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We present the results of an isotopeenabled reactive transport model of a sediment column undergoing active microbial sulfate reduction to explore the response of the sulfur and oxygen isotopic composition of sulfate under perturbations to steady state. In particular, we test how perturbations to steady state influence the cross plot of
Microbial sulfate reduction (MSR), where sulfate is respired in the absence of oxygen by microbial communities, is a key reaction in the global biogeochemical sulfur cycle and is understood to have been an important microbial metabolism over the course of Earth’s history (
The overall rate of MSR itself is linked to the relative degree of chemical equilibrium across four major intracellular steps, each of which are understood to be reversible and the resulting branch points within the cell themselves respond to changes in environmental conditions (
The measurement of the sulfur and oxygen isotope compositions of sulfate (
The exchange of oxygen atoms with the surrounding water that dominates the
The slope of the apparent linear phase (SALP) is defined by how quickly the
Indeed, the SALP may be one of the foremost geochemical tools available to interpret rate and mechanism of MSR, alongside ^{35}SO_{4} based assays (
In sediments and sedimentary pore fluids, one concern is that changes in the SALP may not reflect changes in the mechanism of MSR and may instead reflect nonsteady state conditions that influence the relative slope of one isotope composition over another (
We will present the results of four different endmember test cases that we have identified to demonstrate how the system responds to perturbations. Each case will have similar boundary conditions: a fixed sulfate concentration at the top of the sediment column and a concentration gradient of zero at the bottom of the sediment column. In this context, a boundary condition of zero concentration gradient at the bottom of the column implies that there is no flux of sulfate out of the bottom of the column. We choose this condition for our model because we know from past experience that the sulfate concentration at the bottom of these profiles approaches zero. Therefore, we assume that the only way that sulfate can be removed is through MSR, and that transport out of the bottom of the sediment column by diffusion is not possible. The first case will start with a uniform sulfate concentration throughout the column. Microbial sulfate reduction is then imposed everywhere in the sediment at a rate that is directly proportional to the concentration of sulfate at that point in the sediment column. The pore fluid sulfate concentration,
The model for this system was written using the isotopeenabled reactive transport software CrunchTope based upon the earlier CrunchFlow (
We outline the methodology in three sections, in the first we derive and explain how the chosen model parameters and reaction scheme represent the system of interest. In section two we detail the remaining necessary model parameters relating to system geometry and conditions. In section three we explain our chosen test cases and their motivation.
In this section, we will derive the isotope fractionation behavior that will be incorporated into the model. We then explain the reaction scheme and accompanying rate laws that we implement in CrunchTope to model MSR. We conclude by detailing how we incorporate our desired isotope fractionation and isotope exchange behavior into the specified reaction scheme by manipulating the rate constants and other input parameters.
Simplified sulfate reduction mechanism used in our model. We have combined steps 2, 3, and 4 described in the text into a single internal reduction step. All of the fractionation factors have been combined into a single overall sulfur fractionation factor in the final step, now made irreversible. The rate of oxygen exchange with the ambient water is tied to the reversibility of the single overall reduction step, as detailed in the text body.
We have combined the reduction of sulfate to sulfite via APS into a single, internal, reversible reduction step (
Steady state plots of
We model the oxygen and sulfur isotope compositions independently, using parallel MSR reaction schemes with rate constants that fractionate each isotope system separately. Each isotope system is subject to the same overall sulfate reduction rate. We now proceed to derive the fractionation factors, for each pair of isotopologues (
We calculate the sulfur and oxygen isotope fractionation factors by modifying the approaches of
Here
We assume that the intracellular metabolism is in steady state. At steady state, the difference between the forward and backwards reaction fluxes at each step in the mechanism (
Substituting
This allows us to rephrase
Thus, when approximately all of the intermediate valence state sulfur species are recycled, then the isotope fractionation takes on its minimum value, determined by ^{
f
}
For oxygen isotope fractionation we determine the value of ^{18}
We have assumed that the kinetic oxygen isotope fractionation factor is not influenced by the fraction of intracellular recycling.
The boundary and initial conditions of the individual sulfate and oxygen isotopologues have been calculated using the isotopic compositions of seawater. A full list of model parameters and their values is given in
A list of model parameters and values. In the cases where the parameter has a dependence on the value of r, we have used
Parameter  Symbol  Value  Units  References 

