Supplement to: Revisiting the dissimilatory sulfate reduction
pathway
Alexander S. Bradley, William D. Leavitt, David T. Johnston
Quantitative models for sulfur isotope fractionation (Rees, 1973 ; Brunner &
Bernasconi, 2005 ; Johnston et al., 2007) have treated flux through a system
at steady-state equilibrium, and we continue this approach. We also provide a
framework for a more dynamic model that allows for non-steady state
behavior. This will be critical in future implementations, since experimental
work has shown transient accumulation and subsequent consumption of
trithionate and thiosulfate pools during batch growth of sulfate reducers
(Broco et al., 2005). These results indicate that any dynamic model must be
flexible enough to account for changes in both fluxes and reservoir sizes. Any
model must also include values for fractionation factors associated with each
reaction in the network. Most of these have not yet been experimentally
constrained, so our discussion here is limited to an exploration of the
consequences of the network structure, allowing for addition of
experimentally derived fractionation factors as they are obtained. This
exercise also provides a check on our new models, as we can set flow
parameters such that we recreate published model results. We provide two
examples, and for the purposes of an initial exploration use fractionation
factors that have previously proposed, while emphasizing that these may not
be experimentally verified.
To implement the network as a model, we write a differential equation
for the isotopic composition of each intermediate under non-steady-state
conditions (see below). In the non-steady-state approach, solutions require
quantification of intracellular metabolite concentrations, as well as an
understanding of the dependence of mass fluxes on metabolite concentrations
as dictated by enzyme kinetics. Detailed investigations are required to
constrain these variables, so for the present we simplify the solutions and
exclude the transient terms to solve for the steady-state equations.
The network and terms included in our model are shown in Figure S1.
Each intermediate has an isotope ratio R, equivalent to the ratio of 3xS/32S (3x
= 33, 34, or 36). Each reaction in the network has an associated flux term j.
This term may be in any units, but for simplicity we define j0 = 1 and model
other fluxes as a fractional proportion of this flux (ensuring mass balance).
Each reaction in this network also has an associated fractionation αλ, where
34λ
= 1, 33λ = 0.515 and 36λ = 1.90 (Farquhar et al., 2003). For each
simulation a set of fractionation factors are chosen, as is the reversibility, the
relative importance of each side reaction, and the proportion of each intermediate that
leaks out of the cell (Table S1). Several other rules can be applied to constrain
our model. For example, the stoichiometry of each reaction places
constraints on several of the fluxes, e.g. the flux of sulfite to trithionate (j7)
must be twice that of the reduced intermediate S2+ (j6). We apply these rules
to solve for each flux term for any given set of parameters.
The steady-state solution for the isotope ratio R of any given
intermediate is given as
ss
n
R
(  R ) j


(  R )
i
i
o
i
(1)
o
where  ( i Ri ) j i is the sum of all contributions into the the pool of

intermediate n. i.e. this term represents the summed contribution of all

reactions that form intermediate n, while accounting for their isotopic
compositions and the associated fractionation. For example, the isotopic
composition of intracellular sulfate is the product of both imported
extracellular sulfate, modified by its associated fractionation factor (RSO4out, j0
and α0), and the back reaction from APS (RAPS, j1r and α1r). The flux and
composition of material moving out of a given reservoir are treated similarly,
and appear in the denominator of Equation 1. The full suite of equations for
both the steady state and non-steady state solutions are given below.
Our model suggests that the challenge to understanding sulfur isotope
fractionation lies in understanding the magnitude of the fractionation factors
at each step, and the conditions that change the flux through various parts of
the network.
We define each fractionation factor α as

Rreactant
Rproduct
or alternatively,

R product   * Rreactant
For simplicity in the following equations, we have omitted superscripts

indicating the isotope to which we are referring. The equations are identical
for each isotope, except for variation in the value of λ and R. In these
equations, each Rn can be interpreted as 3xRn, and each α can be interpreted
as 
3x

For the intermediate thiosulfate, the model distinguishes between the

two sulfur atoms. The sulfanyl sulfur is denoted as ‘S2O3-S1’, while the
sulfonyl is denoted as ‘S2O3-S2’. Similarly for trithionate, the sulfanyl sulfur
is denoted ‘S3O6-S1’, while each of the two sulfonyl sulfurs are assumed to be
identical, and denoted as ‘S3O6-S2’.
Solution to the model for any given set of parameters requires two steps.
In the first step, we find the solution to the set of fluxes for the given
parameters. In the second step, we apply these fluxes and the given
fractionation factors to solve for the isotopic composition of each
intermediate.
Table S1 lists the model parameters, and Table S2 shows how they are
varied in the two examples given here.
Fluxes
Fluxes are solved by the following set of equations, which incorporate
both the steady-state requirement that the concentration of each intracellular
metabolite remain unchanged, and reaction requirements given by the
stoichiometry of each reaction:
j0 1
j0r  rev0




