Regulatory evolution and voltage-gated ion channel expression in squid axon: selection-mutation balance and fitness cliffs

Abstract

It has been suggested that optimization of either axonal conduction velocity or the energy efficiency of action potential conduction predominates in the selection of voltage-gated sodium conductance levels in the squid axon. A population genetics model of channel gene regulatory function was used to examine the role of these and other evolutionary forces on the selection of both sodium and potassium channel expression levels. In this model, the accumulating effects of mutations result in degradation of gene regulatory function, causing channel gene expression to fall to near-zero in the absence of positive selection. In the presence of positive selection, channel expression levels fall to the lowest values consistent with the selection criteria, thereby establishing a selection-mutation balance. Within the parameter space of sodium and potassium conductance values, the physiological performance of the squid axon model showed marked discontinuities associated with conduction failure and excitability. These discontinuities in physiological function may produce fitness cliffs. A fitness cliff associated with conduction failure, combined with the effects of phenotypic noise, can account for the selection of sodium conductance levels, without considering either conduction velocity or metabolic cost. A fitness cliff associated with a transition in axonal excitability, combined with phenotypic noise, can explain the selection of potassium channel expression levels. The results suggest that voltage-gated ion channel expression will fall to low levels, consistent with key functional constraints, even in the absence of positive selection for energy efficiency. Channel expression levels and individual variation in channel expression within the population can be explained by regulatory evolution in combination with genetic variation in regulatory function and phenotypic noise, without resorting to more complex mechanisms, such as activity-dependent homeostasis. Only a relatively small region of the large, nominally isofunctional parameter space for channel expression will normally be occupied, because of the effects of mutation.

Fig 1. Schematic representation of the regulatory evolution model.

The phenotype (channel peak conductance) of each individual was determined by random selection of two alleles from a pool of alleles (gametes). Channel conductance was directly proportional to the sum of the strength of the two alleles (see Methods), with or without a component of phenotypic noise. Fitness functions based on axonal electrophysiological performance were used to determine which individuals contributed genetic material to the next generation. This cycle repeats indefinitely. Random mutation of gamete DNA (channel gene promoter/CRM function) was biased towards a reduction in regulatory strength (reduced rate of mRNA transcription). In the simulations, mutation was performed immediately before selection in order to minimize computational costs but the results are equivalent to the natural order, where mutations would occur during gamete formation, as shown in the figure.

Fig 2. Effect of variations in sodium and potassium channel expression on physiological function.

A. Region of sodium and potassium conductance parameter space that supports action potential conduction up to 26°C (filled green area). Upper (red) line corresponds to boundary above which action potential conduction fails to propagate. Lower (blue) line corresponds to boundary below which the action potential fails to repolarize. The experimentally observed combination of conductance values is marked with a red dot. B. Dependence of conduction velocity on peak sodium conductance. The sodium conductance at which conduction velocity peaks (465 mS/cm²) is marked with an arrow (labeled ‘peak’), as is the experimentally observed value (120 mS/cm²) (labeled ‘squid G_Na’). C. Dependence of conduction velocity on peak potassium conductance. The experimentally observed value (36 mS/cm²) is marked (labeled ‘squid G_K’). D. Dependence of sodium ion flux during the action potential on sodium channel conductance. E. Dependence of sodium ion flux during the action potential on potassium channel conductance. For the simulations shown in panels B-E, one voltage-gated conductance was kept constant (either potassium conductance = 36 mS/cm² or sodium conductance = 120 mS/cm²), while the other conductance was swept over the range of conductance values for which action potential conduction did not fail at 26°C. The leak conductance was kept constant (0.3 mS/cm²) for all simulations. The simulation temperature was 18.5°C.

Fig 3. Effect of variations in sodium and potassium channel expression on firing patterns and action potential duration.