δ^{18}O of sulfate in seawater  δ^{18}O_{seawater}  8.6  ‰  ( 
δ^{34}S of sulfate in seawater  δ^{34}S_{seawater}  21.1  ‰  ( 
S isotope ratio in seawater  ξ_{S}  0.045094380  —  — 
O isotope ratio in seawater  ξ_{O}  0.002022444  —  — 
δ^{18}O composition of ambient water  δ^{18}O_{H2O}  0  ‰  — 
Sulfate concentration in seawater  [SO_{4}]_{seawater}  28  mM  ( 
Temperature 

14  °C  — 
Coefficient of molecular diffusion 

0.0275  cm^{2} h^{−1}  ( 
Total sulfate reduction rate 

0.00002  mM h^{−1}  — 
Porosity  Φ  0.8  —  — 
Space step  Δ 
0.4  Cm  — 
Steady state limit 

0.000001  mM  — 
Oxygen exchange extent 

1  —  — 
Oxygen exchange enrichment 

0  ‰  — 
Fixed^{34}S fractionation factor 
^{
f
}

0.975  —  ( 
Maximum change in apparent^{34}S fractionation factor  Β  0.05  —  ( 
^{34}S fractionation factor ( 
^{34}

0.9283  —  Calculated 
^{18}O fractionation factor ( 
^{18}

0.9938  —  ( 
O isotope ratio of ambient water  ξ_{ambient}  0.002047  —  — 
Fraction of sulfite recycled ( 
Γ  0.9333  —  — 
^{32}SO_{4} MSR rate constant ( 

