Patricia Garnier

V Pot

X Portell

W Otten

O Monga

P C Baveye

Accounting for soil architecture and microbial dynamics in microscale models: Current practices in soil science and the path ahead

Keywords: bacteria models, fungi models, microbial dynamics, microscale models, soil architecture, soil micro-environments, soil organic matter decomposition

Macroscopic models of soil organic matter (SOM) turnover have faced difficulties in reproducing SOM dynamics or in predicting the spatial distribution of carbon stocks. These models are based on a largely inadequate linear response of soil microorganisms to bulk concentrations of nutrients and it is clear that a new approach to SOM modeling is required. Introducing explicit microbial activity and OM reactivity in macroscopic models represents a challenge because of the fine spatial scales at which the processes occur. To get a better grasp on interactions that take place at the micro-scale, a new generation of SOM models have been developed at the spatial scale of the soil micro-environments

Highlights:

• We review the state of the art in the development of microscale models of soil microbial processes

• Microscale models have integrated the tremendous progress in knowledge of OM cycling in soils

• We categorize modeling approaches in how they deal with complexity of soil architecture • Modeling benchmarks are required to properly balance complexity and data uncertainty

1.Introduction

Over the last few years, a significant body of research has been devoted to improving the macroscopic models of the turnover of SOM that are integrated in large-scale C-cycling or climatic models of the Earth. To try to account for sub-macroscopic heterogeneity, these macroscopic SOM turnover models make use of empirical coefficients that are introduced to constrain turnover constants as a function of soil water content and, in rare cases, of physical separation between OM and microbes.

In general, these tuning coefficients appear to provide sufficient degree of freedom to reach good estimates of turnover dynamics under stable pedoclimatic situations. However, these models tend to fail in transient situations, e.g., in response to increases in nitrogen fertilization or soil temperature [START_REF] Sierra | Predicting decadal trends and transient responses of radiocarbon storage and fluxes in a temperate forest soil[END_REF]. A common overall assessment is that these models are not capable of predicting SOM dynamics accurately [START_REF] Davidson | The dual Arrhenius and Michaelis-Menten kinetics model for decomposition of soil organic matter at hourly to seasonal time scales[END_REF][START_REF] Sulman | Multiple models and experiments underscore large uncertainty in soil carbon dynamics[END_REF].

As a result, the assumptions embodied in these models have become increasingly challenged in recent years. Various authors have argued that the idea of splitting SOM into distinct pools based only on chemical composition and degree of recalcitrance no longer corresponds to the latest picture available of the nature of SOM [START_REF] Abramoff | The Millenium model: In search of measurable pools and transformations for modeling soil carbon in the new century[END_REF][START_REF] Basile-Doelsch | Reviews and syntheses: The mechanisms underlying carbon storage in soil[END_REF][START_REF] Blankinship | Improving understanding of soil organic matter dynamics by triangulating theories, measurements, and models[END_REF][START_REF] Dungait | Soil organic matter turnover is governed by accessibility not recalcitrance[END_REF][START_REF] Sainte-Marie | C-STABILITY an innovative modeling framework to leverage the continuous representation of organic matter[END_REF][START_REF] Yang | 4D imaging reveals mechanisms of clay-carbon protection and release[END_REF]. Among the pieces of evidence that led to this perspective, [START_REF] Kleber | Old and stable soil organic matter is not necessarily chemically recalcitrant : implications for modeling concepts and temperature sensitivity[END_REF] showed that the oldest SOM contains a greater proportion of easily decomposable organic molecules, while the younger one contains organic structures that are supposed to be more stable, suggesting that "C age is not necessarily related to molecular structure or thermodynamic stability". Other factors may explain the failure of macroscopic models to predict SOM turnover. Among them, are the physical protection of SOM through associations with poorly crystalline minerals [START_REF] Kleber | Poorly crystalline mineral phases protect organic matter in acid subsoil horizons[END_REF][START_REF] Basile-Doelsch | Reviews and syntheses: The mechanisms underlying carbon storage in soil[END_REF] and carbon saturation controlled by organo-mineral interactions [START_REF] Kleber | A conceptual model of organo-mineral interactions in soils : self-assembly of organic molecular fragments into zonal structures on mineral surfaces[END_REF]. Another avenue that has been explored to improve macroscopic models consists of accounting for microbial growth explicitly [START_REF] Wieder | Explicitly representing soil microbial processes in Earth system models[END_REF] in order to better predict the transient behavior of carbon fluxes during environmental changes [START_REF] Hadas | Reduced CO2 release from decomposing wheat straw under N-limiting conditions: Simulation of carbon turnover[END_REF][START_REF] Lawrence | Does adding microbial mechanisms of decomposition improve soil organic matter models? A comparison of four models using data from a pulsed rewetting experiment[END_REF][START_REF] Sainte-Marie | C-STABILITY an innovative modeling framework to leverage the continuous representation of organic matter[END_REF][START_REF] Woolf | Microbial models with minimal mineral protection can explain longterm soil organic carbon persistence[END_REF]. Another possible explanation for the failure of simple macroscopic models to describe the transient dynamics of carbon and nitrogen fluxes in soils is that these models, from the start, are missing crucial aspects of the functioning of soils. This perspective is in line with the conclusion, reached by microbiologists sixty years ago, that microbial processes in soils are strongly influenced by the physical and chemical properties of microenvironments in which microorganisms reside. Alexander (1965), in particular, opined that without somehow taking into account the microscale heterogeneity of soils and the characteristics of these microenvironments, now also referred to as hotspots [START_REF] Kuzyakov | Microbial hotspots and hot moments in soil : Concept & review[END_REF], it would be impossible to describe accurately the multivarious microbial processes occurring in soils. This opinion has been confirmed by later experiments (e.g., [START_REF] Parkin | Soil microsites as a source of denitrification variability[END_REF], and essentially the same message has been reiterated by [START_REF] Young | Interactions and self-organization in the soil-microbe complex[END_REF].

In the last fifteen years, it has become increasingly acknowledged that the spatially heterogeneous architecture of soil, the uneven distribution of water and air in the soil pores, and the heterogeneous spatial distribution of trophic resources and microorganisms contribute to the emergence of microbial processes in soils that cannot be accounted for solely with the macroscopic measurements that are currently carried out [START_REF] Baveye | Emergent properties of microbial activity in heterogeneous soil microenvironments: Different research approaches are slowly converging, yet major challenges remain[END_REF][START_REF] Heitkötter | Soil zymography as a powerful tool for exploring hotspots and substrate limitation in undisturbed soil[END_REF][START_REF] Joergensen | Alive and kicking: why dormant soil microorganisms matter[END_REF]. During the same time frame, it has also become increasingly clear that the static picture of microorganisms confined to isolation in discrete microenvironments is inaccurate because microorganisms are able to move in order to find their sustenance (e.g., [START_REF] Berg | Chemotaxis in Escherichia coli analysed by three-dimensional tracking[END_REF][START_REF] Juyal | Control of pore geometry in soil microcosm and its effect on the growth and spread of Pseudomonas and Bacillus sp[END_REF][START_REF] Lacal | Bacterial chemotaxis towards aromatic hydrocarbons in Pseudomonas[END_REF][START_REF] Kohlmeier | Taking the fungal highway : mobilization of pollutant-degrading bacteria by fungi[END_REF][START_REF] Van Loosdrecht | Influence of interfaces on microbial activity[END_REF].

To our knowledge, [START_REF] Kuka | Modelling the impact of pore space distribution on carbon turnover[END_REF] were the first to have attempted the development of a model of SOM dynamics that accounts for the influence of pore space distribution. However, their model did not account for positional relationships between the different classes of pores. Although model predictions match remarkably a set of observations in an independent field trial, the parameterization of fluxes necessarily lumps together the interactions between microorganisms and trophic resources, their spatial disconnection, and the role of diffusion and/or advection between the different classes of pores.

To avoid this lumping together a new generation of SOM decomposition models has been developed subsequently, accounting for the three-dimensional architecture of soil and the heterogeneity of the distributions of microorganisms, water, air, SOM and reactive surfaces at the level of microbial habitats [START_REF] Baveye | Emergent properties of microbial activity in heterogeneous soil microenvironments: Different research approaches are slowly converging, yet major challenges remain[END_REF]. In two recent mini-reviews, [START_REF] König | Physical, chemical and biological effects on soil bacterial dynamics in microscale models[END_REF] reported on microscale models with a special focus on spatiotemporal dynamics of bacteria and bacterial functions in soil microhabitats, and [START_REF] Golparvar | Pore-scale modeling of microbial activity: What we have and what we need[END_REF] reported on microscale models of reactive transport modeling in porous media.

In this general context, the key objective of the present review article is to assess the current practices for considering soil architecture and microbial dynamics in microscale models of soil microbial processes. We encompass models aimed at predicting SOM decomposition and greenhouse gases emission, but also models whose aim is to better understand ecological interactions between microbes as well as the emergence of the spatial organization of microbial colonies (bacteria and fungi) and biodiversity. Indeed, these studies participate to the general understanding of SOM turnover since for instance production of enzymes that are key factor in the depolymerization step of large polymers is dependent on ecological and spatial interactions and biodiversity. Although not related to SOM turnover, we also report some studies dealing with biofilm growth in artificial porous media for their advanced understanding of the interactions between bacteria cycle of life and water flow. We focus on i) the main modeling strategies to describe soil architecture according to a complexity gradient in capturing the spatial heterogeneity at the scale of microhabitats (Section 2), and on ii) the main strategies adopted to describe soil microbial dynamics, including that of soil fungi (Section 3). Then, we discuss the advantages and disadvantages of these various strategies in Section 4. We address two major limitations of efforts in this area to date, and discuss critically several options that could be explored to fill knowledge gaps in the modeling of soil microbial functions (Section 5).