A. Effect of reducing potassium conductance on excitability. At the normal potassium conductance of 36 mS/cm², the axon model displays Type 3 excitability properties and is refractive after firing a single action potential in response to a sustained depolarizing current step. The axon is converted to Type 2 excitability and begins to fire repetitively following a reduction in potassium conductance (23 mS/cm²). The stimulus current was twice the size of the just-threshold current required to trigger an action potential and the sodium conductance was held constant at 120 mS/cm². B. Region of sodium and potassium conductance parameter space that has Type 3 (light green area) or Type 2 (dark green area) firing properties. C. Dependence of action potential duration on peak sodium conductance. Action potential duration at 50% (red) and 90% (blue) of peak height are shown (APD50 and APD90, respectively). D. Dependence of action potential duration on peak potassium conductance. For the simulations shown in panels C and D, one voltage-gated conductance was kept constant (either potassium conductance = 36 mS/cm² or sodium conductance = 120 mS/cm²), while the other conductance was swept over the range of conductance values for which action potential conduction did not fail at 26°C (Fig 2A). The leak conductance was kept constant (0.3 mS/cm²) for all simulations. The simulation temperature was 18.5°C for panels C and D and 6.3°C for panel A (see Methods).

Fig 4. Effect of selection for action potential conduction on sodium channel expression over successive generations.

A. Evolution of sodium conductance phenotype in the absence of any selection over 10000 generations (population size = 5000). Histograms show starting (top) and final (middle) phenotype distribution. The bottom panel shows how the average sodium conductance within the population evolves over time (10000 generations). The blue line is the average value of ten simulation runs and five independent runs are shown in light grey on the same graph. The average final conductance value was 1.2 ± 0.1 mS/cm² (mean ± S.D., n = 10). It takes approximately 3000 generations to reach steady-state. B. Evolution of sodium conductance phenotype in the presence of positive selection, based on the fitness function shown in (C). Other parameters are the same as in (A). Average final conductance value was 92.7 ± 2.21 mS/cm². C. The fitness function used in (B). This was based on the minimum sodium conductance required to maintain conduction along the axon at a temperature of 26°C. The plateau value of the step fitness functions has no significant effect on the results and a value of 1 was chosen for computational efficiency. D. Dependence of mean conductance on population size for the model shown in (B). Each ten-fold increase in population produced a 2.7 mS/cm² increase in average conductance. Data points are means of ten simulation runs and error bars are S.D.

Fig 5. Effect of selection for action potential conduction on sodium channel expression over successive generations in the presence of phenotypic noise.

A. Evolution of sodium conductance phenotype with the fitness function shown in Fig 4C in combination with phenotypic noise. The average sodium conductance within the population remains stable over time (10000 generations, population size = 5000). The blue line is the average value of ten simulation runs and five independent runs are shown in light grey on the same graph. The average final conductance value was 120.4 ± 4.4 mS/cm² (mean ± S.D., n = 10). B. Histogram showing representative final phenotype distribution translated directly from the final genotype without the addition of phenotypic noise, in order to show the variation in the underlying sodium channel cis-regulatory genotype. C. Histogram showing representative final phenotype distribution, including the effect of phenotypic noise. This is the distribution of sodium channel protein expression values that was subject to selection. Phenotypic noise was modeled as a binomial distribution (N = 28) with an expected standard deviation of 13.2 mS/cm². The N value for the noise distribution was fitted by iteration of the simulations in order to stably maintain the average conductance value close to the starting value. Note that the distributions in (B) and (C) come from the same simulation run.

Fig 6. Effect of different selection criteria on sodium channel expression over successive generations.

A. Evolution of average sodium conductance phenotype under selection for conduction velocity over time (10000 generations, population = 5000). The blue line is the average value of ten simulation runs and five independent runs are shown in light grey on the same graph. The average final conductance value was 116 ± 9 mS/cm² (mean ± S.D., n = 10). B. Histogram of representative final phenotype distribution for a single simulation run. C. Evolution of average sodium conductance phenotype under selection for both conduction velocity and sodium conductance cost over time (10000 generations, population = 5000). The blue line is the average value of ten simulation runs and five independent runs are shown in light grey on the same graph. The average final conductance value was 120 ± 3.5 mS/cm² (mean ± S.D., n = 10). D. Histograms of representative phenotype distribution for a single simulation run. E. Fitness function used in (A) and (B). Conduction as a function of sodium conductance (shown in Fig 2A) was scaled and combined with the fitness cliff shown in Fig 4C. Note the magnified scale. F. Fitness function (blue line) used in (C) and (D) was created by subtracting a cost of action potential generation (grey dotted line) from the conduction velocity (red dotted line).

Fig 7. Effect of different selection criteria on potassium channel expression over successive generations.

