Dynamics of gene duplication and transposons in microbial genomes following a sudden environmental change

A variety of genome transformations can occur as a microbial population adapts to a large environmental change. In particular, genomic surveys indicate that, following the transition to an obligate, host-dependent symbiont, the density of transposons first rises, then subsequently declines over evolutionary time. Here we show that these observations can be accounted for by a class of generic stochastic models for the evolution of genomes in the presence of continuous selection and gene duplication. The models use a fitness function that allows for partial contributions from multiple gene copies, is an increasing but bounded function of copy number, and is optimal for one fully adapted gene copy. We use Monte Carlo simulation to show that the dynamics result in an initial rise in gene copy number followed by a subsequent falloff due to adaptation to the new environmental parameters. These results are robust for reasonable gene duplication and mutation parameters when adapting to a novel target sequence. Our model provides a generic explanation for the dynamics of microbial transposon density following a large environmental change such as host restriction.

Figure 1

Schematic: Gene activity following a change in environmental conditions. Each gene gives rise to a gene product that has a number of multiple activities that effect organismal fitness differently. Some of these activities are beneficial, while others are deleterious. In general, the selected genes tend to maximize the benefit while minimizing any harmful “side effect.” However, when an organism undergoes a change in its environmental conditions, previously deleterious activities may become beneficial and vice versa.

Figure 2

Schematic: Adaptation to a changing environment. We consider organisms experiencing a large change in environmental conditions. Due to the role of gene duplications in adapting novel functions, these changed conditions result in an enhancement of the IS element density within the genome. Free-living bacteria experience a fluctuating environment, which results in the maintenance of IS density. Host restriction represents a large change in environmental conditions, resulting in an initial boom in IS density, such as that seen in the recent obligate organisms. However, a host also represents a stable and consistent environment. The lack of any need for rapid adaptation leads to near-zero IS density by a process of slow decay.

Figure 3

Schematic: Gene deletion and duplication. (Upper) Deletion can occur via homologous recombination or mismatch repair between nearly identical IS elements by using the IS elements as templates for homologous recombination. (Lower) Gene duplication occurs when a composite transposon that is made up of two flanking IS elements replicates and inserts itself in a different part of the genome.

Figure 4

Average gene number $\langle n\rangle$ with and without gene duplication for simulations with the fitness function given by Eq. (1). The average is given by the darker lines, while the lightly shaded areas represent the area covered by the average standard deviation. In the initial phases of adaptation, gene duplication dominates as the primary mode for enhancing average fitness. As time passes, the slower mode of adaptation provided by sequence mutation refines the genes, and the average number of genes per genome decreases. In the case with no gene duplication or deletion the gene number remains constant. Simulations were carried out with a duplication rate of $1$ per generation, a gene deletion rate of 0.2 per gene per generation, a mutation rate of 0.01 per gene per generation, $N=10$, $c=10$, $\mu =0.05$, and $d=0.2$. We considered a population of $10\hspace{0.5em}000$ organisms and averaged across $100$ replicate simulations with the same parameters but different initial seeds. In the case without gene duplication, we set the gene duplication and deletion rate to $0$ without change to the other parameters.

Figure 5

Average organismal fitness $\langle \mathcal{F}\rangle$ with and without gene duplication. With gene duplication the initial rate of adaptation is faster then in the single-gene case. However, at long times the single-gene case results in genes that are closer to the target sequence $\mathcal{T}$. Parameters are the same as given in Fig. 4.

Figure 6

Average gene fitness $\langle g\rangle$ with and without gene duplication for simulations with the fitness function given by Eq. (1). With gene duplication the initial rate of gene adaptation is faster than in the single-gene case. However, at longer times the single-gene case results in genes that are closer to the target sequence $\mathcal{T}$. Parameters are the same as given in Fig. 4. Note that the horizontal axis scale differs from that shown in Fig. 4.

Figure 7

Average gene fitness $\langle g\rangle$ with and without gene duplication for noise parameter $d=0.0$, $0.2$, $0.4$, and $0.6$ (from top to bottom). With gene duplication the initial rate of gene adaptation is faster then in the single-gene case. However, at longer times the single-gene case results in genes that are closer to the target sequence $\mathcal{T}$. Other parameters are the same as given in Fig. 4.

Figure 8

Rescaled plot of organismal fitness for different mutation rates without gene duplication. Average organismal fitness $\langle F\rangle$ across 100 simulations plotted against a rescaled $x$ axis of generations times mutation rate. Other parameters are the same as given in Fig. 4. This plot shows that the rate of organismal adaptation scales in proportion to the rate at which point mutations are produced.

Figure 9

Rescaled plot of organismal fitness for different mutation rates with gene duplication. Average organismal fitness $\langle F\rangle$ plotted against a rescaled $x$ axis of generations times mutation rate. Other parameters are the same as given in Fig. 4. This plot shows that the rate of organismal adaptation scales in proportion to the rate at which point mutations are produced.

Figure 10

Rescaled plot of organismal fitness for different system sizes with and without gene duplication. Average organismal fitness $\langle F\rangle$ plotted against a rescaled $x$ axis of generations times the natural logarithm of the system size. “No Duplication” indicates a duplication rate of $0.0,$ and “With Duplication‘’ indicates a duplication rate of $1.0$. Mutation rate used in both cases was $0.001$. Other parameters are the same as given in Fig. 4. This plot shows that the rate of organismal adaptation scales in proportion to the logarithm of the system size.

Figure 11

Average gene number $\langle n\rangle$ with and without gene duplication for simulations with fitness function given by Eq. (3). Average was taken over the 418 and 1000 simulation runs that adapted to the target sequence with and without gene duplication, respectively. In the initial phases of adaptation, gene duplication dominates as the primary mode for enhancing average fitness. As time passes, the slower mode of adaptation provided by sequence mutation refines the genes, and the average number of genes per genome decreases. In the case with no gene duplication or deletion the gene number remains constant. Parameters not mentioned above are the same as given in Fig. 4.

Figure 12

Average organismal fitness $\langle {\mathcal{F}}^2\rangle$ with and without gene duplication. Average was taken over the 418 and 1000 simulation runs that adapted to the target sequence with and without gene duplication, respectively. Adaptation with gene duplication is faster then in the single-gene case. However, at long times the single-gene case catches up to the gene duplication case. Parameters are the same as given in Fig. 4.

Figure 13

Average gene fitness $\langle g\rangle$ with and without gene duplication for simulations with the fitness function given by Eq. (3). Average was taken over the 418 and 1000 simulation runs that adapted to the target sequence with and without gene duplication, respectively. Note that the gene duplication scheme requires an initial gene with greater fitness in order to adapt toward the target sequence $\mathcal{T}$. With gene duplication the initial rate of gene adaptation is faster than in the single-gene case. Despite the initial advantage present in the gene duplication case, at longer times $\langle g\rangle$ for the single-gene case reaches similar levels. Parameters are the same as given in Fig. 4.

Figure 14

Gene number $\langle n\rangle$ with gene duplication under changing and constant environments. The fitness function is given by Eq. (1). Example trajectory for changing environments was plotted by changing the target sequence $\mathcal{T}$ to a random sequence at randomly selected intervals according to a $0.04$ probability of change per generation. The curve for constant environment comprises the averaged data from Fig. 4 and is plotted here for comparison purposes. Other parameters are the same as given in Fig. 4. Notice that the number of genes in the changing environment are generally larger than in the case for a constant environment.