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THE INHERITED EFFICIENCIES MODELTo examine the evolutionary potential of a non-genomic system, we have employed a simple, computationally tractable model which is still capable of capturing the essential biochemical features of the real system. In this model, protocellular walls are permeable to amino acids but not to oligopeptides of any length. Within the protocell, the formation and destruction (also called hydrolysis) of bonds between consecutive amino acids in oligopeptides (peptide bonds) occur through catalyzed, albeit possibly very inefficient, pathways. A peptide of any length can act in a double role as a substrate for polymerization or hydrolysis, or as a catalyst of chemical reactions. Since only two reactions are considered in the present model, all peptides are characterized by three traits: their length and their efficiencies as catalysts of ligation and hydrolysis of peptide bonds. These efficiencies can be interpreted as the inverse of turnover rates and are currently assumed to be independent of each other. In a system composed of different types of amino acids, peptides of the same length but
different composition vary in their catalytic ability. In a detailed model, this can be
accounted for by providing microscopic rules that relate the peptide sequence to its
catalytic efficiency. Since these rules, however, are not known at present, we adopt a
stochastic model, in which the specific identities of amino acids are not considered.
Instead, the dependence of the catalytic efficiency on the sequence, The position of the maximum of each distribution function increases, in a sigmoidal fashion, with the length of the polymer: The parameter When two peptides are joined together, the catalytic efficiencies of the product of
this reaction are related to the efficiencies of the reactants. For example, the product
of the addition of a small peptide to a much longer peptide has efficiencies which closely
resemble the efficiencies of the longer ``parent''. To underscore this relationship the
model is called an Inherited Efficiencies Model. Statistically, catalytic efficiencies of
the product of a ligation reaction are chosen from a conditional probability, A similar approach is taken to define a conditional probability for the products of
hydrolysis reactions. However, since hydrolytic enzymes act more efficiently on disordered
peptides than on ordered peptides, not all peptide bonds are equally likely to be
hydrolyzed. Although our model does not explicitly include the degree of ordering of
different polymers, we exploit the relationship between structure and function: without a
stable three-dimensional structure, high efficiency protein catalysis is impossible. In
the current implementation of the model, the degree of structure of a peptide, s,
is computed using: When a peptide is hydrolyzed to form two new peptides, the catalytic efficiencies of
the ``offspring'' are, once again, chosen from a conditional probability, Simulations of the Inherited Efficiencies Model are carried out using a Monte Carlo method. Each Monte Carlo cycle consists of three stages: (1) the reaction to be performed (ligation or hydrolysis) is chosen, (2) the substrate or substrates are chosen from the list of peptides present in the system, (3) the properties of the product or products of the reaction are sampled from the appropriate distributions and the list of polymers is updated. The number of monomers in the protocell is held fixed to reflect the equilibrium between the concentrations of amino acids inside the protocell and in the environment, facilitated by the permeation properties of the protocellular boundary. The probabilities for the two reaction types are computed from the corresponding total catalytic capabilities of the peptides within the protocell. Once the reaction type is chosen, the probabilities of individual reactions are used to choose the substrate(s) of the reaction. Finally, the properties of the products of the reactions were chosen from the conditional probabilities described above.
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, is captured by assuming that the
efficiencies of peptides of length n for catalyzing ligation and hydrolysis
reactions are distributed with probabilities 
and 
, respectively. In the current
implementation, these probability distributions are Gaussian: 
(
) sets the rate at which the
mean efficiencies vary between their minimum value of 
(
) and their maximum value of 
(
) and 
(
) is the length at which the mean efficiency
is halfway between its maximum and minimum. This relationship captures the biochemically
plausible property that initially the efficiencies increase, on average, only slightly
with the length of the polymer. Only when peptides reach lengths sufficient for them to be
able to adopt an ordered three-dimensional structures do the average efficiencies start
increasing markedly. Then, for even longer polymers, the average efficiencies again
stabilize, since gaining additional length no longer produces significant improvement in
catalytic properties. The widths of the distributions 
(
) are chosen such that probabilities of
sampling negative efficiencies are quite small. If such instances occur, the efficiencies
are reflected across the origin. 
, which gives the probability
of creating a peptide of length n=k+l with efficiency 
and 
,
respectively. Since this probability is a property of the ligation process, the same form
is used to assign efficiencies of ligation and hydrolysis to the product. In the present
implementation, this probability has the form of a multivariate Gaussian: 
(
) and 
for the ligation
(hydrolysis) properties of the substrates or the product. 
.
As stipulated by the model, this implies that efficient catalysts are less likely to be
hydrolyzed than inefficient, presumably disordered, peptides. The maximum value of the
bias, almost always equal to unity, will be denoted by 
and the mininum value by 
. The degree of structure for
which the bias is halfway between its maximum and minimum values will be denoted 
and the rate of decrease of
the bias will be controlled by a parameter denoted as 
. 
, of creating peptides of
lengths k and l, with catalytic efficiencies 
