10/15/2020
diffusion_model
https://static-content.springer
.com/esm/art%3A10.1038%2Fs41396-020-00757-1/MediaObjects/41396_2020_757_MOESM19_ESM.html
1
/
4
Supplemental File 2
In [1]:
import
numpy
as
np
from
numpy
import
ma
import
matplotlib
import
matplotlib.pyplot
as
plt
from
matplotlib
import
cm
We hypothesize that preference for ammonium assimilation over nitrogen fixation consumed ammonium
diffusing inwards into the ANME-SRB consortia shown in Figure 8, forcing cells in the consortium interior to fix
additional nitrogen from
to compensate for the decrease in N source from ammonium uptake. Here, we
model ammonium diffusion into an ANME-SRB consortium and simultaneous ammonium assimilation as a
classic 1D diffusion problem with boundary conditions describing continuous input at a fixed location (x = 0)
with decay:
where
is the concentration of ammonium at time
,
is the diffusivity for dissolved ammonium, and
is the
first-order rate constant for ammonium assimilation.
1
5
푁
2
=
퐷
−
푘
퐶
∂
퐶
∂
푡
퐶
∂
2
∂
푡
2
퐶
푡
퐷
푘
Solving this differential equation at steady state
gives:
where
is the concentration of ammonium in the porewater and thus the concentration at the exterior of the
consortium.
(
=
0
)
∂
퐶
∂
푡
퐶
=
e
x
p
(
−
푥
)
퐶
0
푘
퐷
̅
̅
̅
√
퐶
0
Re-arranging to solve for
yields
푥
푥
=
−
l
n
(
)
퐶
퐶
0
푘
퐷
̅
̅
√
In our approach, we explore
holding
constant at 25 μM, the threshold value of ammonium
concentration above which diazotrophy is inhibited in incubations of methane seep sediments (
Dekas, et al.
2018
(https://sfamjournals.onlinelibrary.wiley.com/doi/full/10.1111/1462-2920.14342)
).
will thus
describe the depth within a consortium at which ammonium will be depleted sufficiently to induce diazotrophy,
described in the figure below as 'diazotrophy frontier depth'. We have calculated
= 0.0004-0.0009 hr
from
time-series measurements of ammonium concentration in methane seep sediment incubations under
ammonium-replete (≥ 25 μM) conditions (Dekas, et al. 2018, Fig. S5, "Mat-774", "Mat-794").
푥
(
,
퐷
)
퐶
0
퐶
푥
(
,
퐷
)
퐶
0
푘
−
1
10/15/2020
diffusion_model
https://static-content.springer
.com/esm/art%3A10.1038%2Fs41396-020-00757-1/MediaObjects/41396_2020_757_MOESM19_ESM.html
2
/
4
In [2]:
C
=
25
k
=
0.0004
We explore a range of
between 0.01 and 316 μM, representative of porewater ammonium concentrations
measured
i
n
s
i
t
u
(Dekas, et al. 2018).
퐶
0
In [3]:
C_0
=
np
.
logspace
(
-
2
,
2.5
,
1000
)
We use an estimate for the diffusivity of ammonium derived from the literature (
Krom and Berner, 1980
(https://aslopubs.onlinelibrary.wiley.com/doi/pdf/10.4319/lo.1980.25.2.0327)
) of
= 3.5 * 10^6 to constrain
maximum possible diffusivity, and explore a range of parameter values for
down to
, representing the
limitations on diffusion imposed by diffusion between cells within the consortium. A large range of values for
was employed here to reflect the challenges of measuring this parameter within ANME-SRB consortia.
퐷
퐷
1
0
1
퐷
In [4]:
D
=
np
.
logspace
(
1
,
6.7
,
1000
)
In [5]:
c_0
,
d
=
np
.
meshgrid
(
C_0
,
D
)
x
=
np
.
zeros
((
1000
,
1000
))
x
=
-
np
.
log
(
25
/
c_0
)
*
1
/
np
.
sqrt
(
k
/
d
)
10/15/2020
diffusion_model
https://static-content.springer
.com/esm/art%3A10.1038%2Fs41396-020-00757-1/MediaObjects/41396_2020_757_MOESM19_ESM.html
3
/
4
In [7]:
fig
,
ax
=
plt
.
subplots
()
x
=
ma
.
masked_where
(
x
<=
0
,
x
)
norm
=
cm
.
colors
.
LogNorm
()
cs
=
ax
.
contourf
(
c_0
,
d
,
x
,
50
,
norm
=
norm
)
cbar
=
fig
.
colorbar
(
cs
)
plt
.
xlim
(
25
,
c_0
.
max
())
plt
.
xlabel
(
'Porewater [NH$_4$$^+$] (μM)'
)
ax
.
set_yscale
(
'log'
)
ax
.
set_xscale
(
'log'
)
ax
.
set_xticks
([
30
,
50
,
100
,
300
])
ax
.
get_xaxis
()
.
set_major_formatter
(
matplotlib
.
ticker
.
ScalarFormatter
())
plt
.
ylabel
(
'Diffusivity (μm$^2$ h$^{-1}$)'
)
cbar
.
set_label
(
'Diazotrophy frontier depth (μm)'
)
plt
.
rcParams
[
'figure.figsize'
]
=
[
10
,
10
]
plt
.
rcParams
[
'font.size'
]
=
12
plt
.
show
()
10/15/2020
diffusion_model
https://static-content.springer
.com/esm/art%3A10.1038%2Fs41396-020-00757-1/MediaObjects/41396_2020_757_MOESM19_ESM.html
4
/
4
Plotting the results, we see that the depth at which we observe significant
incorporation and thus
diazotrophic activity in our consortia (1 to 10 μm into consortia, Fig. 8) is possible at diffusivities near that
measured for bulk marine sediment (≈
μm
hr
) for porewater ammonium concentrations that approach 25
μm. Thus, the simple model presented here broadly supports our hypothesized mechanism for the observed
gradient in diazotrophic activity presented in Figure 8.
푁
1
5
1
0
6
2
−
1