Main modeling strategies to describe soil architecture at the microbial habitat scale

To review how the soil micro-environments are considered in modeling efforts, it is convenient to classify the microscale models according to two criteria (Figure 1): i) the nature of the attributes of the grid nodes, and ii) the degree of complexity of the description of soil architecture. From the point of view adopted in this review, microscale models are classified as implicit when the node attributes consist of bulk soil averaged variables: porosity, water content, and effective diffusion coefficients. By contrast, microscale models are classified as explicit, also called "pore-scale" models, when the attributes are Boolean values associated with only one phase, i.e., water, air, or solid, and, for the former two phases, the diffusion coefficients are set equal to the corresponding molecular diffusion coefficients in water or air. Implicit models only allow a simplified description of soil architecture (Section 2.1), whereas the explicit strategy makes it feasible to reproduce actual, detailed soil architectures as well as generate idealized soil architectures (Section 2.2). A mixed strategy considering both implicit and explicit attributes is also sometimes adopted to describe detailed soil architectures (Section 2.3). For each of these distinct approaches, we report how models account for the abiotic processes that distribute air and water in the pore space, or lead to the transport of solutes (dissolved organic carbon (DOC), NO3 -, O2, etc) and gaseous products (CO2 , CH4, N2O), as well as the spread of reactive solutes on soil solid surfaces. Finally, we report how solid and dissolved SOM and microorganisms are spatially located in the soil architecture (Section 2.4). Some of the studies cited in this review sometimes use the same model, either identically the same, or a slightly modified (simpler or more complex) version of it. The technical description of the models, their equations, and the parameters involved, are not always detailed enough to determine precisely where the differences reside. We therefore decided to present this analysis considering each literature source as an independent study.

Implicit microscale models

This strategy maintains the classical macroscopic approach, but it is applied at the level of microhabitats. The micro-environments are spatially discretized in a regular grid and the size of the grid nodes ranges from about 1 μm 3 , i.e., roughly the average size of bacterial cells in soils, to about a few mm 3 (Table S1). This strategy is adopted in 25 out of the 78 studies reported on this review.

Ignoring totally the underlying soil architecture and diffusion processes, [START_REF] Masse | MIOR: An individual-based model for simulating the spatial patterns of soil organic matter microbial decomposition[END_REF] and [START_REF] Nunan | The ecology of heterogeneity: Soil bacterial communities and C dynamics[END_REF] only distributed spherical spots of OM and of microorganisms in a 3D volume.

Then, based on population ecology, the models rely on physical contacts between microbial spots and OM spots to achieve consumption and growth. To account for the spatial heterogeneity of the soil and the variable accessibility of the resource for the microbial decomposers, the size of the microbial spots can be varied [START_REF] Garnier | Modeling the influence of soil-plant residue contact on carbon mineralization: Comparison of a compartmental approach and a 3D spatial approach[END_REF][START_REF] Nunan | The ecology of heterogeneity: Soil bacterial communities and C dynamics[END_REF].

Using a different approach, [START_REF] Folse | Cooperation, competition, and coalitions in enzyme-producing microbes: Social evolution and nutrient depolymerisation rates[END_REF] model in a pseudo-2D grid the cooperation, competition, and formation of coalitions in a community of enzyme-producing microbes. Nutrients (carbon, nitrogen, and phosphorus) are unlimited and do not diffuse, whereas enzymes produced by microorganisms diffuse. Each node of the grid lacks information on its nature (pore or solid), or on its water and air content. All this information is encapsulated in a single effective diffusion coefficient of the soluble substrates. Other studies on the emergence of a spatial structure of bacterial colonies repeat this same approach [START_REF] Kaiser | Microbial community dynamics alleviate stoichiometric constraints during litter decay[END_REF][START_REF] Kaiser | Social dynamics within decomposer communities lead to nitrogen retention and organic matter build-up in soils[END_REF][START_REF] König | Modelling functional resilience of microbial ecosystems: Analysis of governing processes[END_REF][START_REF] König | Spatiotemporal disturbance characteristics determine functional stability and collapse risk of simulated microbial exosystems[END_REF][START_REF] König | Disturbance Size Can Be Compensated for by Spatial Fragmentation in Soil Microbial Ecosystems[END_REF][START_REF] Wang | Emergent properties of organic matter decomposition by soil enzymes[END_REF].

In two other similar studies [START_REF] Chakrawal | Dynamic upscaling of decomposition kinetics for carbon cycling models[END_REF][START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF], the effective diffusion coefficients are calculated via the macroscopic relationship proposed by [START_REF] Millington | Permeability of porous solids[END_REF], which considers hydration. However, spatial information about the water content at each grid node is not provided. This particular aspect is addressed by [START_REF] Evans | Synergistic effects of diffusion and microbial physiology reproduce the Birch effect in a micro-scale model[END_REF], who simulate variably saturated conditions to reproduce the Birch effect. However, soil water content is kept constant across all nodes of the grid. [START_REF] Knudsen | Individual-based approach to modeling hyphal growth of a biocontrol fungus in soil[END_REF] develop a 3D model of hyphal growth under similar conditions.

Prescribed values for water content, nutrient availability, and temperature are attributed to each grid node. However, like [START_REF] Evans | Synergistic effects of diffusion and microbial physiology reproduce the Birch effect in a micro-scale model[END_REF], [START_REF] Knudsen | Individual-based approach to modeling hyphal growth of a biocontrol fungus in soil[END_REF] do not allow these attributes to vary spatially. Thus, although these approaches go a step further to include spatial heterogeneity of hydration conditions in the simulated domain, neither explores this potential. This further step is taken in models studying the coexistence of competitive species under wet-dry cycles [START_REF] Long | Aquatic habitats and diffusion constraints affecting microbial coexistence in unsaturated porous media[END_REF], 2009[START_REF] Šťovíček | Microbial community response to hydrationdesiccation cycles in desert soil[END_REF]. [START_REF] Long | Aquatic habitats and diffusion constraints affecting microbial coexistence in unsaturated porous media[END_REF], 2009) vary the porosity and water content attributes spatially over their grid node according to a log-normal distribution of the variances of the parameters of the water retention curve proposed by [START_REF] Van Genuchten | A closed-form equation for predicting the hydraulic conductivity of unsaturated soils[END_REF]. Later, to describe more complex ecological interactions (competition and mutualism), [START_REF] Kim | Individual-based model of microbial life on hydrated rough soil surfaces[END_REF] calculate the water content of each node with the help of a fractal dimension to describe the roughness of the porous surface and assume that pores have a conical geometry.

In the majority of implicit microscale models, physico-chemical processes are not considered at all.

In the few articles that are exceptions [START_REF] Archie | Spatiotemporal simulations of 2,4-D pesticide degradation by microorganisms in 3D soil-core experiments[END_REF][START_REF] Banitz | Individual-based modelling of soil organic matter in NetLogo: Transparent, user-friendly, and open[END_REF][START_REF] Gras | Individual-based modeling of carbon and nitrogen dynamics in soils:Parameterization and sensitivity analysis of abiotic components[END_REF][START_REF] Gras | Individual-based modelling of carbon and nitrogen dynamics in soils: Parameterization and sensitivity analysis of microbial components[END_REF][START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF], the sorption of dissolved OM components, organic pollutant (pesticide), and/or NH4 + are described as mass transfer processes.

Explicit microscale models

Case of detailed actual soil architecture

The description of soil architecture generally relies on cutting-edge imaging techniques (e.g., [START_REF] Baveye | Emergent properties of microbial activity in heterogeneous soil microenvironments: Different research approaches are slowly converging, yet major challenges remain[END_REF][START_REF] Lucas | Combination of imaging infrared spectroscopy and X-ray computed microtomography for the investigation of bio-and physicochemical processes in structured soils[END_REF], such as through the use of X-ray computed tomography (CT), either at synchrotron facilities or using table-top laboratory equipment. The quality and resolution of the images determines the extent to which they may provide 3D spatial information about, e.g., the geometry of the pore space, or of water and air interfaces, and about the location of solid particulate organic matter (POM). Segmentation of CT images (e.g., [START_REF] Hapca | New local thresholding method for soil images by minimizing grayscale intra-class variance[END_REF][START_REF] Houston | Effect of scanning and image reconstruction settings in X-ray computed micro-tomography on quality and segmentation of 3D soil images[END_REF][START_REF] Schlüter | Image processing of multiphase images obtained via X-ray microtomography: A review[END_REF] is usually carried out in order to assign to each voxel one of the phases of the soil: void or solid, with voids filled with water or air, solid being OM, minerals or matrix.

Of the 78 references reported in this review, only 9 studies encompass the actual architecture of soils, using CT soil images as direct input data (Table S1). Attempts to model fungal growth have considered the architecture of both repacked soil samples [START_REF] Falconer | Biomass recycling: A key to efficient foraging by fungal colonies[END_REF](Falconer et al., , 2015;;[START_REF] Pajor | Modelling and quantifying the effect of heterogeneity in soil physical conditions on fungal growth[END_REF] and undisturbed soil samples [START_REF] Cazelles | Soil fungal dynamics: Parameterisation and sensitivity analysis of modelled physiological processes, soil architecture and carbon distribution[END_REF][START_REF] Falconer | Emergent Behavior of Soil Fungal Dynamics : Influence of Soil Architecture and Water Distribution[END_REF][START_REF] Kravchenko | Fungal colonization in soils with different management histories: Modeling growth in three-dimensional pore volumes[END_REF]), whereas two studies on SOM decomposition by bacteria have used CT information obtained in undisturbed soil samples [START_REF] Portell | Microscale heterogeneity of the spatial distribution of organic matter can promote bacterial biodiversity in soils: Insights from computer simulations[END_REF][START_REF] Vogel | Modeling the effect of soil meso-and macropores topology on the biodegradation of a soluble carbon substrate[END_REF][START_REF] Vogel | To what extent do uncertainty and sensitivity analyses help unravel the influence of microscale physical and biological drivers in soil carbon dynamics models?[END_REF].

Measurements of air-water interfaces in soil using Synchrotron CT are available under very specific conditions (e.g., [START_REF] Brusseau | Measuring air-water interfacial areas with Xray microtomography and interfacial partitioning tracer tests[END_REF][START_REF] Pot | Three-dimensional distribution of water and air in soil pores: Comparison of two-phase two-relaxation-times lattice-Boltzmann and morphological model outputs with synchrotron X-ray computed tomography data[END_REF]. However, models based on an actual soil architecture so far have described unsaturated conditions via a mechanistic approach using a two-phase lattice-Boltzmann model (LBM) that considers Van der Waals forces [START_REF] Shan | Lattice Boltzmann model for simulating flows with multiple phases and components[END_REF], 1994) and capillary forces [START_REF] Raiskinmäki | Spreading dynamics of threedimensional droplets by the lattice-Boltzmann method[END_REF]. Phase separation simulations under static conditions [START_REF] Appert | A liquid-gas model on a lattice[END_REF] can be performed for different water saturation conditions following the approach proposed by [START_REF] Genty | Numerical simulation of 3D liquid-gas distribution in porous media by a twophase TRT lattice Boltzmann method[END_REF]. A 3D map of a continuous LBM fluid density is obtained, the highest and the lowest densities of which can be associated to the water and air phases respectively [START_REF] Portell | Microscale heterogeneity of the spatial distribution of organic matter can promote bacterial biodiversity in soils: Insights from computer simulations[END_REF][START_REF] Vogel | Modeling the effect of soil meso-and macropores topology on the biodegradation of a soluble carbon substrate[END_REF]. Then the soluble carbon substrate diffuses only in the nodes filled with water according to the molecular diffusion coefficient of the substrate in water. The coefficient according to which a fungal colony is assumed to diffuse through soil pores is maximal in the case of lowest density [START_REF] Falconer | Emergent Behavior of Soil Fungal Dynamics : Influence of Soil Architecture and Water Distribution[END_REF].