j1  j 0  j 0r  1  rev1
j1r  j1 * rev1
j 2  j 0  j 0r  1  rev2
j2r  j2 * rev2

j3  j2  j12  j14  j2r  j11  j7  j9  j15

j4  j 3  j6

j5  j 4  j8

j6  j3 * dsrs1

j7  2* j6

j8  j4 * dsrs2

j9  j 8

j10  j6  j16

j11  j10 *(1 TR1)

j12  j10  j10 *(1 TR1)

j13  j10  j8  j17

j14  j13

j15  j 2  j 2r * leakSO3



j16  j6 * leakS3O6
j17  j 8  j10 * leakS2O3
j18  j5  j13


The solutions for the fluxes j0, j0r, j1, j1r, j2, j2r are functions of the
parameters rev0, rev1, and rev2. To simplify the mathematical treatment, we
currently allow reversibility only on the reactions between sulfate and sulfite,
as outlined in the Rees model, and do not consider abiotic reactions. Therefore
this model may not yet appropriately treat back reactions from sulfide. A full
treatment will incorporate this when more information is available regarding
appropriate fractionation factors.
Under the given conditions, the fluxes j4 – j18 can each be rewritten in
terms of the input parameters and the flux j3, and are easily solved once the
solution to j3 is obtained. The solution to j3 is more complicated to find – it
depends on the parameters, but in the equations given above relies on the
solutions to j4 – j18, which are in turn dependent on j3. In other words, the
equation for the flux j3 is recursive.
We therefore write the equation for flux j3 as a non-homogenous
recurrence relation, of the form
j3 (n 1)  r * j3 (n)  k .
In this expression, the solution j3 at recursion depth (n+1) is a function

of the solution at depth n and the parameters r and k. The parameters r and
k are functions only of the input parameters, which include the reversibility
of the steps between sulfate and sulfite, the flux to each side reaction, and the
leakage of each intermediate. The closed-form solution to the recurrence
relation depends only on the input parameters and n. Models runs indicate
that a stable solution for j3 is typically found at a recursion depth of less than
100. Therefore, for a given set of parameters, we then find the solution for j3
for an arbitrarily large n >> 100 (typically 10,000). Remaining fluxes j4 – j17
are then solved in terms of j3.
Isotope ratios- steady state solutions
For the nth metabolic intermediate, we can write the equation for the
change with time in the ratio 3xR (excluding the ‘3x’ superscript for
simplicity):

Rn
  R j     R j



i
o
i
i
i
o
o
Mn
where, as explained in the main text,

into a pool, and
(1)
o
  R j
o
o
o
o
  R j
i
i
i
i
is the sum of contributions
is the sum of all fluxes out of a pool. Mn is the

reservoir size of metabolic intermediate n.

Equation 1 can be solved for the value of each Rn as a function of time:
 iRi ji
  j  / M t
Rn (t)  Rn (0) e  o o n 
 oRo
(2)
This transient term in this equation is dependent on the intracellular

concentration (Mn) of intermediate n. If we take the limit of this equation as t
approaches infinity, the transient term approaches zero and we have the
steady state solution to the isotope ratio Rn:
Rnss 
 R j
 R
i
i
o
i
o
These equations can be solved for each intermediate. Based on the

network shown in Figure S1, the steady state solutions are given as follows:









ss
RSO4

( 0 RSO4 out ) j 0  (1r RAPS ) j1r
1 j1   0r j 0r
ss
RAPS

(1RSO4 ) j1  ( 2r RSO3 ) j 2r
 2 j 2  1r j1r
ss
RSO3

( 2 RAPS ) j 2  (12RS 3O 62 ) j12  (14 RS 2O 32 ) j14
 2r j 2r   3 j 3   7 j 7   9 j 9  11 j11  15 j15
RSss3O 6S1 
( 6 RS 2 ) j 6
10 j10  j16
RSss3O 6S 2 
( 7 RSO3 ) j 7
12 j12  2 j16
RSss2O 3S1 
( 8 RS ) j 8  (10RS 3O 6S1 ) j10
13 j13  j17
RSss2O 3S 2 
( 9 RSO3 ) j 9  (TR1* RS 3O 6S 2 ) j10  (11RSO3 ) j11
14 j14  j17
RSss 
( 4 RS 2 ) j 4
 5 j 5   8 j8
RSss2 
( 3 RSO3 ) j 3
 4 j 4   6 j6
R Hss 2S 

( 5 R5 ) j 5  (13RS 2O 3S1 ) j13
18 j18
These steady-state equations form a discrete dynamical system of
recurrence relations, each of which is similar in form to the equation for flux
j3. The system of equations can be written in matrix form such that
r
r
x(n 1)  Ax (n)
r
where A is the matrix of coefficients of the steady state solutions, x is the

vector containing the isotope ratio of each metabolic intermediate, and n is
r

the recursion depth. We define x (0) as a vector consisting of the isotope ratios
where each R = CDT, i.e. each δ3xS=0‰. We wish to find a stable solution
r
r
r 
r
x (n) where x(n)  Ax (n)
. The solution for x (n) is difficult to find explicitly,
since in this application A is a singular matrix, with fewer eigenvalues than



the matrix dimension. Therefore, we find the stable solution by
computationally raising A to an arbitrarily large power n (typically 106) such
r
r
that x(n 1)  A n x(n) . The matrix A n reduces all isotope effects and fluxes to a

r
single term for each intermediate that multiplied by x (0) yields the steady


state isotope model.

