Leading an effective response to the accelerating crisis of
anthropogenic climate change will require improved understanding of global
carbon cycling. A critical source of uncertainty in Earth system models
(ESMs) is the role of microbes in mediating both the formation and
decomposition of soil organic matter, and hence in determining patterns of
CO

The current crisis of anthropogenic climate change is expected to accelerate
during the 21st century. Despite considerable effort to better constrain
global biogeochemical models, considerable uncertainty remains about how
best to represent emerging mechanistic understanding of soil element cycling
into process-based models
(Wieder
et al., 2015; Todd-Brown et al., 2018). This is a critical gap in knowledge
because variations among models predict hugely varying responses to global
change drivers such as temperature, soil moisture, and CO

While the fields of population and community ecology have long confronted
the challenges posed by non-linearity and heterogeneity in spatiotemporal
scaling of ecological dynamics (Chesson,
2009; Levin, 1992), ecosystem ecology and biogeochemistry have tended to
approach the challenge of scale either by (1) utilizing mean-field
assumptions or (2) addressing the challenge of scaling via grid-based
computational/numeric methods. While there is nothing inherently wrong with
either approach, they unfortunately cannot yield theoretical insight into
the consequences of non-linearity and heterogeneity for scaling. Briefly,
the combination of non-linearity and heterogeneity means that aggregated
behavior differs systematically from mean-field predictions, a special case
of Jensen's inequality. In mathematical notation:

Although Jensen's inequality is well-known from basic probability theory
(Ross, 2002) its implications for ecological dynamics under
heterogeneity were not well-appreciated until the pioneering work of Peter
Chesson in the 1990s (Chesson, 1998). In the case
of carbon cycle science, there are a few immediate and critical
applications. For instance, most trace gas emission processes are well-known
to be non-linear functions of underlying drivers such as temperature and
soil moisture. For example, ecosystem respiration is an exponential function
of temperature (usually expressed in

Chakrawal et al. (2020) provide a detailed and compelling first-pass application of scale transition theory to biogeochemical modeling. Our contribution here complements their laudable effort by providing a more generic mathematical analysis of the scale transition, equally applicable to both forward and reverse Michaelis–Menten microbial kinetics. As in Chakrawal et al. (2020), we address the consequences of heterogeneity in both substrate/microbes (“biogeochemical heterogeneity”) and in the kinetic parameters (“ecological heterogeneity”). However, we diverge from their approach in that, rather than explore detailed simulation models, we derive a completely non-dimensionalized expression for aggregating non-linear microbial kinetics over both types of heterogeneity simultaneously. We illustrate the clarity this brings in several special cases of our full analysis. Altogether, our approach provides new insight into the properties of the scale transition and enables clear conclusions to be drawn across systems in terms of the role of spatial variances and covariances in shaping ecosystem carbon efflux. Our work provides a simplified, yet systematic framework around which to base subsequent empirical and simulation-based studies.

A variety of microbially explicit process-based models have been proposed in
the literature, starting with the classic enzyme pool model of Schimel and
Weintraub (2003). In order to elucidate universal properties of the scale
transition, we focus here on the CO

Our specific model for

Following the terminology of Chesson (1998, 2012), the above is our “patch”
model, and our goal is to understand how spatial variances and covariances
impact the integrated flux, which represents the spatial expectation or

Analytically, an exact solution would require specification of a joint
distribution for C, MB and parameters,

However, following Chesson (2012) and
Chakrawal et al. (2019), we are free to approximate the solution for
arbitrary distributions using a Taylor series approximation expanded to the
second moment. Specifically, we take the expectation over a multivariable
Taylor series expansion, centered around the mean-field values of all
parameters

Expanding Eq. (7) out, we have five terms involving the variances of C,
MB,

We define a dimensionless quantity

We divide all of the terms in Eq. (6) by their mean-field value and represent the
whole equation as a product:

We notice that

Similarly, since the covariance terms can be rewritten as

Applying steps 1–5 to all the terms in the equation, we end up with a fully
dimensionless equation:

Note that by symmetry, we have also solved for the case of the forward
Michaelis–Menten kinetics. This can be expressed simply by interchanging C
and MB and by correspondingly altering

Having fully non-dimensionalized Eq. (7), we are in a much better
position to gain analytical insight into the scale transition. To begin, we
note the pivotal role played by the quantity

Accordingly, in our setup, the multiplicative factor for the scale
transition correction approaches a simplified expression, as

This is quite remarkable. Despite invoking the situation where microbial
biomass (and its enzyme supply) is effectively infinite – thus linearizing
the underlying patch models – we cannot eliminate the possibility of a
potentially substantial deviation from mean field when scaling decomposition
kinetics. We note that in this resulting expression, we have reduced the
situation to a set of three critical correlations involving two microbial
physiological parameters (

Returning to the situation where

More generally, starting with our dimensionless Eq. (10) puts modelers
and empiricists in a better position to assess the quantitative significance
of the scale transition correction across systems compared to expressions
with opaque second partial derivatives and cross derivatives and
arbitrarily scaled variance terms. By re-expressing

To illustrate these advantages in interpretability, we first take the
special case of a model where we treat all parameters as constant (and
known) except substrate and microbial biomass. This corresponds to setting
the other CV and

deviation from mean-field behavior declines, and

first-order kinetics are approached.