Case of simplified actual soil architecture

The models described in the previous subsection rely on a huge amount of information contained in soil images, which is constantly increasing due to steady improvements of imaging techniques. Today, 3D CT images that are routinely acquired require several tens of gigabytes of storage for a size of up to about 2000 3 voxels [START_REF] Helliwell | The role of plant species and soil condition in the structural development of the rhizosphere[END_REF][START_REF] Koebernick | High-resolution synchrotron imaging shows that root hairs influence rhizosphere soil structure formation[END_REF][START_REF] Quigley | Patterns and Sources of Spatial Heterogeneity in Soil Matrix From Contrasting Long Term Management Practices[END_REF]San Jose Martinez et al., 2018). The numerical solvers of the pore-scale models that directly include CT images as input data, such as LBM or finite volume resolutions, require large computational and memory capabilities. Thus, a more efficient strategy in terms of computer usage consists of reducing the information amount by simplifying soil architecture. Two directions have been adopted to that end (Figure 1). Either models retain the salient connectivity and size features of the pore space morphology using morphological or irregular pore network models, or they generate grids or pore networks that reproduce the statistical properties of actual soil samples as given by soil bulk measurements like particle size distribution or percolation experiments. The studies adopting this strategy, 18 out of the 78 references, are reported in increasing order of simplification of soil architecture.

Morphological models are based on paving the pore space with an ensemble of simplified geometrical primitives (usually balls, e.g., [START_REF] Vogel | Comparison of a lattice-Boltzmann model, a full-morphology model, and a pore network model for determining capillary pressuresaturation relationship[END_REF]. This is done after having detected the median axes of the pore space. Only the minimal set of maximal balls that make a piecewise approximation of the pore space is retained (e.g., [START_REF] Al-Kharusi | Network extraction from sandstone and carbonate pore space images[END_REF][START_REF] Monga | Representing geometric structures in 3D tomography soil images: application to pore space modelling[END_REF][START_REF] Silin | Pore space morphology analysis using maximal inscribed spheres[END_REF]. This set constitutes the nodes of an adjacency graph considering the neighborhood information of each ball or node where the decomposition of OM is calculated [START_REF] Monga | 3D geometric structures and biological activity: Application to microbial soil organic matter decomposition in pore space[END_REF][START_REF] Monga | Using pore space 3D geometrical modelling to simulate biological activity : Impact of soil structure[END_REF][START_REF] Monga | Simulating microbial degradation of organic matter in a simple porous system using the 3-D diffusion-based model MOSAIC[END_REF][START_REF] Ngom | Extraction of three-dimensional soil pore space from microtomography images using a geometrical approach[END_REF]. This ensemble is much more compact in terms of numerical data than all the voxels of the CT image. Other 3D geometric primitives such as ellipsoids and generalized cylinders [START_REF] Ngom | 3D shape extraction segmentation and representation of soil microstructures using generalized cylinders[END_REF][START_REF] Kemgue | From spheres to ellipsoids: Speeding up considerably the morphological modeling of pore space and water retention in soils[END_REF] have also been proposed, however they have not yet been used to simulate SOM decomposition. In this strategy, unsaturated conditions are modeled using the Young-Laplace law. The geometric primitives whose size does not exceed the size that would be expected to drain at a prescribed water potential are filled with water leading to an explicit 3D distribution of water and air at the nodes of the graph [START_REF] Pot | Three-dimensional distribution of water and air in soil pores: Comparison of two-phase two-relaxation-times lattice-Boltzmann and morphological model outputs with synchrotron X-ray computed tomography data[END_REF]. The soluble carbon substrate diffuses only in the water-filled nodes according to the molecular diffusion coefficient of the substrate in water [START_REF] Monga | Simulating microbial degradation of organic matter in a simple porous system using the 3-D diffusion-based model MOSAIC[END_REF].

Irregular pore-network (PN) models are another valuable alternative to reduce the amount of information contained in the soil images to a more manageable size. These models consist of an ensemble of nodes connected to each other by links whose length, radius and number can be varied [START_REF] Blunt | Flow in porous media--pore-network models and multiphase flow[END_REF][START_REF] Xiong | Review of pore network modelling of porous media: Experimental characterisations, network constructions and applications to reactive transport[END_REF]. The generated networks also keep most of the characteristics of the pore space morphology: the pore size distribution, a mean degree of pore branching or connectivity (e.g., [START_REF] Vogel | Quantitative morphology and network representation of soil pore structure[END_REF], and the spatial heterogeneity of the clustering of pores and connectivity (e.g., [START_REF] Blunt | Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow[END_REF]. While these irregular PN models are often used nowadays in computer fluid mechanics (e.g., [START_REF] Martínez-Mendoza | Capillary pressure and relative permeability estimation for low salinity waterflooding processes using pore network models[END_REF], the model of Perez-Reche et al. ( 2012) is the only one that takes advantage of the potential of this approach to simulate microbial invasion in soils.

Simpler PN models based on a regular arrangement of the nodes are more often used, e.g., facecentered cubic grids that retain only the pore size distribution and the mean degree of pore branching of a soil. To simulate growth in cases involving a single bacterial cluster to several aerobic and anaerobic communities, [START_REF] Ebrahimi | Microbial dispersal in unsaturated porous media: Characteristics of motile bacterial cell motions in unsaturated angular pore networks[END_REF], 2015, 2016, 2017, 2018) generate 3D regular angular PNs where the coordination number of the nodes, the size distribution of the links and the porosity are defined according to natural soil systems values or fitted in order to recover the van Genuchten (1980) water retention curve parameters of a given soil (Table S1). The width and spanning angle of triangular links can also be calibrated to soil water retention curves [START_REF] Rosenzweig | A Channel network model as a framework for characterizing variably saturated flow in biofilm-affected soils[END_REF][START_REF] Rosenzweig | Modeling biofilm dynamics and hydraulic properties in variably saturated soils using a channel network model[END_REF]. This strategy is much less representative of soil architecture than the approaches using irregular PN models or morphological models, as noted by [START_REF] Blunt | Detailed physics, predictive capabilities and macroscopic consequences for pore-network models of multiphase flow[END_REF] and [START_REF] Ebrahimi | Genetic algorithm-based pore network extraction from micro-computed tomography images[END_REF]. However, since regular PNs do not describe the spatial arrangement of pores, some links can be randomly removed to create spatial heterogeneity [START_REF] Ebrahimi | Microbial dispersal in unsaturated porous media: Characteristics of motile bacterial cell motions in unsaturated angular pore networks[END_REF]. The length of the links can be changed to account for tortuosity using helical bonds [START_REF] Ebrahimi | Microbial community dynamics in soil aggregates shape biogeochemical gas fluxes from soil profiles -upscaling an aggregate biophysical model[END_REF]. As with morphological models, the Young-Laplace law is used to explicitly distribute water and air in the PN. The choice of a triangular crosssection of the links makes it possible to more realistically simulate variable water contents than would be possible if the links were spherical or cylindrical. In the latter cases, the application of the Young-Laplace law leads to a binary response (empty or water-filled) and can thus promote greater fragmentation of the liquid phase.

The flexibility of the latter approach permits also to generate multi-scale porous systems. Taking advantage of a commercially available and widely used PN simulator (Xpore), and its predecessor (Pore-cor), [START_REF] Laudone | A model to predict the effects of soil structure on denitrification and N2O emission[END_REF] develop a dual-porosity PN model. The positional relationships of the macroporosity and microporosity are generated in order to fit several fundamental properties of soil, namely water retention, aggregate size distribution, porosity of the microporous and macroporous zones, and percolation characteristics [START_REF] Laudone | A dual-porous, inverse model of water retention to study biological and hydrological interactions in soil[END_REF][START_REF] Matthews | Use of a void network model to correlate porosity, mercury porosimetry, thin section, absolute permeability, and NMR relaxation time data for sandstone rocks[END_REF]. We observe that it is also one of the few models that consider diffusion of gases (CO2, N2O and N2) emitted within the pore space.

Finally soil architecture can be described by using a regular grid whose node attributes are associated to pore or solid phases on the basis of statistical estimates of the pore sizes derived from a set of published CT images of soil aggregates [START_REF] Resat | Modeling Microbial Dynamics in Heterogeneous Environments: Growth on Soil Carbon Sources[END_REF]. However, in this approach, the actual arrangement of pores is lost as pores are randomly placed on the grid with an additional constraint that pores be continuously connected. Initial random assignment of nodes into pore and solid phases are also considered in two pore-scale models that simulate temporal modifications of soil architecture under microbial activity. Over time, solid nodes with a weak structure are spatially rearranged to maximize the overall stability of solid node clusters [START_REF] Crawford | Microbial diversity affects self-organization of the soil -microbe system with consequences for function[END_REF][START_REF] Ray | Discrete-continuum multiscale model for transport, biomass development and solid restructuring in porous media[END_REF]. This stability is defined according to the local rate of production of stabilizing agents, whose concentration is a function of the local respiration rate.

Case of idealized soil architecture

Columns filled with glass beads have occasionally been used to physically mimic soil architectures.

