​To study the annual invasion of a tree pathogen (Melampsora larici-populina, a fungal species responsible for the poplar rust disease), Xhaard et al (2012) had conducted a spatiotemporal survey along the Durance River valley in the French Alps over nearly 200 km, measuring sampled leaves and twigs from 40 to 150 trees at 12 evenly spaced study sites at seven-time points. By combining Bayesian genetic assignment and a landscape epidemiology approach, they were able to estimate the genetic origin and annual spread of the plant pathogen during a single epidemic.

The observed temporal variation in the spatial pattern of infection rates allowed Saubin et al (2023) to estimate the key factors that determine the speed of the invasion dynamics. In particular, it is crucial to estimate the probability and extent of long-distance dispersal. The dynamics of the macroscale population density was formulated by the reaction-diffusion (R.D.) model and by the integro-difference (I.D.) model. Both consist of the diffusion/dispersal component and the reaction component. In the I.D. model, the kernel function represents the distribution of the dispersion. The likelihood function was obtained by coupling the mathematical model of the population dynamics and the statistical model of the observational process.

Saubin et al (2023) considered a thin-tailed Gaussian kernel, a heavy-tailed exponential kernel, and a fat-tailed exponential power kernel. The numerical simulation reflecting the above survey confirmed the identifiability of the propagation kernel and the accuracy of the parameter estimation. In particular, the above survey had the high power to identify the model with frequent long-distance dispersal. The data from the survey selected the exponential power kernel with confidence. The mean dispersal distance was estimated to be 2.01 km. The exponential power was 0.24. This parameter value predicts that 5% of the dispersals will have a distance > 14.3 km and 1% will have a distance > 36.0 km. The mechanistic-statistical approach presented here may become a new standard for the field-based studies of invasion dynamics.

References

Saubin, M., Coville, J., Xhaard, C., Frey, P., Soubeyrand, S., Halkett, F., and Fabre, F. (2023). A mechanistic-statistical approach to infer dispersal and demography from invasion dynamics, applied to a plant pathogen. bioRxiv, ver. 5 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://doi.org/10.1101/2023.03.21.533642

Xhaard, C., Barrès, B., Andrieux, A., Bousset, L., Halkett, F., and Frey, P. (2012). Disentangling the genetic origins of a plant pathogen during disease spread using an original molecular epidemiology approach. Molecular Ecology, 21(10):2383-2398. https://doi.org/10.1111/j.1365-294X.2012.05556.x

In standard mutation-selection models for describing the dynamics of phenotypically structured populations, it is often assumed that the mutation rate is constant across the phenotypes. In particular, this assumption leads to a constant diffusion coefficient for diffusion approximation models (Perthame, 2007 and references therein).   

Patout et al (2021) study the dependence of the mutation rate on the birth rate, by introducing some diffusion approximations at the population level, derived from the large population limit of a stochastic, individual-based model. The reaction-diffusion model in this article is of the “cross-diffusion” type: The form of “cross-diffusion” also appeared in ecological literature as a type of biased movement behaviors for organisms (Shigesada et al., 1979). The key underlying assumption for “cross-diffusion” is that the transition probability at the individual level depends solely upon the condition at the departure point. Patout et al (2021) envision that a higher birth rate yields more mutations per unit of time. One of their motivations is that during cancer development, the mutation rates of cancer cells at the population level could be correlated with reproduction success.   

The reaction-diffusion approximation model derived in this article illustrates several interesting phenomena: For the time evolution situation, their model predicts different solution trajectories under various assumptions on the fitness function, e.g. the trajectory could initially move towards the birth optimum but eventually end up at the survival optimum. Their model also predicts that the mean fitness could be flat for some period of time, which might provide another alternative to explain observed data. At the steady-state level, their model suggests that the populations are more concentrated around the survival optimum, which agrees with the evolution of the time-dependent solution trajectories.   

Perhaps one of the most interesting contributions of the study of Patout et al (2021) is to give us a new perspective to model the mutation rate in phenotypically structured populations and subsequently, and to help us better understand the connection between mutation and selection. More broadly, this article offers some new insights into the evolutionary dynamics of phenotypically structured populations, along with potential implications in empirical studies.   

References

Perthame B (2007) Transport Equations in Biology Frontiers in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-7643-7842-4_2

Patout F, Forien R, Alfaro M, Papaïx J, Roques L (2021) The emergence of a birth-dependent mutation rate in asexuals: causes and consequences. bioRxiv, 2021.06.11.448026, ver. 3 peer-reviewed and recommended by Peer Community in Mathematical and Computational Biology. https://doi.org/10.1101/2021.06.11.448026

Shigesada N, Kawasaki K, Teramoto E (1979) Spatial segregation of interacting species. Journal of Theoretical Biology, 79, 83–99. https://doi.org/10.1016/0022-5193(79)90258-3