Indeed, our Eq. (16) reveals the exact speed of this convergence in terms
of dimensionless

We illustrate the scale transition solutions to Eq. (16) as a function of

Scale transition correction for models given spatial colocation between microbes and substrate across a gradient of

In the case of pure spatial colocation, with no variation in the kinetic
parameters, the scale transition correction factor varies from a maximum of
1.5 to a minimum around 0.75 and in all cases indeed converges to 1 as

Another benefit is that it is mathematically tractable to see how the
variance and covariance terms can balance each other and to solve for where
they are equal. If we introduce a new term

If we fix

So far, we have analyzed in depth the role of variability in microbes and
their substrate but not in the ecological drivers underlying maximal
reaction rates (i.e.,

To make matters clear, we re-express the rate limiting maximal reaction
velocity

Using the Taylor expansion again to second order, we have

Scale transition correction for

As is clear in Fig. 2, the scale transition for temperature is extremely convex. Integration of fluxes over ecosystems with significant heterogeneity in temperature invokes substantial deviation from a mean-field model. For instance, at a CV of 0.2, the scale transition correction is 1.10, but by a CV of 0.5 it is 1.66. Obviously, the significance of this depends on the scale and heterogeneity over which an accurate flux model is desired. For a smaller footprint eddy covariance tower (e.g., Gomez-Casanovas et al., 2018) over a uniform habitat type, soil (and near surface) temperatures probably do not vary by much more than 20 %. Regardless, our general mathematical analysis quantifies and clarifies exactly how the scale of variation influences the degree of the scale transition correction.

Notably, the only difference between the scale transition correction for
first-order and for reverse Michaelis–Menten kinetics is that in the latter
there would be additional correlation terms to consider, e.g., the
correlation between temperature and

Unlike soil temperature, we expect heterotrophic respiration to respond in a
unimodal fashion to soil moisture. At low levels of soil moisture, microbes
are moisture limited, and at high levels they are oxygen limited, with some
optimum range of values in the middle. Although a considerable amount of
work has gone into developing soil moisture functions, including both
empirical and theoretical derivations
(Yan et al., 2018; Tang and Riley,
2019), there is no clear consensus on an optimal representation. Moreover,
many of the candidate functions complicate analysis considerably by virtue
of stepwise formulation (Linn and
Doran, 1984). Therefore, to study the implications of the scale transition,
we proceed via a powerful simplifying abstraction and simply represent the
soil moisture response as a quadratic of the form

We seek the scale transition:

As shown in Fig. 3, where mean field soil moisture is close to the optimal
value, scale transition effects are expected to be quite large. For
instance, by the time the coefficient of variation is 0.5, efflux would only
be 75% and declines rapidly to 0 as the coefficient of variation
approaches 1. Clearly, this latter outcome is not necessarily biologically
realistic, and a more detailed numerical experiment should be done to
explore scenarios with that much variation. However, our abstractions yield
the simple insight that mean-field solutions invariably

Our results on soil moisture relate to the argument by Tang and Riley (2019)
that heterotrophic respiration arises from a two-step process whereby
substrate must diffuse into the vicinity of microbes and then be taken up
– the latter by a Michaelis–Menten kinetic. However, microscale variations
in soil moisture mediate and regulate the first step of the process, so that
the “effective substrate affinity” (the

We proceed by first holding all terms constant except allowing the half-saturation constant

More broadly, our analysis highlights that, under non-linear
Michaelis–Menten kinetics for representing carbon processing, the impact of
environmental heterogeneity acting on the substrate affinity parameter is
opposite of when it acts on the

Thus, for the analysis of upscaling fluxes in the presence of soil moisture heterogeneity, we expect the concave corrections of Fig. 3b to hold, regardless of the fine-scale details of the soil moisture function used.

Scale transition factor for variations in the substrate affinity (“

We close our discussion by considering the implications of the scale transition for advancing the state of biogeochemical modeling. Critically, the representation of non-linear (microbial driven) kinetics is a crucial modeling choice with large implications for long-term SOC forecasts. Traditional first-order process-based models dodge explicit representation of these kinetics but nonetheless have worked well in practice. This state of affairs persists because both non-linear and linear kinetics are capable of representing coarse-scaled biogeochemistry reasonably well, at least in certain respects. Since first-order kinetics are known to be a crude approximation, the crucial question for practice is not whether they are “true”, but rather whether there is significant, systematic information loss inherent to their use. Fortunately, the scale transition offers a clear, clean path to discriminate between these alternative model formulations.

As noted throughout, the dimensionless term

Previous work (Sihi et al. 2016) has approached this question theoretically,
from first principles. Here, we point out that demonstrating substantial
deviation from the mean-field model when fitting non-linear kinetics to data is
both a necessary and sufficient condition for inferring that

In addition to the role of

In addition to fitting fully parameterized flux models (as above), simpler
statistical models could be fit examining the role of variations in
microbial biomass, or colocation of microbial biomass and SOC, in explaining
across-site variations in ecosystem respiratory fluxes (

We further note that the scale transition presented here is closely related to global sensitivity analysis (GSA; Saltelli et al., 2010). In its fundamental setup, a GSA tests effects of variability in parameters. While GSA has been typically used towards characterizing the uncertainty of parameters, it is directly applicable to spatial and temporal variability. For example, the first-order results of a GSA (or the result of one at a time parameter substitution) provide the contribution of that parameter to the scale transition. Similarly, the “all but one” perturbation offers insights into how the net effect of all parameters (and variables) violates the mean field approximation. Therefore, a computationally expensive GSA can be leveraged to garner further insights on top of sensitivity effects, allowing for the characterization of the scale transition. Indeed, a computationally intensive approach to simulating scale transitions was utilized by Chakrawal et al. (2020) to good effect. However, we suggest future computational studies build off of the dimensionless approach studied here, including those extended to multiple microbial populations which would result in multiple dimensionless lambdas and corresponding multiplicative contributions to the scale transition. Obviously, the parameter space needs to be properly chosen (or subsetted) to reflect appropriate means, variabilities and perhaps most challenging – correlations. Equation (10) would then provide analytical, albeit approximate, insight into the scale transition effects, while the GSA would enable study of any shortcomings from approximation and also allow for quantification of individual variable importance for those parameters that enter into the dynamics in multiple places.

Finally, our analysis of environmental factors including temperature and soil moisture leads to readily testable predictions. For temperature, the scale transition is convex, and thus, ceteris paribus, variation in soil temperature should lead to greater effluxes than mean field models would predict. The implications of this for climate feedback should be studied in greater detail. For soil moisture, which varies considerably across both space and especially time, our analysis based on an idealized quadratic representation yields a concave scale transition correction, i.e., the mean-field soil moisture will overestimate efflux. Likewise, when represented in both substrate affinity and multiplicative active microbial biomass fraction terms, as in Tang and Riley (2019), the scale transition remains concave. However, environmental factors that act only through substrate affinity would result in a convex correction as in Fig. 4. Once again, we highlight that the nature of scaling corrections, wherever it is possible to be studied empirically, can provide insight into the most productive representations of our models.

Here, we have illustrated how the spatial scale transition can be expressed
in dimensionless form, yielding insight into the systematic operation of
Jensen's inequality in upscaling microbial decomposition kinetics. Our
analysis has identified the central role of the dimensionless quantity

This dual sense of convergence also provides opportunity to empirically test for the presence of significant non-linear microbial dynamics in upscaled field data: to the extent that upscaled fluxes deviate from the flux estimated at mean-field conditions, we have ipso facto evidence for the importance of formulating our biogeochemical models with these non-linear terms. Conversely, where there is close agreement between mean-field and upscaled fluxes, there are arguably stronger reasons for retaining first-order process model formulations.

In closing, we would like to point out how this mathematical analysis
illustrates the challenge of scaling quite nicely. In the context of
non-linear models, for each parameter that is allowed to vary in space,
there is not only a new variance parameter, but also a number of new covariance
terms are induced, growing as the factorial of the number of varying
parameters

Model complexity grows exponentially with number of spatially varying parameters. We argue to keep models as simple as possible for both analytical and computational tractability.

Even with a maximally generic and simplified expression, fitting such non-linear time series models to field data still represents quite a challenge, especially while adequately accounting for and propagating uncertainty. Modelers and theoreticians should appreciate the complexity of the task at hand. Fortunately, our analysis has identified a potentially robust route to limiting model complexity: screen systematically for the importance of various correlations in explaining variations in fluxes. Accordingly, we recommend that research focus first upon spatial colocation of MB and C, which is readily measured, and then to thoughtfully and carefully expand models with additional terms as needed.

All code used to generate the figures in this paper, which visualize our theoretical results and mathematical derivations, is publicly available at the first author's GitHub repository here:

No data sets were used in this article.

CHW conceived the original concept, developed the mathematical analysis and wrote the manuscript. SG developed the concepts, contributed to the mathematical analysis, and co-authored and edited the manuscript.

The contact author has declared that neither they nor their co-author has any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We stand on the shoulders of giants: Peter Chesson's research program on scale transition was enormously influential. We thank Timothy Trevor Caughlin for introducing us to Chesson's papers many years ago and everyone who has humored discussions of Jensen's inequality ever since. Will Wieder and the anonymous reviewer provided constructive reviews that improved our manuscript, and Kathe Todd-Brown provided valuable discussion as we worked through revisions.

This paper was edited by Jens-Arne Subke and reviewed by William Wieder and one anonymous referee.