Example 1: A Rees-like model
In this example, we maintain the full network structure, but simplify the
fluxes so that sulfite only reacts with DsrAB or back-reacts toward sulfate
(Figure S1). Reactions between sulfate and sulfite are reversible, while the
reduction of sulfite to sulfide (by DsrAB and DsrC) is irreversible. For
comparison to previous models, we examine two sets of parameters. In the
first set, we assign α0 = 1.003, and α2 = α3 = 0.975 (i.e. Rees). In the second set,
we assign α0 = 1.003, α2 = 0.975, and α3 = 0.947 (i.e. BB05). All other α values
are 1.000. It is important to realize that in this approach the full
fractionation between sulfite and sulfide is assigned to α3 (which corresponds,
approximately, to the formation of a Fe-S bond between sulfite and
siroheme). This may not reflect the actual location of fractionation in vivo but
is maintained here as an example.
With the fractionation factors from the Rees model, sulfide varies from
δ34SSO4-H2S = +3‰ for the case of an irreversible reaction, to δ34SSO4-H2S = -47‰
for a reaction that is 99% reversible (Figure S2a). These results replicate
Rees, but extend the calculations over the full range of reversibility, and
additionally calculate the δ34S values of intracellular intermediates, as in
Johnston et al. (2007). With this network structure, the network parameters
rev0 and rev2 are varied between 0 and 0.99 (rev = 1 would imply zero
forward reaction and mathematically be undefined), and represent the
proportion of sulfate import into the cell that can flux back out (rev0), and the
proportion of sulfite that regenerates APS (rev2). We assume that rev1,
representing the back reaction of APS to sulfate is fully reversible (rev1 =
0.99). In the absence of this assumption, the solution field collapses so that
the only expressed fractionation is α0. As expected, if the input values are
changed to mimic BB05, 99% reversibility for both rev0 and rev2 produces
δ34S SO4-H2S = -72‰, in accordance with the BB05 results, and a slightly higher
Δ33SH2S than Rees, with values up to +0.205 (Figure S2b). The features
illustrated with this network structure are fully consistent with previously
published work (Brunner and Bernasconi, 2005; Johnston et al., 2005; 2007).
Example 2: Sulfate reduction with thionate loop
Important features of the potential thionate loop have not been fully
addressed in previous models, and are only partially addressed here. For
instance, while the BB05 model included thionates, it did not recognize
intramolecular distinctions between sulfur atoms in trithionate or
thiosulfate, nor did it not incorporate branching (the BB05 network was
linear). For example, we recognize the importance of intramolecular
differences between the oxidized and reduced moities of thionates. But in the
absence of information regarding the kinetic or equilibrium isotope effects in
the production and consumption of these molecules, it is difficult quantify the
effect this side pathway will have on the isotopic composition of sulfide. If the
reactions involved in creating and then reducing thionates are irreversible,
then the effect of this pathway on the isotopic content of sulfide will be
minimal. However, if these species are in isotopic equilibrium there could be
a significant contribution. Figure S3 shows the results of a model in which
the Rees fractionations are applied to the central pathway, while including
the thionate loop and applying α13 = 0.995 and α14 = 0.985 (Smock et al., 1998)
to these reactions. While the fractionation factors may differ from the
estimate presented here, this principally demonstrates the prediction that
oxidized moities of thiosulfate should be enriched in 34S relative to the
reduced moities. The effect on sulfide will vary as a result of allowing
exchange reactions with the extracellular environment. Figure S4 shows the
results of the same model with varying amounts of produced thionates (up to
30%) allowed to leak from the cell to the extracellular environment. Our
purpose here is again not to offer a quantitative prediction, but to
demonstrate that these parameters can impact the isotopic composition of
products, and are currently unquantified in most experimental approaches to
understanding sulfur isotope fractionation.
Figures & Tables
Figure S1: Network of reactions in the isotope model. Similar to Figure 1 in
the main text, but including the model parameter associated with each
reaction.
Figure S2: a) Model results for the Rees-like pathway showing the isotopic
compositions of intracellular sulfide, sulfite, and sulfate with Rees
fractionation factor applied to α3. b) same result with BB05 fractionation
factor applied to α3.
Figure S3: Isotopic composition of intracellular sulfide (thin black dotted
field), sulfite (thin grey dotted field), and the sulfanyl (thick black field), and
sulfonyl (thick grey field) of trithionate in a reaction network with a thionate
loop and no leaks. Thiosulfate reductase is assumed to have the fractionation
factors given by Smock et al. (1998).
Figure S4: Isotopic composition of sulfide produced by a cell with a thionate
loop and various proportions of thionate leaks.
Table S1: Parameters in the branching network model
Table S2: Assigned values for parameters in the examples given in the text
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