To describe these artificial packings, imaging (Graf von der [START_REF] Graf Von Der Schulenburg | Three-Dimensional Simulations of Biofilm Growth in Porous Media[END_REF][START_REF] Peszynska | Biofilm growth in porous media: Experiments, computational modeling at the porescale, and upscaling[END_REF][START_REF] Pintelon | The effect of biofilm permeability on bio-clogging of porous media[END_REF] or 2D mathematical algorithms of grain arrangement [START_REF] Kapellos | Hierarchical simulator of biofilm growth and dynamics in granular porous materials[END_REF] can be used as input data. With such algorithms it is also easy to introduce spatial heterogeneity, such as the occurrence of fractures [START_REF] Tartakovsky | Pore-Scale Model for Reactive Transport and Biomass Growth[END_REF]. A regular PN has also been generated based on pore size distribution of a column packed with glass beads [START_REF] Gharasoo | A reactive transport modeling approach to simulate biogeochemical processes in pore structures with pore-scale heterogeneities[END_REF][START_REF] Gharasoo | How the chemotactic characteristics of bacteria can determine their population patterns[END_REF]. These geometries are mostly used to study the permeability reduction due to the accumulation of biomass and its metabolic by-products in coarse-textured porous media under water-saturated conditions. Water flow, advection, and reaction-diffusion of soluble substrate are thus key processes in addition to biological processes. To consider the higher viscosity of the biomass and associated exopolymers compared to that of water, the coefficient of diffusion of the solutes through it can be reduced compared to that in water (e.g., [START_REF] Tang | An improved pore-scale biofilm model and comparison with a microfluidic flow cell experiment[END_REF].

To create synthetic rough, porous surfaces, 2D regular PN models with triangular channels are generated based on typical pore sizes and connectivity. Variable water-saturated conditions can be calculated with the Young-Laplace law. This approach has been adopted to model the motility of bacteria and expansion of bacterial colonies [START_REF] Wang | Aqueous films limit bacterial cell motility and colony expansion on partially saturated rough surfaces[END_REF], as well as the coexistence [START_REF] Borer | Spatial organization of bacterial populations in response to oxygen and carbon counter-gradients in pore networks[END_REF][START_REF] Long | Microbial growth on partially saturated rough surfaces: Simulations in idealized roughness networks[END_REF][START_REF] Wang | A Hydration-Based Biophysical Index for the Onset of Soil Microbial Coexistence[END_REF][START_REF] Hecht | New Developments in FreeFem++[END_REF] and self-organization [START_REF] Wang | Trophic interactions induce spatial self-organization of microbial consortia on rough surfaces[END_REF] of bacteria. [START_REF] Wang | A Hydration-Based Biophysical Index for the Onset of Soil Microbial Coexistence[END_REF][START_REF] Hecht | New Developments in FreeFem++[END_REF], simplify the time-consuming explicit calculation of solute diffusion-reaction processes in the links of the PN by directly calculating an effective diffusion coefficient at each node using the [START_REF] Millington | Permeability of porous solids[END_REF] expression using the node water status as input. This simplification brings this contribution closer to an implicit description of soil architecture (see Section 2.1).

Random assignment of pore and solid nodes is also commonly used to generate soil analogs. [START_REF] Falconer | Modelling interactions in fungi[END_REF] create porous media of porosity varying between 0 and 1 and simulate fungal invasion. Other authors have considered instead a random assignment of collections of nodes made impervious to mycelium and nutrient diffusion [START_REF] Boswell | The Development of Fungal Networks in Complex Environments[END_REF][START_REF] Boswell | Modelling mycelial networks in structured environments[END_REF][START_REF] Boswell | Linking hyphal growth to colony dynamics: Spatially explicit models of mycelia[END_REF][START_REF] Vidal-Diez De Ulzurrun | Modelling three-dimensional fungal growth in response to environmental stimuli[END_REF]. In a simpler form, the fungus models of [START_REF] Boswell | Growth and Function of Fungal Mycelia in Heterogeneous Environments[END_REF] and [START_REF] Boswell | Linking hyphal growth to colony dynamics: Spatially explicit models of mycelia[END_REF] do not consider impermeable materials but a spatially heterogeneous initial distribution of the carbon substrate in the form of different tessellations. In these cases, diffusion of the substrate external to the fungus is not permitted.

Different sizes and lengths of geometrically highly simplified, linear pores in 2D are used to assess the role of salient soil pore geometry, from shallow pores to deep crevices with small opening, on the growth and survival of bacteria [START_REF] Resat | Modeling Microbial Dynamics in Heterogeneous Environments: Growth on Soil Carbon Sources[END_REF]. Using the COMSOL Multiphysics software applied to a pore of uniform geometry, [START_REF] Schmidt | Elucidating the impact of micro-scale heterogeneous bacterial distribution on biodegradation[END_REF] assess the influence of the spatial location of bacterial colonies along the pore on the biodegradation of an organic substrate that moves through it.

Finally, experimental micro-models of diverse geometrical forms like a succession of rectangular water-saturated microhabitats [START_REF] Keymer | Bacterial metapopulations in nanofabricated landscapes[END_REF], packing of cylinders, and etched-glass networks [START_REF] Borer | Modeling metabolic networks of individual bacterial agents in heterogeneous and dynamic soil habitats (IndiMeSH)[END_REF] have been reproduced in a number of computational models [START_REF] Borer | Modeling metabolic networks of individual bacterial agents in heterogeneous and dynamic soil habitats (IndiMeSH)[END_REF][START_REF] Centler | Modeling population patterns of chemotactic bacteria in homogeneous porous media[END_REF][START_REF] Knutson | Pore-scale simulation of biomass growth along the transverse mixing zone of a model two-dimensional porous medium[END_REF][START_REF] Tang | An improved pore-scale biofilm model and comparison with a microfluidic flow cell experiment[END_REF]. Due to simplification of the reproduced micro-models into a 1D grid, [START_REF] Centler | Modeling population patterns of chemotactic bacteria in homogeneous porous media[END_REF] use effective diffusion coefficients for the substrate and bacteria, together with effective chemotactic sensitivity constants towards the substrate and the bacteria.

Mixed explicit/implicit microscale models

A final strategy mixing implicit and explicit attributes of the grid nodes to describe actual soil architecture from soil imaging is adopted in two studies [START_REF] Yan | Pore-scale investigation on the response of heterotrophic respiration to moisture conditions in heteregeneous soils[END_REF][START_REF] Yan | Microscale water distribution and its effects on organic carbon decomposition in unsaturated soils[END_REF]. In this approach, a dual threshold segmentation of the CT images is used to obtain ternary images where the gray voxels whose value lies between the two thresholds that define voids and solids are associated with a mix of pore and solid phases. These latter nodes have an implicit porosity attribute whose value depends on the gray level scale (Figure 1). The diffusion coefficients of the dissolved OM, O2 and CO2 are equal to their molecular diffusion coefficient in liquid and air phase for the explicit water and air voxels, respectively. Whereas for the implicit unsaturated voxels, the diffusion coefficients are calculated according to empirical laws [START_REF] Archie | Spatiotemporal simulations of 2,4-D pesticide degradation by microorganisms in 3D soil-core experiments[END_REF][START_REF] Hunt | Continuum percolation theory for water retention and hydraulic conductivity of fractal soils: Estimation of the critical volume fraction for percolation[END_REF]. Mixed explicit and implicit diffusion coefficient attributes of the grid nodes are thus used in this approach. Additionally, the sorption processes of OM are described by mass transfer rates using effective surface reactivity parameters.

Spatial description of the initial distribution of SOM and microorganisms in the reviewed models

It is critical to consider how OM is distributed within the soil architecture. Although combined staining and/or chemical and imaging techniques are promising tools (e.g., [START_REF] Maenhout | Chemical staining of particulate organic matter for improved contrast in soil X-ray μ CT images[END_REF][START_REF] Peth | Localization of soil organic matter in soil aggregates using synchrotron-based Xray microtomography[END_REF][START_REF] Roose | Challenges in imaging and predictive modeling of rhizosphere processes[END_REF] they are far yet from routinely providing data. Therefore, the placement of SOM has been mainly considered through modeling scenarios. When SOM consists of dissolved nutrients, it is most of the time homogeneously applied at the nodes of the grid (e.g., [START_REF] König | Spatiotemporal disturbance characteristics determine functional stability and collapse risk of simulated microbial exosystems[END_REF] and/or applied at specific locations in the simulated domain to produce gradients or counter gradients between carbon and O2 (e.g., [START_REF] Ebrahimi | Hydration and diffusion processes shape microbial community organization and function in model soil aggregates[END_REF]. Metabolites or enzymes produced by microbial activity are located at the nodes of the producers (e.g., [START_REF] Folse | Cooperation, competition, and coalitions in enzyme-producing microbes: Social evolution and nutrient depolymerisation rates[END_REF], from where they diffuse.

When SOM consists of solid pieces, different distribution scenarios have been used ranging from aggregated to dispersed distributions (e.g., Falconer et al., 2015). Likewise, microorganisms are either homogeneously located in the simulated domain (e.g., [START_REF] Gharasoo | How the chemotactic characteristics of bacteria can determine their population patterns[END_REF], randomly distributed (e.g., [START_REF] Portell | Microscale heterogeneity of the spatial distribution of organic matter can promote bacterial biodiversity in soils: Insights from computer simulations[END_REF], or locally placed (e.g., Ebrahimi andOr, 2015, Falconer et al., 2015;[START_REF] Kravchenko | Fungal colonization in soils with different management histories: Modeling growth in three-dimensional pore volumes[END_REF], and the effect of different distributions can be explored. They can also be placed according to the size of pores (e.g., [START_REF] Monga | Using pore space 3D geometrical modelling to simulate biological activity : Impact of soil structure[END_REF]. Models using explicit attributes and describing actual soil architectures tend to locate bacteria in water-filled nodes where they are assumed to live (e.g., [START_REF] Vogel | Modeling the effect of soil meso-and macropores topology on the biodegradation of a soluble carbon substrate[END_REF] while fungi preferentially invade air-filled nodes (e.g., [START_REF] Falconer | Emergent Behavior of Soil Fungal Dynamics : Influence of Soil Architecture and Water Distribution[END_REF]. Only one model [START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF] uses different probabilistic distributions based on [START_REF] Raynaud | Spatial ecology of bacteria at the microscale in soil[END_REF] who integrate data in the visualization of microorganisms in soils [START_REF] Nunan | Spatial distribution of bacterial communities and their relationships with the micro-architecture of soil[END_REF].

Main modeling strategies to describe the microbial dynamics in microscale models

In this section, we focus first on the nature of SOM pools, the type of microorganisms and the strategies used to distribute them spatially in their microhabitats. Then we focus on biological processes, encompassing substrate decomposition, microbial substrate uptake, microbial growth, microbial metabolic processes and mobility. It would make sense to report the microbial dynamics modeling approaches in terms of the soil spatial representation since the latter is at the heart of the reported microscale models. However, the number and the degree of details of the biological processes are not linked with the degree of representativeness of actual soil architecture or the choice of implicit or explicit grid node attributes. Rather, it is the research question to which all these microscale models try to answer that has dictated the strategy. This is evidenced in Table 1 which lists the references from the most to the least accurate soil spatial representation. Table 1 provides details of each of the biological processes, the nature of SOM pools, microorganisms and modeling strategy to describe microorganisms colonies.

