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These different noise sources were treated collectively as single source, referred to as phenotypic noise throughout, that was introduced at the point of translation of each individuals genotype into the channel expression phenotype Fig 1. The combination of a fitness cliff and phenotypic noise can maintain a stable sodium conductance indefinitely Fig 5A , producing a final average sodium conductance of The spread in the conductance values in the final population Fig 5C is due to two sources of variation, variation in the sodium channel cis-regulatory genotype shown in Fig 5B plus variation due to the effect of phenotypic noise.

The range of conductance values in the final population is comparable to the two to four-fold ranges of channel conductance values that have been observed in identified neurons from natural populations [ 37 ], where both genotype variation and other noise sources would also be expected to contribute to variation in channel protein expression. Repeated iteration of the simulations was required in order to fit the appropriate noise distribution that could maintain a particular stable conductance value. Lower phenotypic noise levels produced lower average final conductance values and higher noise levels resulted in higher conductance values.

Evolution of sodium conductance phenotype with the fitness function shown in Fig 4C in combination with phenotypic noise.


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The average final conductance value was 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. Histogram showing representative final phenotype distribution, including the effect of phenotypic noise.

Excitable Cells

This is the distribution of sodium channel protein expression values that was subject to selection. 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. The other two mechanisms that will allow purifying selection to maintain a stable sodium conductance phenotype require the incorporation of positive selection for conduction velocity into the fitness function.

The first of these approaches assumes that increasing sodium conductance is cost-free so that the fitness function will reflect a scaled version of the conductance-velocity curve Fig 6E.

In the second approach, positive selection for conduction velocity is combined with a cost for increasing sodium conductance Fig 6F. Histogram of representative final phenotype distribution for a single simulation run. Histograms of representative phenotype distribution for a single simulation run. 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. 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.

Incorporating a contribution of conduction velocity together with a fitness cliff due to conduction failure can successfully maintain the average sodium conductance close to the initial phenotype Fig 6A. This is an example of the classic mutation-selection balance that is believed to underlie the establishment of a broad range of phenotypic properties [ 16 ]. A balance is established between positive selection for increasing conduction velocity and the negative effect of mutation Fig 6E.

A steeper fitness curve, implying stronger selection for conduction velocity, results in a higher average conductance value as the conductance average is drawn closer to the conductance that produces peak conduction velocity data not shown.


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A fitness function incorporating an energy cost for action potential generation is shown in Fig 6F. The fitness cost for the ion fluxes was approximated by assuming that the energetic cost increases linearly with sodium channel conductance, which is an approximation of the results from the axon model Fig 2D. In this case, increasing conduction velocity red line is now balanced by increasing cost grey line so that there is a distinct peak in the fitness function blue line.


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  6. This form of the fitness function can also maintain the average conductance values close to the starting phenotype over multiple generations Fig 6C. Because the slopes of the fitness curve are now steeper, the effect of random drift is diminished and both the time courses of independent trials and final phenotypes have a tighter distribution Fig 6C and 6D.

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    In this case, the balance between the two competing selection constraints is the primary determinant of the final phenotype, although mutational effects will still cause the average conductance to fall below the peak of the fitness function. The approach for understanding how purifying selection may maintain stable potassium conductance values was similar to that used for the sodium conductance. Selection for increased conduction velocity or reduced metabolic cost will only act to reduce the potassium conductance Fig 2C and 2E , suggesting that neither of these properties can make a significant contribution to the fitness function for the potassium conductance.

    Two mechanisms can account for the observed potassium conductance values. Incorporation of this fitness cliff into the model by itself results in selection of a potassium conductance of Addition of a phenotypic noise distribution can maintain a stable potassium conductance Under these conditions there is reduced variation in the underlying potassium channel cis-regulatory genotype Fig 7B.

    Evolution of potassium conductance phenotype with the fitness function shown in Panel F in combination with phenotypic noise. The blue line is the average value of ten simulations and five independent runs are shown in light grey on the same graph. 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. It is this distribution of potassium channel protein expression values, which varies slightly with each successive generation, that is subject to selection.

    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. Evolution of potassium conductance phenotype under selection for reduced action potential duration using the fitness function shown in Panel G. 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. 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. Fitness function used in D and E. Alternatively, it is possible that there is positive selection for a relatively short action potential duration Fig 3D.

    A fitness function Fig 7G that incorporates the dependence of action potential duration on potassium conductance with appropriate scaling can also result in selection of an average potassium conductance close to the observed value As seen for the sodium conductance, a fitness function with a shallow slope results in greater variation in the cis-regulatory phenotype both over time Fig 7D and within the population Fig 7E. Natural selection is conservative and incremental so that channel expression levels can remain stable across wide phylogenetic distances [ 12 ].

    Nonetheless, specific channel expression levels can also evolve relatively quickly when selection pressures change [ 12 , 13 ]. How sodium and potassium conductances might evolve from a range of starting values towards the observed values maintained by purifying selection was examined. Phenotypic noise distributions for both the sodium and potassium conductance that could maintain average conductance values close to the experimental values over infinite generations were first fitted by iteration, similar to that described for the simulations shown in Fig 5B and 5C and Fig 7B and 7C.

