Nonlinear fitness landscape of a molecular pathway

Abstract

Genes are regulated because their expression involves a fitness cost to the organism. The production of proteins by transcription and translation is a well-known cost factor, but the enzymatic activity of the proteins produced can also reduce fitness, depending on the internal state and the environment of the cell. Here, we map the fitness costs of a key metabolic network, the lactose utilization pathway in Escherichia coli. We measure the growth of several regulatory lac operon mutants in different environments inducing expression of the lac genes. We find a strikingly nonlinear fitness landscape, which depends on the production rate and on the activity rate of the lac proteins. A simple fitness model of the lac pathway, based on elementary biophysical processes, predicts the growth rate of all observed strains. The nonlinearity of fitness is explained by a feedback loop: production and activity of the lac proteins reduce growth, but growth also affects the density of these molecules. This nonlinearity has important consequences for molecular function and evolution. It generates a cliff in the fitness landscape, beyond which populations cannot maintain growth. In viable populations, there is an expression barrier of the lac genes, which cannot be exceeded in any stationary growth process. Furthermore, the nonlinearity determines how the fitness of operon mutants depends on the inducer environment. We argue that fitness nonlinearities, expression barriers, and gene-environment interactions are generic features of fitness landscapes for metabolic pathways, and we discuss their implications for the evolution of regulation.

Figure 1. Schematic representation of the lac pathway.

The lac operon is composed of three genes controlled by the same promoter: lac Z, lac Y, and lac A. The lac pathway also involves the constitutively expressed repressor LacI. It represses the transcription from the lac promoter by attaching to the operator sequence lacO1. Inducers, such as IPTG, deactivate the repressor LacI and thus stimulate the synthesis of the gene products LacZ, LacY, and LacA. The rate of production of the three lac proteins is denoted by and it depends on the sequence of lac O1, on the presence of inducer inside the cell, and on the growth rate. All three lac genes are transcribed with the same rate, hence LacZ can be used as a reporter for the whole operon. LacY transports molecules such as IPTG inside the cell with a rate , which depends on the concentration of these molecules. One proton is transported with each substrate molecule . Growth (measured by the Malthusian fitness ) dilutes the internal molecules, thus lowering their concentrations. The strains used in this study differ by the lacO1 sequence and are grown in various IPTG concentrations.

Figure 2. Fitness of lac regulatory mutants in different environments.

Measured fitness cost of each mutant strain, plotted against LacZ concentration (normalized to the fully induced wild-type value). Measurements are obtained in minimal medium with 0.1% glycerol in the absence of IPTG (blue dots) and in the same medium with 1 mM IPTG (mauve squares). Fitness is measured by competition against a reference strain which has a deletion of the whole lac locus and of lac I . The fitness cost of a given strain is defined as the reduction in growth rate (Malthusian fitness) compared to the reference strain (see Materials and Methods for details). In presence of 1 mM IPTG, a control strain with deleted lac Y gene has an expression level comparable to the wild type, but a fitness close to that of constitutive mutants in absence of IPTG (red dot). All points show the average of 12 replicates for fitness and at least 3 replicates for protein concentration, with error bars giving the standard error. Lines show model predictions (the dashed line represents an unstable solution, see main text).

Figure 3. From genotype and environment to pathway phenotypes and fitness.

Environment and genotype determine the function of the lac pathway, which is described by the two phenotypes of protein production and protein (transport) activity. These phenotypes are coupled by a pathway-specific positive feedback loop (blue circle). The pathway itself is coupled to growth (fitness) by a generic positive feedback loop: protein production and protein activity are fitness costs, and cell growth reduces protein concentration and activity by dilution (red circle). In addition, growth can affect the rate of gene expression (dashed arrow). These feedback loops generate strong nonlinearities in the phenotype-fitness map and the genotype-environment-fitness map; see Figure 4 and Figure 5.

Figure 4. Phenotype-fitness map.

The fitness cost of the lac pathway is shown as a function of the protein production rate and the transport rate . The fitness landscape obtained from our model (shaded surface) is strongly nonlinear and has two branches. The stable part of the landscape (solid shading) ends at a fitness cliff (solid blue line), beyond which populations cannot maintain growth. The remaining part of the lower fitness branch is unstable (striped shading). Protein expression and activity of viable populations are bounded by a barrier (dotted blue line). Model predictions of pathway phenotypes and fitness for individual strains under varying inducer concentrations are shown as a family of red lines (light red: wild type, dark red: operator mutant strains). Experimental fitness values are shown as dots (the offset from the model surface is marked by gray lines).

Figure 5. Genotype-environment-fitness map.

The fitness cost of the lac pathway is shown as a function of the operator genotype summary variable (maximum rate of protein production at a growth of 1 cell division/hr, see text) and the external inducer concentration . The model fitness landscape is again strongly nonlinear: it has a stable upper branch (solid shading) and an unstable lower branch (striped shading) separated by a fitness cliff (blue line), similar to the phenotype-fitness map of Figure 4. Model predictions for individual strains under varying inducer concentrations are shown as a family of red lines (light red: wild type; dark red: operator mutant strains with equal to wild type value, see text). Experimental fitness values are shown as dots (the offset from the model surface is marked by gray lines).