Nature of substrates and microorganisms

Most of the studies (51 out of the 78 references) consider a single simple soluble substrate (monomer) that is directly taken up by microbes (Table S1, e.g., [START_REF] Long | Microbial growth on partially saturated rough surfaces: Simulations in idealized roughness networks[END_REF]. Other studies (24 references) consider a preliminary step of decomposition (depolymerization) of complex solid substrate (polymers) into soluble organic compounds that are taken up by microbes (e.g., [START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF]. Of the 78 references, two studies do not specify the nature of microorganisms [START_REF] Perez-Reche | Prominent effect of soil network heterogeneity on microbial invasion[END_REF][START_REF] Crawford | Microbial diversity affects self-organization of the soil -microbe system with consequences for function[END_REF], and only 13 relate to filamentous fungi, while the others relate to bacteria, including 9 references that deal with accumulation of bacterial biomass and associated exopolymers.

Bacteria models generally simulate a single specific species or strain (in 47 references, e.g., [START_REF] Knutson | Pore-scale simulation of biomass growth along the transverse mixing zone of a model two-dimensional porous medium[END_REF], or up to eight different genotypes [START_REF] Folse | Cooperation, competition, and coalitions in enzyme-producing microbes: Social evolution and nutrient depolymerisation rates[END_REF] or 10 different taxa [START_REF] Nunan | The ecology of heterogeneity: Soil bacterial communities and C dynamics[END_REF]. Fungi models simulate a single species except for [START_REF] Falconer | Modelling interactions in fungi[END_REF] that are the only ones to study more than one fungal species, comparing compatible and/or antagonist species. Most bacterial models describe i) generic functional groups such as anaerobic bacteria, aerobes, methanogens (e.g. [START_REF] Ebrahimi | Dynamics of soil biogeochemical gas emissions shaped by remolded aggregate sizes and carbon configurations under hydration cycles[END_REF], ii) generic competitive species such as competitive, generalists, or cheaters for the same resource (e.g. [START_REF] Kaiser | Microbial community dynamics alleviate stoichiometric constraints during litter decay[END_REF][START_REF] Monga | 3D geometric structures and biological activity: Application to microbial soil organic matter decomposition in pore space[END_REF][START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF] or from a range of specialists of a single resource to generalists of up to ten different resources [START_REF] Nunan | The ecology of heterogeneity: Soil bacterial communities and C dynamics[END_REF]. Three studies consider commensal or mutualistic relations in microbial consortia. In their simplest form, the models consider a chain of up to three substrates and microbe species, with one of the species dependent on one substrate produced by other species [START_REF] Wang | Trophic interactions induce spatial self-organization of microbial consortia on rough surfaces[END_REF][START_REF] Kim | Individual-based model of microbial life on hydrated rough soil surfaces[END_REF]. In its most complex form, the model of [START_REF] Borer | Modeling metabolic networks of individual bacterial agents in heterogeneous and dynamic soil habitats (IndiMeSH)[END_REF] which is based on the COMETS model of [START_REF] Harcombe | Metabolic resource allocation in individual microbes determines ecosystem interactions and spatial dynamics[END_REF], encompasses a reduced metabolic network involving more than a hundred metabolites and reactions.

Modeling strategies to describe microbial colonies

Case of bacteria

About a third of the references (21 out of 62 studies dealing with bacteria) model the life cycle processes at the scale of bacterial colonies that are spatially distributed in the soil pore space. This population-based strategy calculates the growth of a single (lumped) mass of bacteria representative of the microorganisms contained in a given microhabitat. Cell division is not considered and mortality processes (starvation or predation) are assumed to be continuous processes (e.g., [START_REF] Vogel | Modeling the effect of soil meso-and macropores topology on the biodegradation of a soluble carbon substrate[END_REF].

The updating scheme is synchronous, i.e., uptake, growth, mortality occur at the same time for the whole microcolony.

Another strategy, adopted by 37 references, models the life cycle processes at the scale of individuals, which are the bacteria cells (e.g., Graf von der [START_REF] Graf Von Der Schulenburg | Three-Dimensional Simulations of Biofilm Growth in Porous Media[END_REF]. In this individualbased, or agent-based modeling approach [START_REF] Hellweger | A bunch of tiny individuals -Individual-based modeling for microbes[END_REF], cell division processes are discrete events, a feature that facilitates the discretization of the spatial expansion of the population.

Starvation mortality or predation processes are discrete events (the whole cell dies, not a fraction of it).

In addition, interspecific and intraspecific variability in the individual properties are considered (e.g., size or mass threshold to start cell division). Synchronous updating (e.g., [START_REF] Kreft | BacSim, a simulator for individual-based modelling of bacterial colony growth[END_REF] or asynchronous updating (e.g, [START_REF] Ginovart | Individual-based modelling of microbial activity to study mineralization of C and N and nitrification process in soil[END_REF] schemes can be used. Despite the fact that, in the presence of recycling processes, differences can appear between both schemes [START_REF] Caron-Lormier | Asynchronous and synchronous updating in individual-based models[END_REF], the type of updating of individual-based models is not always stated, which complicates reimplementation and, ultimately, reproducibility of the research. Meta-agents, or "superindividuals", i.e., ensembles of single bacterial cells, are used to mitigate the large computation costs associated with the description of processes at the single-cell level [START_REF] Banitz | Individual-based modelling of soil organic matter in NetLogo: Transparent, user-friendly, and open[END_REF][START_REF] Borer | Modeling metabolic networks of individual bacterial agents in heterogeneous and dynamic soil habitats (IndiMeSH)[END_REF][START_REF] Masse | MIOR: An individual-based model for simulating the spatial patterns of soil organic matter microbial decomposition[END_REF]. This coarse-grained description is also used by [START_REF] Long | Aquatic habitats and diffusion constraints affecting microbial coexistence in unsaturated porous media[END_REF] who adopt the random walker model developed by [START_REF] Ben-Jacob | Generic modelling of cooperative growth patterns in bacterial colonies[END_REF]. Using this approach, the question remains how to identify the size of the meta-agents that makes it possible to retain the influence of the individual behavior of bacteria.

Case of fungi

The fungal "individuality" is much more difficult to define as compared to bacteria. Fungi are multicellular organisms that spread in the pore space. Fungi grow as an interconnected network of hyphae, named mycelium or fungal colony. The shape of the whole fungal colony thus needs to be considered. None of the reviewed fungal models considers single cells. Continuous models and individual-based models of fungal growth have been distinguished [START_REF] Boswell | Linking hyphal growth to colony dynamics: Spatially explicit models of mycelia[END_REF], Table S1). In the former approach, the fungal mycelium is considered as a continuous biomass (e.g., [START_REF] Boswell | Growth and Function of Fungal Mycelia in Heterogeneous Environments[END_REF][START_REF] Falconer | Biomass recycling: A key to efficient foraging by fungal colonies[END_REF], whereas the individual-based approach considers individual hyphae explicitly and describes the discrete (random) events involved when they grow and branch (e.g., [START_REF] Knudsen | Individual-based approach to modeling hyphal growth of a biocontrol fungus in soil[END_REF][START_REF] Boswell | The Development of Fungal Networks in Complex Environments[END_REF].

Description of biological processes

SOM hydrolysis

In some models (e.g., [START_REF] Long | Aquatic habitats and diffusion constraints affecting microbial coexistence in unsaturated porous media[END_REF], the depolymerization step of SOM is assumed to be independent of microbial growth and be described by a fixed hydrolysis rate. Often, models assume that depolymerization depends on enzyme production (e.g., [START_REF] Allison | Cheaters, diffusion and nutrients constrain decomposition by microbial enzymes in spatially structured environments[END_REF] or microbial growth (e.g., Falconer et al., 2015). Complex biogeochemical reactions including mixed kinetics and thermodynamic equilibrium reactions, such as oxic degradation of SOM [START_REF] Aguilera | A knowledge-based reactive transport approach for the simulation of biogeochemical dynamics in Earth systems[END_REF], are calculated for the soluble substrate, using the Biogeochemical Reaction Network Simulator in one study [START_REF] Gharasoo | A reactive transport modeling approach to simulate biogeochemical processes in pore structures with pore-scale heterogeneities[END_REF].

The mathematical description of SOM hydrolysis follows the three conceptual classes described by [START_REF] Wutzler | Colimitation of decomposition by substrate and decomposers -a comparison of model formulations[END_REF], which represent respectively no-, linear-and non-linear dependencies on decomposer biomass. In the first class, constant rates or more often Michaelis-Menten kinetics (e.g., [START_REF] Kaiser | Microbial community dynamics alleviate stoichiometric constraints during litter decay[END_REF] are used. The latter permits an indirect dependence on biomass uptake and growth through microorganisms that produce enzymes. [START_REF] Chakrawal | Dynamic upscaling of decomposition kinetics for carbon cycling models[END_REF] use the three conceptual classes in a comparative study.

Microbial substrate uptake and growth

In its simpler form, and depending on the research question, growth of biomass is neglected (in six references, e.g., [START_REF] Laudone | A model to predict the effects of soil structure on denitrification and N2O emission[END_REF][START_REF] Crawford | Microbial diversity affects self-organization of the soil -microbe system with consequences for function[END_REF] and a constant biomass value is used. A large number of studies (36 references) use the Monod equation to describe the non-linear growth of decomposer biomass (e.g., [START_REF] Banitz | Assessing biodegradation benefits from dispersal networks[END_REF] whereas linear kinetics is adopted in eight references (e.g., [START_REF] Boswell | Modelling mycelial networks in structured environments[END_REF][START_REF] Gharasoo | How the chemotactic characteristics of bacteria can determine their population patterns[END_REF]. The logistic growth is also described in two studies [START_REF] Centler | Modeling population patterns of chemotactic bacteria in homogeneous porous media[END_REF][START_REF] Gharasoo | How the chemotactic characteristics of bacteria can determine their population patterns[END_REF]. Fixed rates are used to calculate growth in the six references that adopt the offer-demand approach (e.g., [START_REF] Masse | MIOR: An individual-based model for simulating the spatial patterns of soil organic matter microbial decomposition[END_REF][START_REF] Monga | 3D geometric structures and biological activity: Application to microbial soil organic matter decomposition in pore space[END_REF]. In five studies, growth is calculated from metabolic pathway reactions (e.g., [START_REF] Gras | Individual-based modelling of carbon and nitrogen dynamics in soils: Parameterization and sensitivity analysis of microbial components[END_REF][START_REF] Borer | Modeling metabolic networks of individual bacterial agents in heterogeneous and dynamic soil habitats (IndiMeSH)[END_REF]. Finally, in one case, growth is implicitly modeled by increasing the size of the area where the cell consumes the resource assuming the increase to be an inverse function of the catabolic breadth of the cell taxon, i.e., the range of organic substrates that can be used [START_REF] Nunan | The ecology of heterogeneity: Soil bacterial communities and C dynamics[END_REF].

Microbial substrate uptake and growth can also be limited by the availability of other nutrients (N, P) and O2. Two approaches are adopted. Either microbe stoichiometry constraint limits uptake (e.g., [START_REF] Allison | Cheaters, diffusion and nutrients constrain decomposition by microbial enzymes in spatially structured environments[END_REF] or the concentration of the limiting factor is introduced in the Monod formalism. In the latter case, the dual-Monod equation is generally used (e.g., [START_REF] Yan | Pore-scale investigation on the response of heterotrophic respiration to moisture conditions in heteregeneous soils[END_REF] or the minimum of the Monod equations established for each substrate [START_REF] Borer | Spatial organization of bacterial populations in response to oxygen and carbon counter-gradients in pore networks[END_REF][START_REF] Kim | Individual-based model of microbial life on hydrated rough soil surfaces[END_REF]. In one reference, a multi-substrate Monod kinetics is used to describe the uptake of small polymers and monomers [START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF].

Microbial metabolism

Almost two third of the references consider microbial maintenance (e.g., [START_REF] Boswell | Growth and Function of Fungal Mycelia in Heterogeneous Environments[END_REF][START_REF] Resat | Modeling Microbial Dynamics in Heterogeneous Environments: Growth on Soil Carbon Sources[END_REF]. Growth respiration is generally implicitly considered by using the assimilation yield, whereas more rarely it is integrated in a lumped respiration term that does not distinguish maintenance and growth respiration processes (e.g., [START_REF] Vogel | To what extent do uncertainty and sensitivity analyses help unravel the influence of microscale physical and biological drivers in soil carbon dynamics models?[END_REF]. The production of metabolites by microorganisms can be directly calculated in the form of enzymes (e.g., [START_REF] Evans | Synergistic effects of diffusion and microbial physiology reproduce the Birch effect in a micro-scale model[END_REF], byproducts [START_REF] Kim | Individual-based model of microbial life on hydrated rough soil surfaces[END_REF], generic inhibitor [START_REF] Falconer | Modelling interactions in fungi[END_REF][START_REF] Kravchenko | Fungal colonization in soils with different management histories: Modeling growth in three-dimensional pore volumes[END_REF] or sticky agents [START_REF] Ray | Discrete-continuum multiscale model for transport, biomass development and solid restructuring in porous media[END_REF], which can undergo transformation (decay) or sorption processes.

The production of greenhouse gases is not always modeled (27 references), and when it is, the majority of studies deal only with CO2 (e.g., [START_REF] Nguyen-Ngoc | Modeling Microbial Decomposition in Real 3D Soil Structures Using Partial Differential Equations[END_REF], whereas a minority also calculate denitrification and emission of N2O [START_REF] Ebrahimi | Hydration and diffusion processes shape microbial community organization and function in model soil aggregates[END_REF], 2016, 2018;[START_REF] Laudone | A model to predict the effects of soil structure on denitrification and N2O emission[END_REF] or CH4 [START_REF] Ebrahimi | Mechanistic modeling of microbial interactions at pore to profile scale resolve methane emission dynamics from permafrost soil[END_REF], 2018).

Microbial death, recycling, and dormancy

Microbial decay processes are mostly described by starving death (48 references, e.g., [START_REF] Allison | Cheaters, diffusion and nutrients constrain decomposition by microbial enzymes in spatially structured environments[END_REF] and additionally by predation (e.g., [START_REF] Portell | Microscale heterogeneity of the spatial distribution of organic matter can promote bacterial biodiversity in soils: Insights from computer simulations[END_REF]. However, only one reference explicitly simulates predator growth [START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF]. The recycling of dead biomass in solid-(e.g., [START_REF] Gras | Individual-based modeling of carbon and nitrogen dynamics in soils:Parameterization and sensitivity analysis of abiotic components[END_REF], soluble-(e.g., [START_REF] Wang | Emergent properties of organic matter decomposition by soil enzymes[END_REF], and in both solid and soluble forms (e.g., [START_REF] Folse | Cooperation, competition, and coalitions in enzyme-producing microbes: Social evolution and nutrient depolymerisation rates[END_REF][START_REF] Monga | Simulating microbial degradation of organic matter in a simple porous system using the 3-D diffusion-based model MOSAIC[END_REF] appears in 17 references. Dormancy is only accounted for in 12 references (e.g., [START_REF] Resat | Modeling Microbial Dynamics in Heterogeneous Environments: Growth on Soil Carbon Sources[END_REF].

Microbial mobility

Colonization is a key behaviour of fungi and it is driven by hyphae invasion. Advective and diffusive movements including apical growth and branching (e.g., [START_REF] Boswell | Modelling mycelial networks in structured environments[END_REF] or solely diffusive movements (e.g., [START_REF] Falconer | Biomass recycling: A key to efficient foraging by fungal colonies[END_REF] are described. The hyphae segments can be constrained to grow and branch on a regular lattice [START_REF] Boswell | Modelling mycelial networks in structured environments[END_REF][START_REF] Boswell | Linking hyphal growth to colony dynamics: Spatially explicit models of mycelia[END_REF][START_REF] Boswell | The Development of Fungal Networks in Complex Environments[END_REF] or the geometrical constraint on orientation of hyphae growth can be relaxed when using a lattice-free approach [START_REF] Knudsen | Individual-based approach to modeling hyphal growth of a biocontrol fungus in soil[END_REF][START_REF] Vidal-Diez De Ulzurrun | Modelling three-dimensional fungal growth in response to environmental stimuli[END_REF]. Fungal networks generated with these lattice-free models are more realistic morphologically than the ones generated with the latticebased approach but they require much more computational resources [START_REF] Boswell | Linking hyphal growth to colony dynamics: Spatially explicit models of mycelia[END_REF].

A large majority of the bacterial models account for movement. Motility can be considered as a simple random motion from one node containing bacterial cells to a neighbor node (e.g., [START_REF] Ginovart | Individual-based modelling of microbial activity to study mineralization of C and N and nitrification process in soil[END_REF]. The choice of the neighbor can be further constrained by hydration status [START_REF] Long | Aquatic habitats and diffusion constraints affecting microbial coexistence in unsaturated porous media[END_REF], 2009) of the targeted node, its chemical nature (cellulose adhesion in [START_REF] Resat | Modeling Microbial Dynamics in Heterogeneous Environments: Growth on Soil Carbon Sources[END_REF] or the size of pore [START_REF] Perez-Reche | Prominent effect of soil network heterogeneity on microbial invasion[END_REF]. Cell motion is described only during the cell division process (e.g., [START_REF] Kaiser | Microbial community dynamics alleviate stoichiometric constraints during litter decay[END_REF] or in situations of excess biomass (e.g., [START_REF] Knutson | Pore-scale simulation of biomass growth along the transverse mixing zone of a model two-dimensional porous medium[END_REF][START_REF] Graf Von Der Schulenburg | Three-Dimensional Simulations of Biofilm Growth in Porous Media[END_REF]. Several studies have accounted for cell propulsion due to flagellated motility where hydration status can reduce the cell velocity (e.g., [START_REF] Wang | Aqueous films limit bacterial cell motility and colony expansion on partially saturated rough surfaces[END_REF], chemotaxis process toward substrate (e.g., [START_REF] Ebrahimi | Hydration and diffusion processes shape microbial community organization and function in model soil aggregates[END_REF] or towards bacteria (e.g., [START_REF] König | Modelling functional resilience of microbial ecosystems: Analysis of governing processes[END_REF]. Two studies [START_REF] Banitz | Assessing biodegradation benefits from dispersal networks[END_REF][START_REF] Banitz | Spatial metrics as indicators of biodegradation benefits from bacterial dispersal networks[END_REF] consider bacterial diffusion along analogs of fungal network hyphae, whereas others account for the effect of shear stress which can reduce the movement of bacteria cells (e.g., [START_REF] Tartakovsky | Pore-Scale Model for Reactive Transport and Biomass Growth[END_REF].

Carbon translocation by fungi and anastomosis

Carbon translocation along fungal mycelium is driven by the concentration gradients of the internal substrate along the hyphae (e.g., [START_REF] Falconer | Biomass recycling: A key to efficient foraging by fungal colonies[END_REF] while an active translocation can additionally be described (e.g., [START_REF] Boswell | Modelling mycelial networks in structured environments[END_REF]. Anastomosis is explicitly considered by removing hyphal tips when they move to a node that already contains a hypha (e.g., [START_REF] Boswell | Modelling mycelial networks in structured environments[END_REF]. Hyphal interactions that can result in the fusion of two individual colonies or maintain mycelial individuality of somatically incompatible genotypes are described in [START_REF] Falconer | Modelling interactions in fungi[END_REF].

Discussion about the main modeling strategies

Soil architecture

In all the reported studies, the inclusion of a spatial component is at the heart of the latest generation of microscale models to make ecological interactions emerge and explain soil microbial activity. The value of studies that use implicit approaches to summarize the environment is to emphasize the role of space and dispersal in ecological interactions, which can lead to patterns and hotspots that would not be observed in well-mixed system. The advantage of this approach is that these models rely on a few, easily measured bulk soil variables (Table 1). However, they cannot establish new mechanistic relationships between soil architecture descriptors and soil microbial response since they are dependent on empirical macroscopic laws like those relating the effective diffusion coefficients to hydration.

At the opposite end of the spectrum, methods that use explicit attributes can obviously capture most of the spatial heterogeneity of the clustering and connectivity of pores. This approach uses new knowledge of the microscale intricacies of soil architecture, which until recently had remained a black box for a long time. It offers the possibility to identify relevant 3D soil architecture descriptors that explain soil microbial activity and in fine will help upscale microbe activity from the micro-scale to the core-scale. The level of accuracy of the description of spatial heterogeneity needed in this approach has opened the brought-together scientists of various and complementary fields (soil science, mathematics, computer vision, ecologists, physicists). However, this strategy needs a huge amount of input data and, as a consequence, simulations require tremendous computing calculations and memory operations, which limits the size of the investigated domain that can be considered (Table 1). In the case of idealized geometry, like those of experimental micromodels or packings of glass beads, explicit attributes permits to exactly reproduce the geometry and helps validate the models (Table 1). The disadvantage is that actual soil architecture cannot be investigated.

The value of the approaches that describe simplified actual soil architecture or idealized porous media (which encompass regular grid and regular PN models) is to explore simplified scenarios of the complexity of pore spatial arrangement to investigate its potential role in soil microbial response (Table 1). The great advantage of methods based on morphological or PN models is that once the simplified networks have been defined, processes can be simulated much more efficiently allowing for larger samples at greater speed to be tested. A disadvantage is that these methods tend to focus on single aspects of the pore space such as connectivity, volume, pore-size distribution, but may not capture well the pore-solid interface area, which is critical for many processes in soil. It is also not clear at this stage whether regular PN are sufficient to catch the complexity of pore arrangement as compared to the more realistic irregular PN. Indeed, it is far from certain that predicted key factors in SOM microbial decomposition like accessibility between OM and decomposers is correctly captured in regular PN.

Furthermore, only a limited number of soil types has been sufficiently described to provide reliable statistics on the pore space morphology. They are necessary data to generate regular PN, and this renders quantification of the soil microbial activity by this approach very questionable.

We observe that the more realistic the description of the 3D soil architecture is, the more mechanistic is the description of the distribution of water and air in the pore space, calculated from physical laws that consider interfacial and capillary forces, giving rise to realistic air-water interfaces (Table 1). It is surprising that none of the PN models reported in this review calculate mechanistically the invasion of the air phase in pores as a function of the evaporation rate, as for example [START_REF] Yiotis | A 2-D pore-network model of the drying of single-component liquids in porous media[END_REF] did. They all rely on the empirical Young-Laplace relation, which is based on a cylindrical representation of pore geometry. Finally, knowledge on the spatial location of microbes in soils (e.g., [START_REF] Eickhorst | Improved detection of soil microorganisms using fluorescence in situ hybridization (FISH) and catalyzed reporter deposition (CARD-FISH)[END_REF][START_REF] Nunan | Spatial distribution of bacterial communities and their relationships with the micro-architecture of soil[END_REF] has not been taken into account so far in microscale models. At best, scenarios based on statistical models of spatial distribution of bacteria [START_REF] Raynaud | Spatial ecology of bacteria at the microscale in soil[END_REF] have been used. Likewise, data on the spatial location of tiny SOM patches using staining techniques (e.g., [START_REF] Peth | Localization of soil organic matter in soil aggregates using synchrotron-based Xray microtomography[END_REF] or multimodal data acquisition on spatial location of chemical elements [START_REF] Hapca | Automated statistical method to align 2D chemical maps with 3D X-ray computed micro-tomographic images of soils[END_REF] and bacteria [START_REF] Schlüter | Correlative imaging reveals holistic view of soil microenvironments[END_REF] have not been used although they can be easily implemented in approaches considering explicit description of soil architecture.

Microbial dynamics

Given the large variety of processes described no consensus seems to emerge on the optimum choice of a spatial modeling strategy. Almost one third of the reported studies used population-based approaches, and the choice of the approach was not related to the size of the grid node volumes that could reach down to a few µm 3 (Table S1). At this scale, a limited number of individuals can be contained in a node and this should in principle call for an individual-based approach rather than a population-based one, meant to represent large numbers of microorganisms. Indeed, population-based and individual-based model results will diverge when there is a low number of individuals, because in this case, the randomness of discrete events can result in a behavior that is different from the average [START_REF] Boswell | Linking hyphal growth to colony dynamics: Spatially explicit models of mycelia[END_REF]. This is particularly true when the microenvironments of soil microhabitats are depleted in nutrients which would prevent the development of a large number of individuals.

Randomness of events, like motility used as a strategy to alleviate starving conditions, could make emerge a different picture than in population-based models.. The same criticism could be levied to a large extent toward individual-based microscale models, which still adopt the Monod equation to describe bacterial growth, even though this usage of the equation for small groups of cells is not warranted at all, and other formalisms exist [START_REF] Bermúdez | On the analysis of microbiological processes by Monte Carlo simulation techniques[END_REF]). Nevertheless, irrespective of the modeling formalism used, encapsulating microbial growth at the single cell level allows to account for interspecific and/or intraspecific variability as well as the influence of random effects, which can have a high influence at low microbial numbers.

In this review we voluntarily omitted to report on numerical schemes used to solve the partial differential equations (PDEs) describing biological as well as physical processes, adopting in this respect the point of view that the methods used have been intensively verified and assessed by numerical analysts(e.g., [START_REF] Ginzburg | Two-relaxationt time lattice Boltzmann scheme: About parametrization, velocity, pressure and mixed boundary conditions[END_REF][START_REF] Hecht | New Developments in FreeFem++[END_REF] and it is not the purpose of this review to evaluate them, in part because benchmarks have to be developed for microscale models simulating microbial activity.

An encouraging finding is that microscale models have integrated the tremendous progress in knowledge of OM cycling in soils. Even though the vast majority of models simulate the uptake of simple monomers, when they take into account the solid form of SOM they all include a depolymerization step to produce dissolved OM that is taken up by microorganisms. The description of this process is generally less complex when actual soil architecture is considered, perhaps to keep the overall complexity of the models within computationally manageable bounds, whereas more complicated reactions of depolymerization are found when soil architecture is simplified or idealized.

Similarly, microscale models have attempted to integrate the knowledge gained on ecological interactions between species. Even though the vast majority of models simulate a single species, when they consider different species, they include ecological interactions between species from the simpler case of competition for food to more complex interactions (mutualism or commensalism). Once again, the simpler representations of ecological interactions tend to be adopted when actual soil architecture is described. We also note that none of the microscale models have so far coupled fungi and bacteria except the models of [START_REF] Kaiser | Microbial community dynamics alleviate stoichiometric constraints during litter decay[END_REF] or [START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF], which do it through different parameterizations of growth and uptake and not through different expansion rules. As fungi and bacteria or archea operate at different spatial scales (a fungal colony can translocate nutrients over cm's), the combination of both in a single model is currently challenging and hampered by computational limitations.

Most of the microscale models have also integrated some of the increasing knowledge about microbial mobility in soil (e.g., [START_REF] Juyal | Influence of soil structure on the spread of Pseudomonas fluorescens in soil at microscale[END_REF]. However, bacterial models that describe in detail the complexity of soil architecture have tended to disregard this process. No consensus has been reached in general on the type of motility that needs to be included in models, as random motion and/or chemotaxis can be found in experiments, mostly depending on the research question being addressed.

Advanced descriptions of spatial distribution of soil microorganisms derived from ecological knowledge have also started to be considered in the latest generation of microscale models [START_REF] Pagel | Spatial control of carbon dynamics in soil by microbial decomposer communities[END_REF].

Although tremendous progress has been made toward understanding the role of O2 on microbial activity (e.g., [START_REF] Keiluweit | Are oxygen limitations under recognized regulators of organic carbon turnover in upland soils?[END_REF], few studies have integrated that knowledge. Most models consider decomposition of SOM under optimal O2 diffusion and do not describe O2 diffusion in the liquid phase. Similarly, although the role of air-filled pores in the emission of N2O has been demonstrated experimentally (e.g., [START_REF] Rabot | Effect of the soil water dynamics on nitrous oxide emissions[END_REF], of the five microscale models that describe denitrification processes, only one includes diffusion of the emitted gases (N2 and N2O) in the liquid and gas phases within the soil. Whereas diffusion of the produced CH4 is ignored at the pore-scale in virtually all models, it is considered by one 1D-upscaled model at the scale of the soil profile [START_REF] Ebrahimi | Mechanistic modeling of microbial interactions at pore to profile scale resolve methane emission dynamics from permafrost soil[END_REF].

Perhaps the most striking feature of existing models is the fact that the majority of them so far have not accounted for the presence of roots, despite the rhizosphere being the most active region in soil.

Models simulating root surface colonization by bacteria have focused on the description of the root itself and its growth while the rhizosphere is either considered as inactive grid nodes, in the sense that bacteria are set inactive until the root invades the soil node [START_REF] Muci | A combination of cellular automata and agent-based models for simulating the root surface colonization by bacteria[END_REF] or it is considered as a boundary influx of soil bacteria [START_REF] Dupuy | Mechanisms of Early Microbial Establishment on Growing Root Surfaces[END_REF]. Contrastedly, models addressing the transport of nutrients have had a stronger focus on the rhizosphere by using as input data 3D CT images of root hairs in soil pores and they have demonstrated the benefits of such image-based modeling approaches in understanding the uptake of phosphorus (e.g., [START_REF] Keyes | High resolution synchrotron imaging of wheat root hairs growing in soil and image based modelling of phosphate uptake[END_REF][START_REF] Roose | Challenges in imaging and predictive modeling of rhizosphere processes[END_REF]. The development of this type of model that includes the growth and metabolism of, e.g., beneficial bacteria, pathogens or mycorrhiza is therefore urgently required to advance our understanding of the rhizosphere and open up opportunities to control it to benefit crop production.

Limitations of microscale models and future prospects

We identified two major limitations that constitute severe challenges to microscale modeling efforts.

On one hand, available input data on soil architecture need to be sufficiently representative of reality and to capture its essential features. On the other hand, the parameterization of biotic components of these models needs to be sufficiently robust. In both respect, we shall identify research avenues that could help fill current knowledge gaps and choose the most appropriate modeling strategies.