    Simulations were then started from combinations of higher or lower average sodium and potassium conductance values. Over successive generations average, conductance values of the population would reliably converge from any higher or lower combination of starting conductances Fig 8 , blue circles towards to the experimental conductance values Fig 8 , red circles , assuming that the starting population began within the set of successful phenotypes.

    At higher starting values, the accumulating effects of mutation cause the average conductance values to fall towards the final stable values. At low starting points, variation in channel expression causes a significant number of individuals to fall over the fitness cliffs, favoring the selection of individuals with higher conductance values.

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    The effect of this positive selection gradually diminishes as the average conductances in the population converge towards the experimental values and a mutation-selection balance is established. The variation in channel expression values within the population Fig 8B reflects a combination of variation in the underlying cis-regulatory genotype of the two channels combined with variation due to phenotypic noise.

    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 colored traces black, orange and brown mark the path of the average conductances over 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 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. 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.

    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.

    The Molecular Properties and Evolution of Excitable Cells - 1st Edition

    It is this distribution of channel protein expression values, which varies slightly with each successive generation, that is subject to selection. 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. The filled ellipses are centered on the average conductances and the length of the two axes correspond to 4 times the S. Increasing dispersion of population conductance values causes a systematic increase in average channel expression levels Fig 8C. This illustrates an interesting trade-off between energy efficiency and robustness.

    The most energy efficient phenotypes lowest average channel expression levels are found in the population with the least phenotypic variation Fig 8C. However, this population is also the most fragile, since a higher proportion of its individuals are found in close proximity to one of the two fitness cliffs. With increasing population variation in channel expression levels more individuals become robust, in the sense that their phenotype lies further and further away from the fitness cliffs and are, therefore, decreasingly likely to fail, even under unusually adverse circumstances. Both of these properties are expected and have been observed experimentally [ 14 ].

    The linkage between the regulatory evolution model and the model of axonal action potential conduction is based on the relationship between physiological performance and evolutionary fitness. Despite the critical importance of this relationship in shaping much of biological function, it has only been studied experimentally in a few very favorable organisms [ 14 ]. Modeling is the only way to study this issue in most organisms. We consider two basic conditions: regions of discontinuity in the relationship between channel expression levels and physiological performance that might result in fitness cliffs and regions where this relationship is continuous and potential increases in fitness are the result of quantitative changes in physiological performance.

    The performance of cellular electrophysiological systems can be a highly non-linear function of voltage-gated ion channel expression levels, resulting in distinct discontinuities in function [ 47 , 48 ]. For the squid axon, there are two clear functional discontinuities: the success or failure of the axon to transmit an action potential and the transition between Type 3 and Type 2 excitability Figs 2 and 3. Failure of action potential conduction in the giant axon will have a significant effect on predator-escape and prey-capture behavior in the squid [ 39 , 40 ] and it is reasonable to assume that this discontinuity in physiological performance will have a large effect on fitness.

    The effect of the transition between Type 3 and Type 2 excitability on fitness is more ambiguous. Type 3 excitability is conserved in cold water squid [ 21 ], suggesting that this property is critical for the performance of the squid motor circuit. Type 3 neurons allow finer temporal precision than do repetitively firing Type 2 neurons and are important for fine-tuning the response to synaptic input and ensuring one-to-one mapping between the firing of pre- and post-synaptic cells [ 24 ].

    The giant axon is the main site of synaptic integration leading to action potential generation so the subthreshold behavior of the axon can be a critical feature in tuning overall motor circuit performance. A low noise axon may be of particular importance due to the unusually direct linkage between axon firing and key behavioral responses [ 40 , 46 ]. Incorporation of fitness cliffs associated with these two discontinuities in physiological performance into the genetic regulatory model does not by itself result in selection of appropriate channel conductance values Fig 4B.

    There are two sources of noise in the system that can affect channel expression independent of variation in the cis-regulatory function of the sodium and potassium channels. First, because the squid are drawn from a wild population comprised of genetically diverse individuals, there will be genetic variation in background genes, both their expression and function, which can act in trans to produce variation in the expression of the two channel proteins.

    Second, even within genetically identical individuals, protein expression levels can still be subject to multiple sources of noise, particularly for a population such as the squid that develops and lives under widely varying environmental conditions [ 35 , 36 ].

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    The combination of fitness cliffs with phenotypic noise can produce good agreement between the model and experimental observation. This outcome leads to the simple hypothesis that the selection of channel expression values can, in many cases, be explained solely by the requirement that: a electrophysiological function does not fail under the most extreme conditions that might be routinely encountered and b physiological function must be sufficiently robust that it does not fail in the presence of the multiple sources of noise typically found in a natural population.

    The ability to avoid failure under all conditions normally encountered, including the effects of environmental and intrinsic noise, may be a sufficient explanation for the selection of channel conductance levels in the squid axon, and possibly other electrophysiological systems.