoichiometric coupling. Stability of fixed-point coexistence tends to decrease with the length of sequences, although this effect is unlikely to be detrimental for sequences below 100 nucleotides. In sum, resource partitioning (niche differentiation) is the default form of competitive coexistence for replicating templates feeding on a cocktail of different nucleotides, as it may have been the case in the RNA world. Our analysis of different pairing and strand orientation schemes is relevant for artificial and potentially astrobiological genetics.

Figure 1. Template and copy are either different (thin) or identical (bold) for complementary (top) and homologous pairing (bottom) due to strand polarity.

Reverse (top left) and direct palindromes (bottom middle) yield two identical templates with complementary and homologous pairing, respectively, just like homologous pairing with parallel polarity. Note, that in case of reverse palindromes, it is not necessary for the sequence itself to be palindromic to make the two strands identical. Cases of complementary pairing with antiparallel polarity (top left and middle) and homologous pairing with parallel polarity (bottom right) are discussed in the main text; homologous pairing with antiparallel polarity (bottom left and middle) is discussed in Text S1. The remaining case (top right) is not discussed here.

Figure 2. Split plot of the coexistence of two complementary sequence pairs with antiparallel strand polarity (4 sequences per pair) of length .

Lower left half: coexistence is marked by green, extinction of the first sequence pair by red and extinction of the second sequence pair by blue. Upper right half: stability of coexistence according to the leading eigenvalue (red indicates more stable, blue indicates less stable coexistence, white indicates extinction of one of the sequence pairs). The upper triangle shows the stability measures of the sequences pairs from the lower one (mirrored and rotated ). From the point of view of coexistence two pairs (e.g. -, -) and their reverse (-, -) are not fully equivalent. The reason for this is that the degradation rates are assigned to sequences, always in the same order within a set (this means that the same rates are assigned to e.g. and in the two cases, respectively). Despite this difference the plot is almost symmetrical since degradation rates are taken from a narrow distribution. For details, see main text, for parameters, see Text S1.

Figure 3. Probability of coexistence in case of non-uniform degradation rates and homologous pairing ().

Each cell is an average of numerical results over 1000 random parameter setups. Purple indicates highly improbable coexistence, green indicates likely coexistence. Sequences are arranged along the axes first according to Hamming distance and secondly according to lexicographic ordering (increasing content towards bottom and right). For parameters, see Text S1.

Figure 4. Coexistence plots of pairs of double-stranded sequences of length (upper panel) and (lower panel) using two monomers (, ) in case of uniform degradation and identical elongation rate constants and non-complementary pairing.

The green indicates stable coexistence, grey indicates structurally unstable coexistence, i.e. compositional identity (no coexistence in a biological sense), while pink indicates that there is no coexistence possible. Sequences along the axes are arranged first according to Hamming distance and secondly according to lexicographic ordering from more (top and left) to more (bottom and right). For parameters, see Text S1.