A. Evolution of potassium conductance phenotype with the fitness function shown in Panel F in combination with phenotypic noise. The average potassium conductance within the population remains stable over time (10000 generations, population = 5000). The blue line is the average value of ten simulations and five independent runs are shown in light grey on the same graph. The average final conductance value was 36.2 ± 0.5 mS/cm² (mean ± S.D., n = 10). B. Histogram showing representative final phenotype distribution translated directly from the final genotype without phenotypic noise to show the variation in the underlying potassium channel genotype. C. Histogram showing representative final phenotype distribution, including the effect of phenotypic noise. It is this distribution of potassium channel protein expression values, which varies slightly with each successive generation, that is subject to selection. Phenotypic noise was modeled as a binomial distribution (N = 16) with an expected standard deviation of 4.0 mS/cm². The N value for the noise distribution was selected by running successive simulations to obtain a best fit to the experimentally observed conductance value. Note that the distributions shown in (B) and (C) come from the same simulation run. D. Evolution of potassium conductance phenotype under selection for reduced action potential duration using the fitness function shown in Panel G. The average potassium conductance within the population remains stable over time (10000 generations, population = 5000). The blue line is the average value of ten simulation runs and five independent runs are shown in light grey on the same graph. The average final conductance value was 35.6 ± 3 mS/cm² (mean ± S.D., n = 10). E. Histogram of a representative final phenotype distribution. In this simulation no phenotypic noise was used so that the final channel phenotype maps directly from the final cis-regulatory genotype. F. The fitness function used in (A, B and C). This was based on the minimum potassium conductance required to produce Type 3 firing properties, i.e. suppress repetitive firing in response to a sustained depolarizing current injection (see Fig 3A and 3B). G. Fitness function used in (D and E). Action potential duration as a function of potassium conductance (see Fig 3D) was inverted, scaled and then combined with a fitness cliff at 3 mS/cm², the value at which action potential conduction fails.

Fig 8. Stability of the end-point for sodium and potassium channel expression from multiple starting points.

A. The region of sodium and potassium conductance parameter space that supports action potential conduction at 26°C and Type 3 firing properties (filled green area) bounded by two fitness cliffs (red and brown lines). Three different populations started from a different combination of average sodium and potassium conductance values (blue dots) that converged over successive generations to stable final values (red dots, overlapping symbols). The average final conductance values were G_Na = 116.5 ± 0.6 mS/cm² and G_K = 36.0 ± 0.3 mS/cm² (mean ± S.D., n = 3). The colored traces (black, orange and brown) mark the path of the average conductances over 4000 generations for each population, from its starting point to the stable final phenotype. The phenotypic noise used in the simulations had an expected standard deviation of 17.3 mS/cm² for the sodium conductance and 4.5 mS/cm² for the potassium conductance. These values were previously selected by iteration in order to fit the experimentally observed conductance values. The ellipse (dotted grey line) is centered on the average conductances and the length of the two axes correspond to 4 times the S.D. of the phenotypic noise used in both dimensions. B. The distributions of conductance phenotype for the final population at the end of a representative simulation run. A contour plot of 2D histograms of sodium and potassium conductance combinations shows the final phenotype distribution. Color bar indicates the number of individuals found in each bin, expressed as a percentage of the total population. Bins with fewer individuals than 0.2% of the total population were blanked. The red dot indicates the average conductance values for the population. The variation in channel expression levels within this population reflects a combination of variation in the sodium and potassium channel cis-regulatory genotype as well as the contribution of phenotypic noise. It is this distribution of channel protein expression values, which varies slightly with each successive generation, that is subject to selection. C. Average conductance values for the population increase as the population variation increases. Three simulations are shown. The three different final average conductance values (red dots) are shown with their corresponding population variation (solid ellipses, distinguished as three different shades of green). The changes in population variation were driven by changing the phenotypic noise values. Final conductance values for the three representative individual populations were: G_Na = 77.5 ± 6.1, 94.3 ± 11.7, 115.6 ± 18.0 mS/cm² and G_K = 19.3 ± 2.0, 26.4 ± 3.5 and 35.7 ± 5.1 mS/cm² respectively (mean ± S.D., N = 5000). The filled ellipses are centered on the average conductances and the length of the two axes correspond to 4 times the S.D. for that population.