Available input data on soil architecture

The approaches that uses imaging of soil architecture are attractive because they preserve as much as is practically feasible the real spatial arrangement of soil pores and provide realistic statistics on soil physical properties. However, different image treatments (noise reduction, conversion into low number of bits, segmentation algorithms) are applied to obtain binarized or ternary CT images on which most of the image analyses are performed. These treatments can produce modifications of the bulk measures of pore morphology such as porosity, specific surface area, or Euler connectivity [START_REF] Houston | Effect of scanning and image reconstruction settings in X-ray computed micro-tomography on quality and segmentation of 3D soil images[END_REF]. Furthermore, small local changes of the size of key throats governing the water flux can dramatically affect simulated hydrodynamic properties [START_REF] Gackiewicz | Saturated water conductivity estimation based on X-ray CT images -evaluation of the impact of thresholding errors[END_REF][START_REF] Pot | Effect of resolution, reconstruction settings, and segmentation methods onthe numerical calculation of saturated soil hydraulic conductivity from 3D computed tomography images[END_REF]. Imaging data are also limited by the pixel resolution attained by the different imaging tools, which can mask a significant portion of soil porosity. At best, existing microscale models simulate microbial activity at a scale of a few microns and while they can only capture a small part of the tremendous complexity of soil architecture they miss out a lot of the sub-resolution pore space in which many bacteria, archea, and viruses reside. Baveye et al. (2017) report different strategies that can be used to consider the unresolved pores from CT data and to describe their role in the carbon transformation. Among them is the use of all the information contained in the gray-level pixels instead of using binarized images, or to mix the traditional, binary approach with gray-scale modeling, as in the work of [START_REF] Yan | Pore-scale investigation on the response of heterotrophic respiration to moisture conditions in heteregeneous soils[END_REF][START_REF] Yan | Microscale water distribution and its effects on organic carbon decomposition in unsaturated soils[END_REF]. However, an implicit approach is necessary to assign a value for instance to the diffusion coefficients in these pixels as a function of hydration or gas status. Multi-fractal approaches were mentioned by Baveye et al. (2017) as a possible way to simulate the statistical distribution of unresolved pores, but further analysis by [START_REF] Perrier | To what extent can multifractal measures provide an accurate model of the porosity of soils?[END_REF] has recently demonstrated that their potential was limited in this respect. Although we are still far from having handle on the sub-resolution porosity, the approaches that have been explored so far at least confirm the importance of 'hidden' smaller pores in understanding soil processes and thereby encourage future research efforts. Recent advances in imaging techniques should bring better knowledge of pore arrangement at the infra-micron scale [START_REF] Felde | Wet sieving versus dry crushing: Soil microaggregates reveal different physical structure, bacterial diversity and organic matter composition in a clay gradient[END_REF][START_REF] Schlüter | Correlative imaging reveals holistic view of soil microenvironments[END_REF] and address their potential role in shaping soil microbial activity.

An area of research that has recently emerged is the utilization of neural networks to predict processes in rocks, using training data sets made of 3D binary CT images [START_REF] Santos | PoreFlow-Net: A 3D convolutional neural network to predict fluid flow through porous media[END_REF]. To improve training, geometrical features such as the Euclidean or transform distance map, the maximum inscribed sphere flow, and tortuosity maps of the global paths along the flow direction are included in the training data set. Rock architecture is far more homogeneous than is soil architecture, however the convolutional neural network was able to deal to a reasonable extent with the higher heterogeneity observed in rocks (dead-end pores, disconnected pores) compared to the initial training data set of bead packs.

Another limitation comes from the fact that the spatial arrangement of soil constituents is highly dynamic, continuously changing under the effect of biotic activity (including roots, macro-, meso-, and micro-fauna) and abiotic stresses (e.g., drying-rewetting cycles). Time-resolved methods to image the dynamics of soil architecture (e.g., [START_REF] Bottinelli | Macropores generated during shrinkage in two paddy soils using X-ray micro-computed tomography[END_REF] provide experimental data that can potentially be used as direct model inputs. Recent attempts to simulate the evolution of soil architecture as a result of microbial activity [START_REF] Ray | Discrete-continuum multiscale model for transport, biomass development and solid restructuring in porous media[END_REF] or physico-chemical processes [START_REF] Rupp | Application of a cellular automaton method to model the structure formation in soils under saturated conditions: A mechanistic approach[END_REF], together with research focusing on the role of roots in modifying the geometrical arrangement of soil particles [START_REF] Aravena | Quantifying coupled deformation and water flow in the rhizosphere using X-ray microtomography and numerical simulations[END_REF]Kolb et al., 2017) will certainly open new avenues to bridge knowledge gaps.

Parameterization of the biotic component

A major concern that has not been discussed in any of the reviewed references deals with the parametrization of the biological component. When models simulate the activity of a well-defined microorganism, parameters describing its physiology are usually extracted from experimental data measured in pure culture system. In doing so, models can only tackle a tiny part of the soil biodiversity.

Representative species of known functional groups are however modeled. A major difficulty in extracting parameters from experiments is that the underlying mathematical formulations for the growth kinetics used to describe the experimental growth curves are unevenly informative and parameters extracted from these experimental studies can thus be in conflict with the formulation adopted in a given microscale model [START_REF] Vogel | To what extent do uncertainty and sensitivity analyses help unravel the influence of microscale physical and biological drivers in soil carbon dynamics models?[END_REF]. We thus advocate to report them in experimental studies. But even so, it is far from certain that the physiology of bacteria remains the same whether the cells live freely or are attached to soil surfaces (e.g., [START_REF] Fletcher | A microautoradiographic study of the activity of attached and free-living bacteria[END_REF][START_REF] Kirchman | Contribution of particle-bound bacteria to total microheterotrophic activity in five ponds and two marshes[END_REF][START_REF] Lerch | Biofilm vs. planktonic lifestyle: Consequences for pesticide 2,4-D metabolism by Cupriavidus necator JMP134[END_REF][START_REF] Ulířová | Growth rate of bacteria is affected by soil texture and extraction procedure[END_REF]. Additionally, other parameters associated to mortality and dormancy are generally obtained by calibration of macroscopic models (e.g., [START_REF] Ingwersen | Micro-scale modelling of carbon turnover driven by microbial succession at a biogeochemical interface[END_REF] to experimental data from soil incubations with composite microbiota and carbon sources [START_REF] Vogel | To what extent do uncertainty and sensitivity analyses help unravel the influence of microscale physical and biological drivers in soil carbon dynamics models?[END_REF]. These limitations render the use of these experimental data to parameterize the biotic component very questionable.

The mobility of bacteria or fungi is also very difficult to parameterize. For bacteria, at best, data on motility exist in artificial media (e.g., agar surfaces, micro-models), which may not be directly transferable to motility in soil pores. Direct measurements in soils have demonstrated the impact of soil architecture on the ability of bacteria to move [START_REF] Juyal | Control of pore geometry in soil microcosm and its effect on the growth and spread of Pseudomonas and Bacillus sp[END_REF]. These data, which describe the movement of a population of bacteria, are directly transferable to models that do not adopt an explicit description of soil architecture. For models accounting explicitly for soil architecture, this information has to be used to calibrate/validate biotic parameters accounting for microbial movement, irrespective of the formalism, either population-or individual-based.

In the case of fungi dispersal rates, hyphal expansion, and branching angles have been observed on 2D surfaces and incorporated into fungal models [START_REF] Stacey | Soil-borne fungal pathogens: Scaling-up from hyphal to colony behaviour and the probability of disease transmission[END_REF]. In theory these types of parameterization can be expanded to 3D, however it has been shown that branching angles in a 3D pore structure can significantly deviate from those reported in literature, suggesting that expansion of fungal colonies has more complex feedback mechanisms with the environment than is currently considered in models [START_REF] Soufan | Pore-scale monitoring of the effect of microarchitecture on fungal growth in a two-dimensional soil-like micromodel[END_REF].

Conclusions

Given the advantages and disadvantages of the different strategies used to describe soil microenvironments, we recommend modeling benchmarks. Indeed, an explicit approach will not be able to cope with the tremendous amount of data needed, and simplified approaches conserving the key spatial arrangement of pores appears as a necessary alternative. Like [START_REF] Vogel | Comparison of a lattice-Boltzmann model, a full-morphology model, and a pore network model for determining capillary pressuresaturation relationship[END_REF] did in comparing a pore-network model, a lattice-Boltzmann model and a morphological model to simulate the capillary pressure of a soil sample whose topology was obtained by CT, and Pot et al. (2015) did in comparing a lattice-Boltzmann model and a morphological model to simulate experimental spatial distribution of air-water menisci as observed in synchrotron CT images, similar benchmark analyses should be carried out with respect to soil microbial activity. Pure diffusion conditions (as those existing in standardized soil incubation experiments) but also addition of intermittent water fluxes should be tested in modeling scenarios. The latter should benefit from recent progress made in simplifying modelling of water flow at pore scale for unsaturated conditions [START_REF] Zhang | A Lattice Boltzmann model for simulating water flow at pore scale in unsaturated soils[END_REF].

In light of the complexity and uncertainty of soil processes as well as of imperfections of data available, we also advocate for the use of a multimodal description of soil microhabitats along with a modeling strategy involving data at different levels of description such as the so-called pattern-oriented modeling [START_REF] Grimm | Pattern-oriented modeling of agent-based complex systems: Lessons from ecology[END_REF]. The non-destructive imaging of soil properties and processes is rapidly evolving and we need to find new ways of combining and including data emanating from these techniques in the modeling process.

In this review we addressed the current practices used in soil science to account for soil architecture and microbial dynamics at the scale of microbial habitats and address their advantages and disadvantages. We will address in a forthcoming paper the insights gained by these models on soil functions and discuss to what extent it is important to include the explicit description of soil architecture in microscale models of soil microbial activity.

Fundings

This work was supported by a grant from the French ANR to project Soil μ-3D (ANR-15-CE01-0006). WO and XP received support from the National Environment Research Council (NE/P014208/1 and NE/S004920/1) 8 Data Sharing and Data Accessibility statement Data sharing is not applicable to this article as no new data were created or analyzed in this study. The first one would be a packing of porous spheres described by explicit nodes (pore) and implicit nodes of constant local porosity and explicit nodes (the pore phase). The second one would be a random assignment of explicit nodes (pore and solid) and implicit nodes of varying local porosity that would fit the porosity of a soil sample. Microscale models using implicit attributes to describe either an idealized or a detailed soil architecture would be unrealistic cases and they are therefore indicated by the mathematical empty set symbol Ø.

Figure 1 :

 1 Figure 1: Classification of pore-scale models of soil processes, according to the degree of representativeness of soil architecture (from idealized porous media to simplified architectures eventually to actual soils) and to the nature of the grid node attributes (implicit, explicit and mixed, i.e., both explicit and implicit). Implicit attributes are bulk porosity, water content and effective diffusion coefficients which values are represented by different levels of gray. Explicit attributes are local characteristics of the soil phases: solid, pores filled with air or pores filled with water, and molecular diffusion coefficients in air and water, which are represented by different colors. PN and CT are pore network and computed tomography, respectively. We added two potential examples of what would be a microscale model made of mixed attributes describing an idealized and a simplified soil architecture.