Embracing multimodal optimization to enhance Dynamic Energy Budget parameterization

Abstract

Parameterization is one of the most challenging steps in the construction of individual-based models, and it is particularly relevant for the case of Dynamic Energy Budget (DEB) theory given that DEB parameters are mapped to a multimodal fitness landscape. This multimodal fitness landscape could correspond to parameterizations that provide the right outcome for the wrong reasons. Given the lack of available data to directly parameterize some aspects of DEB models, mathematical tools are becoming the state-of-the-art approach to estimate or refine unknown parameters. The aim of this study is to explore the use of a novel mathematical algorithm that recognizes the multimodal nature of the fitness landscape as a way to provide alternative equally good parameterizations for DEB models. The Multimodal Optimization for Model CAlibration (MOMCA) framework was used to calibrate a DEB model for the blue mussel Mytilus edulis using datasets that included environmental information, growth, and physiological rates. The inclusion of physiological rates, an uncommon approach in DEB parameterization, allowed for constraining the range of solutions, and reducing parameter uncertainty. The application of the MOMCA framework allowed for the identification of the energy acquisition sub-model as one of the top priorities for improving the mechanistic understanding of mussel bioenergetics, and consequently for enhancing model performance. The MOMCA framework could complement the standard procedures to estimate DEB parameters.

Introduction

Dynamic Energy Budget (DEB) theory (Kooijman 2010) mechanistically explains individual bioenergetics throughout the life cycle under dynamic environmental conditions. Building on thermodynamics first-principles and assuming that the mechanisms responsible for the organization of individual metabolism are not species-specific (Sousa et al. 2008), DEB theory can be applied to all species. The potential of such a mechanistic theory has led to an increase in its popularity in the scientific literature, being applied to more than 2,000 species (Add my Pet, https://www.bio.vu.nl/thb/deb/deblab/add_my_pet/), among which bivalves are one of the most studied group (e.g. Pouvreau et al. 2006, Rosland et al. 2009). Despite the successful application of the theory, DEB has been criticized for not being ‘efficient’ due to the large number of parameters, namely 14 in its standard versions, and the fact that most of them are species-specific (Marquet et al. 2014). The complex and data-demanding procedure to estimate the value of these parameters (Lika et al., 2011a) may also limit its expansion. Contrarily, it has been argued that DEB is efficient given the large number of processes that can be derived from those parameters, for instance, life cycle development, feeding, growth, maintenance, metabolic heating, reproduction, and senescence (Kearney et al. 2015). The large suite of predictions that can be derived is precisely one of the strengths of DEB implementation in ecosystem-scale models (e.g. Guyondet et al. 2010).

The need to parameterize such a relatively large number of parameters, most of them species-specific, is one of the most challenging aspects of DEB modelling. Early on, the parameterization of DEB relied on the empirical and independent approximation to each parameter (e.g. van der Veer et al. 2006). Mathematical calibration such as the use of the Nelder-Mead method was also implemented to estimate the values of unknown parameters or to improve model fit in populations of the same species (e.g. Bacher and Gangnery 2006). A breakthrough in DEB parameterization came with the introduction of the “covariation method” (Lika et al. 2011a), a non-linear least squares regression approach. Lika et al. (2011a) suggested that the most robust parameterization would be to simultaneously estimate all DEB parameters by minimizing the discrepancies between simulations and all available observations. These available observations range from single data-points that represent a single value (zero-variate data in DEB jargon) such as maximum reproduction rate, to arrays of values (uni-variate data) such as growth over time. This method also takes into account the inter-dependency of some parameters, avoiding potentially incoherent combinations of parameters. In addition, Lika et al. (2011a) used DEB theory principles to constraint the potential range of solutions, and introduced the concept of pseudo-data, which are well-known values for a generalized animal that could be scaled to the modelled species. The use of pseudo-data has demonstrated to be beneficial to exploit existing knowledge about parameter values that cannot be extracted from observations, and it constitutes a key component to confine potential solutions to those that make sense biologically (Marques et al. 2019). Marques et al. (2019) has replaced Lika et al. (2011a) as the standard procedure to estimate DEB parameters by adding more advanced loss functions and filters to prevent parameter sets that are not compatible with the general principles of the theory.

Despite this progress, there are still major challenges in DEB parameter estimation, which are common to most mechanistic models in biology. For example, interindividual variability is included through population-level averages, but not directly included in the parameter estimation, and similarly, the uncertainty in the estimated parameters is not usually considered (Johnson et al. 2013). Some mathematical approaches have already included this uncertainty by applying a state-space method to determine the probability distribution of the estimated parameters (Fujiwara et al. 2005). Similarly, the use of a Bayesian inference framework allows for the consideration of this uncertainty (Johnson et al. 2013), and it has already been used to estimate DEB parameters (Boersch-Supan and Johnson 2018). Another challenge for a robust estimation in DEB is related to the large number of parameters that could lead to ‘the curse of dimensionality’ type of problems (Jusup et al. 2017). The challenge of dimensionality is that an increase in the number of parameters requires an exponential increase in available empirical data to avoid sparsity and obtain parameters that are statistically significant. This problem is magnified by the fact that some DEB parameters covariate (Fujiwara et al. 2005, Chica et al. 2017), suggesting that different combinations of values can result in the same solution; which could lead to the simulation of the right solutions for the wrong reasons. For example, the same growth could be observed with high feeding and high metabolic costs or with low feeding and low costs. Therefore, the advantage of DEB regarding the potential to simulate a large variety of processes could become a weakness if the available datasets do not allow for a meaningful parameterization that addresses this multimodal optimization problem, understood as a problem without a unique global optimum solution but multiple optima, either global or local (Ehrgott 2005). The multidimensional space of DEB parameters together with data-gaps could be considered one of the major hurdles for DEB parameterization.

Traditionally, the uni-variate data used for DEB calibration focus on growth curves, biometric data, and datasets that can inform about the effect of temperature on the physiology of the individual. Among these datasets, long-term growth curves are usually preferred given the relative simplicity to collect them, their availability in the literature, and the straightforward calculation of growth using DEB outputs; but they lack the power to inform about short-term individual responses. Recent optimization algorithms consider the coexistence of model parameters that are mapped to a multimodal fitness landscape (e.g. MOMCA framework, Chica et al. 2017). They aim to simultaneously locate more than one optimal solution. Subsequently, such approaches represent the nature of DEB theory and they could thus be an ideal way to move forward on DEB parameterization by providing alternative solutions rather than a single one. In this study, a combined dataset (Strohmeier et al. 2015) of growth and physiological rates (respiration and clearance rates) of the mussel Mytilus edulis from two locations of the Lysefjord (Norway) was used to parameterize a DEB model using a novel optimization approach that considers that DEB parameters are mapped to a multimodal fitness landscape (Chica et al. 2017). The inclusion of physiological rates in the parameterization aims to constrain the range of parameters and inform about short-term physiological responses. The ultimate goal of this study is not to provide a generalized set of parameters for M. edulis, but to demonstrate that this novel optimization can provide alternative solutions that could identify knowledge gaps and research priorities for further improvement of DEB parameterization. Furthermore, the provision of alternative equally good solutions becomes an ideal complement to the standard procedures to estimate DEB parameters (Marques et al. 2019).