## Abstract

Degradation of marine organic carbon provides a major source of atmospheric carbon dioxide, whereas preservation in sediments results in accumulation of oxygen. These processes involve the slow decay of chemically recalcitrant compounds and physical protection. To assess the importance of physical protection, we constructed a reaction-diffusion model in which organic matter differs only in its accessibility to microbial degradation but not its intrinsic reactivity. The model predicts that organic matter decays logarithmically with time *t* and that decay rates decrease approximately as 0.2 × *t*^{–1} until burial. Analyses of sediment-core data are consistent with these predictions.

Nearly half of Earth's primary production of organic carbon occurs in the oceans (*1*). Once fixed, this reduced carbon immediately enters the marine food chain, where nearly all of it—about 99.9%—is eventually oxidized back to CO_{2} via heterotrophic metabolism (*2*). The remainder escapes degradation and is immobilized in sediments (*2*–*4*). Whereas degradation rates influence atmospheric and oceanic concentrations of CO_{2} at short time scales, burial rates influence O_{2} concentrations at long (geologic) time scales (*3*, *4*). Moreover, the impact of this slow leak from the biological to the geological carbon cycle extends well beyond O_{2}. For example, the burial of organic carbon is a natural form of carbon sequestration (*5*) that can lead to the formation of petroleum (*6*), and O_{2} accumulated as a consequence of burial must have preceded the evolution of complex life dependent on aerobic metabolism (*7*).

A fundamental understanding of the rates of decay and burial has been elusive, however. A central problem is that there appears to be no single rate that characterizes the microbial decay of organic matter (*8*, *9*). Indeed, a pronounced slowdown in respiration rates, as organic matter falls through the water column and degrades in sediments, has been unambiguously documented at time scales ranging from days to millions of years (*10*, *11*). However, the mechanisms responsible for this slowdown and their relation, if any, to the small fraction of organic matter that is preserved in sediments remain unresolved.

Many hypotheses have been proposed. Roughly speaking, these appeal to either chemistry or physics. The chemical scenarios typically ascribe the slowdown to changes in the chemical composition of organic matter. In probably the simplest such formulation, intrinsically highly reactive “labile” organic carbon is consumed first and then followed by less reactive compounds. The most recalcitrant organic compounds—for example, algaenans—are then “selectively preserved” (*12*, *13*). An alternative hypothesis suggests that recalcitrance is ultimately not traceable to biosynthesis. It instead results from random repolymerization and condensation of a small nonselective set of incompletely degraded but enzymatically depolymerized compounds, which, over time, constitute an increasingly large fraction of the remaining organic matter (*6*).

The physical scenarios attribute the slowdown and burial to some form of physical protection. This reasoning follows in part from observations showing a strong correlation between the concentrations of organic carbon and clay particles (*14*) or mineral surface area (*15*–*17*) in sediments, suggesting that some physical property of the sediments themselves is promoting the preservation of organic matter. Further evidence of physical protection comes from observations indicating that the chemical composition of organic matter changes little as it sinks to the sea floor (*18*), and from experiments suggesting that when apparently recalcitrant organic matter is physically separated from its mineral matrix, it becomes “labile” and is rapidly consumed (*19*).

Both chemical and physical mechanisms must play a role, but they have not received similar attention in quantitative models. Whereas intrinsic reactivity is commonly invoked (*8*, *9*, *20*), explicit consideration of physical mechanisms has been rare (*21*). To address this, we construct a theoretical model that purposely assumes an extreme case in which the chemical composition of organic matter is uniform. Degradation and preservation rates instead derive only from the physics of diffusion-limited reactions in porous media. As we show below, this simple theory provides predictions that are quantitatively consistent with a variety of observations, thereby suggesting that physical mechanisms play an important role in setting both degradation and preservation rates.

Our model assumes that organic matter is randomly distributed on the mineral surfaces of a porous medium populated by randomly distributed heterotrophic bacteria (Fig. 1). We assume that the typical bacterial spacing *r*_{b} ∼ *n*^{–1/3} associated with the bacterial number density *n* is much greater than the typical pore size of the porous medium, consistent with images obtained by microscopy (*22*). Spatial averaging of the porous medium at a scale greater than the pore size but less than *r*_{b} then provides a continuum approximation in which each microbe is embedded in a smooth porous medium characterized by its void fraction (porosity) ϕ.

The decay of a particular parcel of organic matter is assumed to be limited by its accessibility to hydrolytic enzymes. “Accessibility” is associated with grain boundaries. It is interpreted physically as the frequency *f* with which a mineral surface encounters a diffusing enzyme. We assume that *f* is proportional to the concentration *c* of functional enzymes in the pore fluid near the mineral surface, and that enzymes lose their functionality at rate α. Otherwise, *c* is conserved and diffuses within pores with diffusivity *D*. Averaged over lengths much greater than a pore size, the volume-averaged concentration *c̄* = ϕ*c* of active enzymes therefore evolves according to the reaction-diffusion equation (1) where *D̄* is the effective diffusivity in the porous aggregate.

A complete treatment of this problem would require coupling the diffusion of *c* to the diffusion of organic matter hydrolyzed by its contact with *c* along with the consumption of hydrolysate by bacteria (*21*). It would also include coupling the production of enzymes to energy gained from hydrolysate. We simplify our analysis, however, by assuming that the decay of organic matter is rate-limited by hydrolysis, and that the bacterial population density and the flux of enzymes emanating from each microbe is constant. We then set local decay rates *k* ∝ *f* and assume that the appropriate solution of Eq. 1, *c*(*r*) ∝ *e*^{–βr}/*r*, where *r* is the distance from the nearest microbe and β = (α/*D̄*)^{1/2}, suffices to determine *f* from the relation *f* ∝ *c*. These assumptions correspond to a quasistatic limit in which enzymes diffuse much faster than microbes, hydrolysis is diffusion-limited, and the porosity ϕ is constant. They also imply that decreases in hydrolysate production represent a decreasing surplus. Although such excess solubilization suggests inefficiency at early times, it is consistent with previous model calculations (*21*) and observations of effluxes of dissolved organic matter from marine sediments (*23*) and sinking organic aggregates (*24*).

These simplifications allow us to express the temporal decay of the total concentration of organic matter, *g*(*t*), as a continuous (*20*) superposition of first-order reactions (*8*, *9*) weighted by the concentration ρ(*k,t*) *dk* of organic matter associated with rate *k* at time *t*. Our formulation depends on two phenomenological parameters: a characteristic concentration *g*_{0} ∝ *g*(0) arising from the requirement that *g*(0) = ∫ ρ(*k*, 0) *dk*, and a minimum reaction rate *k*_{min} associated with the typical distance (*r*_{b}) between microbes. Assuming that *k*_{min} is much smaller than the (maximum) reaction rate adjacent to microbes, we obtain the approximation (supporting online text) (2) where τ = *k*_{min}*t, k**= *k*/*k*_{min}, *g**(*k*_{min}*t*)= *g*(*t*)/*g*_{0}, and τ_{min} = *k*_{min}*t*_{min} is the dimensionless finite time associated with the initiation of observable decay at dimensional time *t* = *t*_{min} (i.e., in a core-top sample). Equation 2 is a special case of the reactive continuum models introduced by Boudreau and Ruddick (*20*). Its particular form, which defines the exponential integral *E*_{1}(τ)(*25*), derives from its connections to the reaction-diffusion dynamics of Eq. 1.

For τ ≪ 1, an asymptotic expansion of Eq. 2 predicts that *g**(τ) decays logarithmically (*25*): (3) where γ = 0.5772... is Euler's constant. The characteristic time τ ∼1(i.e., ) marks the termination of the logarithmic decay. Because subsequent decay proceeds slowly, as *e*^{–τ}/τ (*25*), we interpret as the approximate time of burial and the effective cessation of degradation. Consequently approximates the burial concentration and *g*_{0}*E*_{1}(1)/*g*_{max}, where *g*_{max} is the observed concentration at *t* = *t*_{min}, approximates the burial efficiency. Assuming ln τ_{min} ≪ –γ, Eq. 3 implies *g**(τ_{min})∼ –ln τ_{min} and therefore (supporting online text) (4) The second relation derives from *k*_{min} ∝ *e*^{–βrb}/*r*_{b} and the assumption that β*r*_{b} ≫ 1. It approximates burial efficiency [neglecting the factor of *E*_{1}(1) ≅ 0.22] by the ratio of the diffusion length β^{–1} to the bacterial spacing *r*_{b}. In conjunction with the first relation, it also shows that as *r*_{b} increases (with β and *t*_{min} constant), *k*_{min} decreases exponentially—reflecting the longer degradation times associated with lower microbial population densities and less efficient burial.

To test these predictions, we assembled a database of 23 published analyses of dated sediment cores that span a wide range of environmental settings, from the deep ocean to shallow waters, and from aerobic to anoxic conditions at the sediment-water interface (table S1). Each data set provides measurements of the bulk concentration *g*_{i} of particulate organic carbon at time *t*_{i}, where *t*_{i} is obtained by dividing depth by the average accumulation rate *V*_{i} of the overlying sediment. A least-squares fit of each core to Eq. 2 provides a comparison between theory and observation along with estimates of the parameters *k*_{min} and *g*_{0} (supporting online text).