0.175744  yr^{−1}  — 
^{34}SO_{4} MSR rate constant ( 

0.163143  yr^{−1}  — 
S^{16}O_{4} MSR rate constant ( 

0.175202  yr^{−1}  — 
S^{18}O_{4} MSR rate constant ( 

0.174116  yr^{−1}  — 
O exchange rate constant ( 

5.011000 × 10^{−3}  
Ion activity product 

—  —  ( 
VCDT34 

0.04416255  —  — 
VSMOW 

0.0020052  —  — 
In implementing our model system in CrunchTope, we explicitly model the concentration of each individual isotopologue within the sediment. We then applied various reactions subject to the available rate laws in CrunchTope. Sulfate reduction was implemented as an irreversible, aqueous reaction with a first order rate dependence on the concentration of the sulfate isotopologue being reduced. We have eschewed a Monod rate equation scheme in favor of a simple first order rate equation because we draw our major conclusions from system behavior at early times, with high sulfate concentrations, and the Monod formulation is equivalent to a first order concentration dependence in that scenario. The kinetic isotope fractionation of the isotopologues was introduced by differing the rate constants for each reduction rate law. The stoichiometry of the reaction is implemented in the model as:
There are four of these irreversible chemical reactions (
In
The oxygen isotope exchange with water was implemented using a single transition state theory (TST) rate law. Instead of explicitly calculating the rates at which different oxygen isotopologues of sulfate are added to and removed from the extracellular sulfate pool as suggested in previous work (
and the resulting rate law is:
In
We have included a single concentration dependence on the species involved because we want the percentage of intracellular sulfite recycled to be constant (since this is one of the major controls on the SALP (
Because the reservoir of water molecules is so much larger (∼2000 times) than the amount of sulfate in the system,
With the reaction scheme detailed in
We can derive the rate constants for each sulfur isotopologue of sulfate by choosing a time scale for our sulfate reduction reaction. We do this by choosing a maximum rate at which the reaction can proceed. This will be the same for both pairs (^{32}S, and ^{34}S, ^{16}O, and ^{18}O) to ensure that the overall sulfate concentration of each isotope system evolves in the same way. We will term this quantity
The concentrations [SO_{4}]_{raremax} and [SO_{4}]_{commonmax} are in such a ratio so as to reflect the natural isotopic composition of seawater:
The parameters
In
We then recall
Combining
The exchange of oxygen isotopes with the ambient water is modeled using a TST reaction, with a rate law given in
To relate the rate constant
It is important to note that the exchange of oxygen atoms with the ambient water cannot change the total amount of sulfate in the system (
In
In
Much effort has been made to determine the precise value of
To simplify matters, we take
We can now equate
Which allows us to express
We can then link the isotope exchange reaction flux to the overall rate of sulfate reduction by approximating
Substituting
We will present results for two values of
We calculate the molecular diffusion of sulfate in seawater,
In
We will initially assume there is no advection and later we will rerun all of the test cases with advection to see how it affects the SALP.
The mesh is made up of a 1D domain with 1,000 nodes equally spaced, each 0.4 cm, giving a sedimentary column of 4 m. The time stepping is handled dynamically by CrunchTope but is limited to a maximum of 1 year per iteration.
We model four different test cases (hereafter referred to as Cases 1, 2, 3, and 4—
A list of test cases and their boundary conditions. All cases have the same lower boundary condition of a concentration gradient of zero for all species.
Case  Initial Condition  Upper Boundary Condition 


1  [SO_{4}] = 27 mM everywhere. Isotopologues reflect seawater isotopic composition  [SO_{4}] = 27 mM. Isotopologues reflect seawater isotopic composition  0.00002 
2  [SO_{4}] = 0 everywhere. Isotopologues reflect seawater isotopic composition  [SO_{4}] = 27 mM. Isotopologues reflect seawater isotopic composition  0.00002 
3  Case 1 and 2 steady state profile  [SO_{4}] = 27 mM. Isotopologues reflect seawater isotopic composition  0.000002 
4  Case 1 and 2 steady state profile  [SO_{4}] = 10 mM. Isotopologues reflect seawater isotopic composition  0.00002 
In
This is particularly relevant to Cases 1 and 2, which are designed to investigate the differences in cross plot equilibration behavior when approaching steady state from initial conditions of uniformly high or low concentrations of sulfate.
In this case the sulfate concentrations in the sedimentary pore fluids are initially uniform, with concentrations and isotopic composition of pore fluid sulfate the same as seawater. We then initiate MSR throughout the sediment column and allow the system to evolve to steady state. An example of a natural system with a uniform sulfate concentration profile might be a monomictic lake such as Lake Kinneret in Israel or Lake FukamiIke, Japan which experience seasonal onset of MSR (
In Case 2, the sediment is initially without any sulfate in its pore fluid, as for the sediments in a freshwater lake. Then sulfate is added at the top of the sediment column at the concentration and isotopic composition of seawater, similar to what would occur during a marine transgression. The sedimentary porefluids then evolve to steady state, with sulfate supplied from above diffusing into the sediment and undergoing MSR within the sediment.
The initial state of the sediment column in this case is the steadystate profile reached in Case 1 and Case 2. In this case, the rate constants,
Similar to Case 3, the initial conditions in the sediment in this case is the steadystate profile from Case 1 and Case 2. The perturbation in this case decreases the concentration of sulfate at the top of the column from 27 to 10 mM, although still with the isotope composition of seawater. The sedimentary pore fluids are then allowed to evolve with the modified boundary conditions. This could be analogous to a restricted freshwater basin which experiences episodic marine influxes, such as the Songliao basin in northeastern China (
The sulfate pore fluid concentration profiles approach a new steady state as predicted in each of the four test cases (
Plots showing the evolution of the sulfate concentration with time during the approach to steady state. From left to right, test Cases 1, 2, 3, and 4. Increasing from early to late times as a percentage of the total time to steady state, going from blue (early) to red (late). See legend. See
The number of years taken for the system to reach steady state in each case, for a sediment column of 4 m. We note that this is the same regardless of the amount of recycling taking place because the exchange reaction does not affect sulfate concentrations.
Case no.  Time to 90% of Steady State (years)  Time to “full” Steady State (years) 