Figure 2A shows data from a single core beneath oxygenated waters. In this case the fit, which is generally good, yields a value of *k*_{min} that is much less than the inverse of the time associated with the base of the core. Thus, the resulting theoretical curve is nearly identical to the asymptotic logarithmic decay of Eq. 3. Figure 2B shows the data from all 23 cores, after rescaling the *g* and *t* axes by the estimates of *g*_{0} and *k*_{min}, respectively, that derived from each fit. The data generally show a reasonable fit to Eq. 2, including some evidence of diminishing decay rates near .

Figure 3A displays the 23 estimates of the paired parameters *k*_{min} and *g*_{0}. One sees that the cores underlying anoxic waters are associated with higher burial concentrations (∝ *g*_{0}) and earlier terminations (i.e., higher *k*_{min}). The otherwise apparent disarray of the data, however, masks the relation between *k*_{min} and *g*_{0} predicted by Eq. 4. Figure 3B compares this prediction to each of the *g*_{0}, *k*_{min} pairs of Fig. 3A. The good fit derives from the consistency of the degradation data with the theoretical model of Eq. 2.

Figure 3B also shows that the variation of *g*_{0}/*g*_{max} is not large and that there is a tendency for both oxic and anoxic cores to cluster around the mean 〈*g*_{0}/*g*_{max} 〉. Thus, in Fig. 3C, wherewe plot the burial flux *Vg*_{0} versus the flux to the sea floor *Vg*_{max}, we find a good fit to a straight line with a slope of unity in logarithmic coordinates. Although anoxia appears to have little influence on burial efficiency (*26*) (Fig. 3B), it is associated with a higher burial flux (Fig. 3C), which is in turn proportional, on average, to the flux to the sea floor. Insertion of the constant of proportionality, 〈*g*_{0}/*g*_{max} 〉 ≅ 0.18, into Eq. 4 then yields the mean 〈β*r*_{b} 〉 ≅ 5.6, with all values of β*r*_{b} between 2.2 and 11.5. The narrow range of β*r*_{b} = (α/*D̄*)^{1/2}*r*_{b} might be related to the apparent constancy of bacterial abundance with respect to fluid volume (*27*). If we assume *D* ∼ ϕ*D* (*28*), this range is also consistent with the natural variation of the dimensional constants α, *r*_{b}, *D*, and ϕ (*21*), but it suggests that 〈β*r*_{b} 〉 ≅ 5.6 is atypically large. We therefore hypothesize that the effective diffusivity *D* ≪ ϕ*D*, because of sorption of enzymes to mineral surfaces (*9*, *29*), shielding of organic matter (*16*) in clay-rich microenvironments (*22*), or both.

Finally, we return to the slowdown of respiration rates discussed in the introduction. Middelburg *et al*. (*10*, *11*) have shown that the phenomenological effective rate *K*(*t*) ≡–*d*ln *g*/*dt* is well fit by the power law *K* = 0.21× *t*^{–0.99} (*11*). Their fit holds over nearly 10 orders of magnitude and includes water-column sediment-flux data in addition to analyses of sea-floor sediments. Figure 4 shows a similar analysis with the 23 cores in our database. Here, rather than numerically estimating *d*ln*g*/*dt* from noisy data, we calculate it analytically from Eq. 2. Conversion back to dimensional variables then yields (5) The approximation is obtained by averaging the logarithm of the slowly varying factor in parentheses over ln *t*, between the average minimum and maximum values of ln *k*_{min}*t* (–6.95 and –3.19, respectively). Its excellent correspondence to Middelburg's expression is notable, not only because it is derived without any free parameters, but also because it indicates that our analysis may also apply to the decay of organic matter as it sinks, accompanied by ballast minerals (*30*), to the sea floor.

The picture that emerges is one in which the decay and preservation of marine organic matter are strongly influenced by physical constraints. In our model, physical protection (*2*, *15*–*19*) manifests itself as spatially varying reactivity that decays rapidly with distance from the nearest microbe, with a characteristic length scale (β^{–1}) that depends on the active lifetime of enzymes and their effective diffusivity. Because this mechanism appears to be general, it seems likely that it should apply to other contexts, such as soils (*31*), where finely divided media restrict microbial motility.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/316/5829/1325/DC1

SOM Text

Fig. S1

Table S1

References