1  90  540 
2  53  465 
3  390  2,200 
4  53  450 
The time taken to reach a profile that has a 10% root mean square difference with the final steady state profile (i.e., 90% progress to steady state) is about six to eight times less than the total equilibration time (see
Top row,
When the recycling parameter is nonzero, (in
Our model reproduces the anticipated behavior of the cross plot of
One goal of this work is to understand the potential error in measuring SALP in natural pore fluids due to nonsteady state effects. We now present the behavior of the SALP during the approach to steady state in an effort to quantify this potential error.
Plot of the slope of the apparent linear phase for the
This reflects the underlying principle that the processes at work in this system mean that the gradients of interest equilibrate in either an overdamped or underdamped way. The results of the model suggest
We note that the timescale over which SALP might be incorrect due to a perturbation is vanishingly small in most natural environments (Figure 6A). For example, in a lake we’d expect SALP to be out of equilibrium for a few hours after some hypothetical perturbation, while in marginal marine sedimentary pore fluids up to a month; it is unlikely we would capture this transience in sampling (although one may be unlucky). We further quantify the potential error in measured SALP as a function of the percentage sulfite recycling occurring (
We consider our model results to three previously published sites, a river estuary, a shallow marine site, and a deep marine site (
An interesting intermediate case is presented in the work of
We suggest that it is extremely unlikely that any natural samples would record SALP analytically resolvable to be out of steady state; in deepmarine sediments there is not a process to push the system so far out of steady state, while in shallow marine, marginal marine, and terrestrial systems the response time is fast enough that we would not capture it. This means that the maximum error in the SALP is associated with recycling (
We can determine whether advection is important in any porous media by evaluating the Péclet number, Pe:
The terms
To test the effect of advection, we applied two artificially large advective fluxes to t the system,
Top row,
The latter case (Pe = 100, not shown) shows that the entire
We have presented the results of a 1D isotopeenabled reactive transport model to demonstrate the effects of transport and nonsteady state behavior on the sulfur and oxygen isotopic compositions in sediments undergoing MSR. In particular we explored the effects of these phenomena on the
The steady state that the system approaches is unaffected by the choice of boundary conditions or direction of approach, demonstrated in test Cases 1 and 2. However, the time scale over which the system reaches steady state was determined in part by the choice of initial conditions because this dictates whether diffusion or reaction acts as the primary driver toward steady state. Our results suggested that the system will approach 90% stability in approximately half the time between perturbation and the new steady state.
Cases 3 and 4 shows that perturbations within the sediment column have a minimal effect on the
Applying these principles to three previously published profiles in three different environments (estuarine, shallow marine, and deep marine) leads us to conclude that the measurements made of the SALP in those studies can be interpreted in terms of microbial metabolism and not in terms of changes in transport dynamics.
We conclude that perturbations due to transport and nonsteady state dynamics on the plot of
All datasets presented in this study can be found in online repositories. The names of the repository/repositories and accession number(s) can be found below:
GA, AT, XS contributed to conception and design of the study. AF designed and implemented the model and analyzed the results. HB, JD advised the implementation of the model. AF wrote the first draft of the manuscript. AF, AT, GA wrote sections of the manuscript. All authors contributed to manuscript revision, read, and approved the submitted version.
The work was supported by ERC 307582 StG (CARBONSINK) to AT, NERC NE/R013519/1 to HB, ISF (2361/19) to GA, and by a NERC ESS DTP Research Experience Placement to AF.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The Supplementary Material for this article can be found online at: