LIMNOLOGY
AND
June 1997
Volume 42
OCEANOGRAPHY
Number 4
Spatially explicit simulations of a microbial food web
Nicholas Blackburn
Department of Microbiology,
Umea University,
901 87 Umea, Sweden
Farooq Azam
Marine Biology Research Division, Scripps Institution
San Diego, La Jolla, California 92093
of Oceanography, University
of California,
t!ke Hagstriim
Department of Marine Ecology and Microbiology, National Environmental
Frederiksborgvej 399, I?O. Box 358, DK-4000 Roskilde, Denmark
Research Institute,
Abstract
A mechanistic model of a simplified microbial food web was implemented based on spatially explicit transport
of nutrients and energy sources by molecular diffusion. Individual organisms were simulated to interact in a twodimensional arena consisting of 70 X 70 interconnected hexagonal compartmentseach of 2 nl in volume. Release
of dissolved organic matter in conjunction with predation events resulted in patches of increased concentration
lasting long enough to be consumed completely by bacteria before being dispersed. Individual bacteria encountered
concentrations of dissolved organic matter varying up to loo-fold within time scales significant for growth. The
simulations also predicted that given a chemotactic response to nutrient gradients, bacteria could grow 50% faster
on average when gathered in loose clusters within patches. Extending the scenario to the three-dimensional case
and looking at a l-ml volume, patch-generating events were estimated to occur several times per hour and spread
to within a few millimeters in radius before being eroded by bacteria. The erosion time scale is probably longer
than the time between patch appearances resulting in patches overlapping and creating a markedly inhomogeneous
microenvironment. In such a scenario, foraging strategies enabling bacteria and predators to respond to elevated
concentrations of food could represent a significant adaptive advantage.
trolled by diffusion rates and swimming speeds. These
mechanisms are implicit in constants defining MichaelisMenten uptake kinetics (Jumars et al. 1993) and clearance
rates for flagellates (Fenchel 1987; Shimeta 1993) but are
rarely modelled explicitly. Although the physics of diffusion
and transport via shear is understood, it is rarely applied
directly in evaluating ecological processes in microbial food
web studies.
Organic substances are often recycled through events such
as point-source excretion and cell lysis. Although dissipated
by diffusion and shear, such sources create a potentially inhomogeneous environment. This results in an analytical gap
for ecologists studying the microbial food web; current modelling procedures generally assume spatial homogeneity although it is suspected that the microenvironment is nonhomogeneous (Azam and Smith 1991; Azam et al. 1993).
The purpose of this paper is to summarize a simple theory
for evaluating rates of physical transport in an ecological
context and to demonstrate the variable structure of the mi-
Within pelagic microbial food webs, nutrients must be recycled to sustain primary production. Bacteria and flagellates
are the main agents for degrading organic matter to mineral
components used by primary producers (Azam and Smith
1991; Azam et al. 1993). Carbon fixed by primary production flows from autotrophic organisms to bacteria, with an
opposite flow of nutrients to autotrophs via bacterial activity.
The flow occurs via a multitude of mechanisms operating
on different temporal and spatial scales (Andersson et al.
1985; Azam et al. 1993; Fuhrman 1992; Hagstrom et al.
1988; Jumars et al. 1989; Larsson and Hagstrom 1979; Nagata and Kirchman 1992). The transport of matter and
encounter frequencies between organisms are partly conAcknowledgments
We thank G. A. Jackson, an anonymous reviewer, Grieg Steward,
and David C. Smith for help and for comments on the manuscript.
This work was supported by a Swedish NFR grant (B-A-U
04452-320) (A. H.) and NSF grant OCE 92-19864 (E A.).
613
614
Bl&kburn
et al.
croenvironment and the influence that microorganisms have
on it, using spatially explicit simulations together with interactions between individual organisms.
Model of a microbial food web
We used a simple model of the microbial food web (Fig.
I). Small phytoplankton cells fixed carbon by photosynthesis
with limits imposed by the amount of nutrients (N) available.
Bacteria degraded dissolved organic matter (DOM) while
releasing excess nutrients and respiring carbon. Heterotrophic flagellates grazed on both the small autotrophs and bacteria, releasing DOM and nutrients in the process. They too
respired carbon. All organisms and DOM had fixed C : N
ratios so that excretion of nutrients could be simply calculated as the amount in excess after respiration plus growth.
Autotrophic growth was assumed to be nutrient limited, and
autotrophs did not leak either organic or inorganic compounds. The system was completely closed and assumed no
addition of nutrients from outside. Thus, a physical flow of
energy from autotrophs to bacteria in the form of DOM carbon had to occur via predation on autotrophs by flagellates
for bacterial growth. Likewise, a flow of nutrients released
during degradation of DOM by bacteria and predation by
flagellates had to occur for autotrophic growth.
The model contained no competition between bacteria and
autotrophs for inorganic nutrients as a result of equal and
fixed C : N ratios. It was assumed that uptake of DOM by
bacteria and nutrient by autotrophs was purely diffusion limited and could be estimated by using the analytical solution
to the mass flow J (Fick’s second law) for the diffusion of
a solute of concentration C and diffusion coefficient D toward a completely absorbing sphere of radius R (Jumars et
al. 1993):
J = 4mRDC.
(1)
The Stokes-Einstein equation and experimental findings
can be used to estimate diffusion coefficients that are inversely proportional to particle radius, with large molecules
of radius 2.5 nm (1,000 Da) having coefficients of - 10 6
cm2 s -I (Jumars et al. 1993)
We assumed that flagellates released DOM by emptying
food vacuoles (Andersson et al. 1985; Jumars et al. 1989).
Digestion was modelled as occurring instantaneously. Volume-specific clearance rates on bacteria were set to 2 X lo5
h I (Fenchel 1987). The same clearance rate on autotrophs
corresponds roughly to a realistic flagellate swimming speed
of 150 pm s-l (Fenchel 1987).
Detailed models exist of bacterial chemotaxis based largely on the behavior of enteric bacteria that appeared to display
a constant velocity but which regulated the time between
successive tumbles as a response to a concentration gradient
(Brown and Berg 1974). Those models were used by Jackson (1987) and Bowen et al. (1993) in detailed simulations
of bacterial chemotaxis around leaking algal cells. We decided to simplify the problem of chemotaxis to the ability
to form clusters because of limitations in accuracy due to a
two-dimensional approximation and limitations on the maximum allowable bacterial concentration in our simulations.
Fig. I. Flow diagram illustrating the model structure. The five
state variables (nutrient = Nutrient, autotroph = Auto, flagellate =
Flag, DOM = DOM, bacterium = Bact) are representedas stocks.
Mass conserved flow occurs between stocks, e.g. predation and bacterial growth, while nonconserved flows occur to or from clouds,
e.g. photosynthesis and respiration. Rates are depicted as labeled
valves. Dependencies between model components are indicated by
thin arrows. In spatially explicit simulations, each grid cell had its
own set of state variables and the model was simulated in parallel
for each grid cell.
A diffusion coefficient for a nondirected random walk can
be approximated as
V2T
6 ’
(2)
where T is the time between directional changes and V is
the velocity (Berg 1983). A diffusion coefficient of 10 + cm2
615
Food web simulations
s I was assumed, corresponding to V = 100 ,um SK’ and T
= 0.1 s. For flagellates V ==: 150 pm s-l and T = 2.5 s.
These parameters are realistic but somewhat arbitrary, i.e.
different combinations of D, V, and T could be equally valid.
The ratio between T for flagellates and bacteria corresponds
roughly to the ratio between the cube of their radii, consistent with estimations of the rotational diffusion coefficient
due to Brownian motion (Mitchell 1991).
To close the nutrient recycling loops in the model, the fate
of the top predator (the flagellate) must be taken into account. With the onset of starvation, the cell volume of flagellates decreases (Fenchel 1987). In this model, decrease in
the cell mass of any individual below a threshold value (80%
of the standard) was interpreted as the onset of starvation
and poor cell condition resulting in the cell lysing and being
recycled in the form of DOM. Flagellates lysed when prey
became too scarce to maintain a basic metabolism corresponding to levels of respiration measured for organisms
starved (Fenchel 1987).
The model can be visualized in a much more complex
setting where phytoplankton exudation, enzymatic dissolution of the particulate phase, bacteria-phytoplankton
interaction, cell lysis by virus, and other processes occur and can
alter the relative importance of protozoa egestion. The present model is an example, for purposes of illustration and
analysis, of how point-source DOM release events can be
important in DOM distribution and for bacterial growth and
why bacterial chemotaxis is an advantageous phenotyp.e in
that situation.
Simulation
The model (Fig. 1) was simulated on a grid divided into
a 70 X 70 two-dimensional lattice of hexagonal cells of
height d. Each cell had its own set of state variables, and
simulations consisted of parallel executions within each grid
cell for each simulated time step.
Bacteria, flagellates, and autotrophs were simulated as individual organisms that were either present or not within any
particular grid cell. The comparative operators in the rate
equations (Table 1) tested for the presence of organisms
while division by the time step implemented the discrete
events of predation and movement of organisms to neighboring cells.
All stocks were represented in units of mass. GrowthAuto
and Growth,,,,, implemented diffusion-limited uptake (Eq. 1)
of nutrient and DOM, respectively. The term on the right
hand side of the < operator in the predation equations
PredBac,)is the fraction of a grid cell volume that
@-~~AuK~~
can potentially be “cleared” by a flagellate per time step
and is equivalent to the probability of encounter within a
time step. The last factor in Excr,;,,, implemented flagellate
death by starvation.
Molecular diffusion of DOM (Diff,,,)
was an implementation of the finite difference approximation to Fick’s fist law
for the gross mass flow F of a solute at concentration C,
with diffusion coefficient D, through an interface of area A
into adjacent compartments separated by distance d:
The area A was calculated as one-sixth of the area of the
wall of a grid cell. The summation sign represents flow to
each of six neighbors.
Only one individual from each class (bacterium, flagellate,
or autotroph) was allowed per grid cell (third factor in
Diff,,,,). When an individual accumulated a biomass twice
that of its standard, it divided by allowing the movement of
half of the biomass into an adjacent cell (first factor in
Diff,,,). The term on the righthand side of the < operator in
the second factor in Diff,,,, was an estimate of the probability
of moving into an adjacent grid cell. Bacterial chemotaxis
was implemented by allowing movement up the DOM gradient only (second factor in DiffcZh,,,,,), simulating a biased
random walk. Combining this probability of moving to an
adjacent cell with the distance between grid cells, the time
step, Eq. 2, and swimming velocity gives an estimate of the
efficiency of the random walk:
VT
3d’
(4)
The parameters used in simulations gave an efficiency of
3%.
The constant factor in Diff,,,, was the fraction of mass
per unit time that was removed from a grid cell and was
used to estimate a time step. By using the given parameters,
the factor gives a value of -0.004 s-l to each of six ncighbors. On the basis of this, a time step of 30 s was chosen.
Likewise, the diffusion of nutrient would have required a
time step of 3 s. As this would have made simulations too
time-consuming for this particular study, the released nutrient was made to be available uniformly in space (implemented as a global variable).
The simulator that was used to perform the simulations
(Cellmatic) is designed to take equations defining the flow
between stocks as the language for implementing spatially
explicit models (see also Blackburn and Blackburn 1993).
Parallel execution within each grid cell and mass conservation of flow between stocks was guaranteed by the simulator. The grid was wrapped horizontally and vertically corresponding to the surface of a toroid. Important in all stages
of implementation was the ability to view simulations animated in real time in a virtual reality scenario. Stand-alone
versions of the model are available at the WWWeb site http:
Nwww.dmu.dk/PublicFiles/.
Results
An equivalent model to the one given above without any
physical transport was first implemented and run to cquilibrium. The model did not give rise to any predator-prey oscillations. The equilibrium values were subsequently used as
start values for the stocks in a spatially explicit simulation.
An initial simulation was run for 500 h with a time step of
30 s on the 70 X 70 grid (Fig. 2). Deviation of stock levels
from initial values was due mainly to stochastic noise in the
simulation. The jaggedness in the flagellate and autotroph
616
Blackburn et al.
Table 1. Equations for the rates of flow: Division by & simulates discrete events, e.g. predation, death or diffusion of individual
organisms. The function rand ( ) returns a number between 0 and 1. Expressions containing comparative operators (<, >, #) evaluate to
1 if true and 0 if false. The expression if (Bexp) then (Tcxpr) else (Fexpr) evaluates to (Texpr) if (Bexpr) evaluates to 1 and to (Fexpr)
otherwise. All organisms (autotrophs, bacteria, and flagellates) diffused to a neighboring cell chosen at random (by the simulator) and
according to the Diff,, rate (Org = organism). DOM diffused according to the Diff,,,, rate to six neighbors in the hexagonal grid. The
nutrient was simulated to distribute instantaneously throughout the grid. Parametersare given in Table 2.
Equations for the change of state: Differential equations defining the model dynamics can be written in terms of the rates of flow.
Rate of flow
Growth,,,,,, = [Auto(x, Y, 0 > 01 X 47r x R,u,o X Q,utr,c,,tX
Growth,,;,,,= [Bact(x, y, t) > 0] X 47~X RllnL,X D,,,,, X
Pred,,;,,,= [Flag@, y,
Pred,,,,, = [Flag(x, y,
t)
t)
> 0] X Bact(x, y,
> 0] X Auto(x, y,
Nutrient@, y, t) X C . N
v
(‘Cl1
DOM(x, y, t)
VCell
t)
t)
X
Clear,,,,, X hlg x dt
VCdl
,/
dt
X
ClearA,,,,, x Gag X dt
VCdl
,/
dt
RcsP,,;,~,
= Growth,,,,, X RespFrac,,,,,
R=4-4.,,,= W%,,,
+ Pred*,,,,) X RespFrac,,,, + Flag(x, y,
Excr,,, = (PredI,;,c,+ Pred,,,,,,)X ExcrFrac,.,,, +
Uptake =
Release =
t)
X RespO,.,,,
Flag& y, 0
X Flag(x, y,
dt
t)
< 0.8 X
Growth,,,,,,,
C .N
Res~,~,,~,
+ Resp,,,,
C:N
Iwlgon
2
I/W/y
t
d2x DOWx,
y, t>
DifL,,= x ix &a4
1(
Org(x,y, Q else Org(x, y, t)
Diff,,,, = if Org(x, y, t) > 2 then
2
X
4 X Do,,: x dt
d2
rand() < -
X [Org(x + dx, y + dy, t) # 0]
dt
I
DifL,,,,, = Diff,,,, X [DOM@ + dx, y + dy, t) > DOM(x, y, t)]
Change of State
Auto& y, t + dt) = Auto(x, y, t) + (Growth,,,,,, - Pred*,,,,- Diff*,,,,,) X dt
Flag(x, y, t + dt) = Flag(x, y, t) + (PrcdAL,,O
+ Predtl,,, - Resp,.,,,,- Excr, ,ag- Diff,:,.,,) X dt
Bact(x, y, t + dt) = Bact(x, y, t) + (Growthlji,~,- Predn,,,,- Resp,,,,,- Diff,,,,,) X dt
DOM(x, y, t + dt) = DOM(x, y, t) + (Excr, ,‘,y- Growtht,,,, - Diff,,,,,) X dt
Nutrient(x, y, t + dt) = Nutrient(x, y, t) + (Release - Uptake) X dt
biomass was the result of predation events occurring in the
relatively small volume containing only a few individuals.
Results are expressed as bacteria equivalents. The simulated volume was close to 0.01 ml and 10” bacteria ml-, (lo3
per grid) was equivalent to a carbon concentration of 160
nM. Both the flagellates and autotrophs were 500X the mass
of a bacterium (Table 2) and did not divide before reaching
twice that mass.
The population dynamics appeared fairly stable, with two
events of flagellate starvation and subsequent cell lysis oc-
curring after 100 h and 350 h, resulting in the largest of the
DOM peaks. The smaller peaks were a result of predation
on autotrophic cells, while the peaks occurring during bacterial predation were invisible in Fig. 2. A snapshot of the
DOM concentration on the grid 30 min after the predation
of an autotroph is shown in Fig. 3. The central cores of these
regions had concentrations many times over the background
level and also proved to be long-lived, lasting up to a few
hours. Nutrient concentrations stayed around 80 bacteria
equivalents per grid or 2.6 nM (Fig. 2).
617
Food web simulations
Fig. 2. The aynamics of stocks summed over the whole arena
m a spatially explicit
simulation.
Fig. 3. The DOM concentration in bacteria equivalents per grid
cell 30 min after the predation of an autotroph. Spatial scale in
Units of mass are in bacteria
equivalents in the simulated volume of about 10 ~1. Discontinuities
were the results of events of predation and death.
millimeters.
To get an idea of the variation in DOM concentration that
the bacteria encountered under these conditions and how
quickly patches eroded and dispersed, the DOM stock was
sampled with a resolution of 10 min at a single location on
the grid (grid cell 30,30). The result showed peaks with amplitudes 10X-IOOX the background level and widths at half
amplitudes of several hours (Fig. 4). A few of the predation
events on autotrophs gave only low peaks because they occurred some distance away (e.g. at 170 and 420 h). The
maximum distance between events was half the span of the
grid (-5 mm) due to the grid being wrapped at the borders.
Many smaller peaks could be seen on the logarithmic scale
of Fig. 4 that were due to predation on bacteria in the vicinity. The average DOM concentration followed closely the
dynamics of the local concentration (Fig. 4), indicating that
it was the amount of release into the arena more than patch
structure within the arena that determined bacterial growth
characteristics.
Two further simulations were run, both starting with the
final configuration of the first simulation shown in Fig. 2.
One used exactly the same model as the first simulation
(-ChT) while the other had the added chemotactic response
of bacteria to nutrient gradients (+ChT), allowing them to
move up DOM concentration gradients. With this mechanism, the chemotactic bacteria gathered in very distinct clusters around patches (Fig. 5), limited in concentration to 5 X
lo5 ml-l by the model implementation.
The chemotactic response affected DOM dynamics (Fig.
6D), where peaks were much sharper and narrower with the
chemotactic response. Chemotactic bacteria moved to areas
of high DOM concentration and diminished the stock much
quicker than they would have been able to do when evenly
Table 2. Model parameters.
Parameter
dr
RA.,,,
R a‘,
R,.,,,
VP,“,
DN.,.,,.,
&,I4
D,,,,
DA,,,”
DFI,,
d
VC,,,
C’%h,,
Cle=r,,,,,
R-P,,,,,,
R=P,,,,
R=P%,.;
E=,.,,,
C:N
VZd”e
Description
Simulated time step
30 5
Radius of autotroph
2 x 10-q cm
0.25 x lo-” cm
2 x 10-4 cm
3.5 x 10-1 ml
10micm2 s-l
Radius of bacterium
Radius of flagellate
Standard flagellate volume
Diffusion coefficient for nutrient
Diffusion coefficient for DOM
Diffusion
coefficient for bacteria
Diffusion coefficient for autotroph
Diffusion coefficient for flagellate
Height and diameter of a grid cell
Volume of a grid cell
Volume-specific flagellate clearance on bacteria
Volume-specific flagellate clearance on autotrophs
Fraction of bacterial uptake respired
Fraction of predation respired
Cell mass required for metabolism
Fraction of uptake excreted as DOM
Carbon-to-nutrient ratio
10-e cm’s-’
lo-” cm: s-’
IO-’ cm’sml
IO-” cm’s
I
0.013 cm
2.2 x 10-e ml
56 s-1
56 s-’
0.5
0.25
8 x 10 8 s-1
0.25
5
Blackburn
618
et al.
T
1
Fig. 4. From the same simulation as Fig. 2 but showing the
DOM stock within a single grid cell (30,30) sampled with a resolution of 10 min and plotted on a log scale together with the grid
cell DOM content averaged over the whole arena (stippled). Units
in bacteria equivalents per grid cell. The flagellate dynamics (right
scale) arc shown for identification of the source of the peaks.
distributed. The characteristic shapes of the respective peaks
were quite invariable, so an evaluation of the chemotactic
advantage could be done by examining the cell-averaged
specific DOM uptake rates for the bacterial population for
the first DOM peaks (Fig. 7). Integrating the second peaks
(positioned around the 30-h mark) of both simulations for 5
h (the width of the chemotactic peak) gave a time-averaged
specific DOM uptake rate of 0.032 h-l for chemotactic bacteria and 0.022 h ’ for nonchemotactic bacteria--a 48% increase in uptake and hence growth with chemotaxis. The
increase was 63% for the third peak (positioned around the
80-h mark) with time-averaged specific uptake rates of 0.025
h-’ and 0.015 h I, respectively.
Both simulations showed similar responses for the first
100 h, after which a chance occurrence of predation on all
the autotrophs in +ChT within a short timespan of 50 h (Fig.
6A) resulting in the two simulations deviating considerably.
This dramatic predation event was not due to the clustering
of the bacteria because neither the autotrophs nor the flagellates were given any kind of behavioral response to patches. After the extinction of the autotrophs in +ChT, the nutrient level increased quickly (Fig. 6E).
Discussion
Patch dynamics-The
scenario that appeared from simulations was that of point-source release of dissolved organic
matter initially dispersing rapidly by molecular diffusion to
form a patch of a few millimeters in diameter (Fig. 3). after
which diffusional flux slowed down significantly and bac-
Fig. 5. The distribution of bacteria as a result of chemotactic
response to the situation shown in Fig. 3. The value of a grid cell
represents the biomass of the bacteria residing. Bacteria divided
after acquiring a biomass of 2. Note that many of the bacteria in
the center were ready to divide. Spatial scale in millimeters.
terial activity eventually eroded the patch away. Within the
simulated arena, patches occurred about every 20 h but were
consumed by bacteria within -5 h (Fig. 7).
The scenario was created in a virtually two-dimensional
world. Computational limitations made a two-dimensional
arena a natural choice for initial implementations. It is, however, possible to estimate what the same scenario would look
like in three dimensions. Using analytical solutions to concentration for diffusion of a solute with diffusion coefficient
D from a point source into a flat volume of height h (Eq. 5)
and into an infinite volume (Eq. 6), the concentration ratio
was calculated to be independent of position but dependent
on time f (Eq. 7).
(5)
(6)
(7)
For dlftusion from point sources, there IS a sharp boundary
within which all mass is contained and the radius for that
boundary is the same for the two- and three-dimensional
case (Table 3). On time scales of 5 h, the radius of patches
is on the order of 0.5 cm and the concentration ratio 30
(Table 3). At the same time, the number of bacteria within
the same radius would be 60 times higher in the three-dimensional case as would the frequency of patch-generating
events (Table 3). The scenario within a spherical volume of
radius 0.5 cm seen in relation to the two-dimensional case
would be that of DOM patches dispersing rapidly but being
consumed much more quickly due to the higher bacteria
numbers within the equivalent radius, irrespective of any
ability to respond to concentration gradients. Subsequent
patches would be superimposed on previous ones within the
same volume due to a high frequency of occurrence (3 h-’
compared to 0.05 h-l). The three-dimensional scenario is
619
Food web simulations
I
5000 -
2
lOOO-
I
:A
I
.------......
-4
o-1000
0
5500,
2500 -
I
I
I
I
I
1
E
5
5
8
3
E
Time (hours)
Time (hours)
A
B
600
500
‘8
i-4
a
i ;,r
400 200
I
0
I
z
5
I
dsi
I
3
.
400
300
200
100
.
I
I
!2
8
U i
0
I
I
I
I
I
I
2:
g
5
g
6
g
Time (hours)
Time (hours)
C
D
200 -
I
;
I
,
150is
‘5
2
.
0
0
I
I
I
I
I
I
%
83
z4
3
3
E
Time (hours)
E
Fig. 6. Stocks summed over the whole grid in units of bacteria equivalents per grid comparing
simulations with chemotactic response by bacteria (stippled) and without (solid).
therefore more dynamic and contains more intricate spatial
structure than the two-dimensional one.
While released organic matter was given a fixed diffusion
coefficient corresponding to molecules in the 1,OOO-Darange
in this model, it consists most likely of a spectrum in size
class. Stringy and sticky compounds can further reduce diffusion rates and form transparent matrices, which in turn can
act as collectors, further enhancing patchiness and creating
islands that bacteria might move between and adhere to.
Chemotaxis-A
biased random walk of 3% efficiency as
simulated here would allow bacteria to move toward the
cores of patches 3 mm away in -0.25 h. While this is quick
in the two-dimensional arena in relation to the lifespan of
patches, it might be too slow in the three-dimensional case
where patches were estimated to diffuse and erode more rapidly but at a higher frequency. Rather than the “run and
tumble” strategy observed for enteric bacteria (Berg and
Brown 1972), a “dart and stop” strategy might be more
620
Blackburn et al.
1
Table 3. Comparison in concentration and volume between
patches spreading into a disc (C,, V,) and sphere (C,, VJ by molecular diffusion from a point source after time t calculated from
Eq. 7. The patch radii were calculated by numerical integration of
analytical solutions to the concentration fields (Eq. 5, 6). Volume
ratios were calculated based on the disc thickness used in the model
(0.013 cm). Organism numbers and frequency of predation events
are proportional to volume.
Time(s)
Patch radius (cm)
Gf C-3
V,f v2
Time (hours)
Fig. 7. Cell-averaged specific DOM uptake by bacteria with
chemotactic response (stippled) and without (solid).
efficient in such an environment. Something intermediate
has been observed in recent studies of swimming behavior
of small marine bacteria, which display very high swimming
speeds (up to 200 pm s-l) and rapid decelerations (Mitchell
et al. 1995a,b). Energy and time-scale considerations can
partly explain why this should be so (Mitchell 199 1). Those
bacteria were also observed to form swarms (Mitchell et al.
1995a, b).
Jackson (I 989) and Bowcn et al. (1993) used steady-state
solutions for concentration fields around a leaking phytoplankter and concluded that bacteria would have difficulty
detecting exudate from cells much smaller than 10 pm in
diameter. Comparing that scenario with the parameters used
in their models with a point-source excretion situation given
in Eq. 5, one can estimate that the concentration of a patch
such as the one shown in Fig. 3 at distance 0.1 cm from its
source 2,000 s after occurrence corresponds to that from an
exuding phytoplankter of diameter >16 pm. The classical
model for chemotaxis estimates changes in run-lengths of a
few percent under the same circumstances (Brown and Berg
1974), in accordance with our approximation. Together, these
observations support the feasibility of bacterial clustering as
simulated, explaining possible adaptive advantages for bacterial chemotaxis even in oligotrophic
totrophs tend to be small.
ecosystems
where au-
Growth-The
initial organism and solute concentrations
for the two-dimensional simulations define much the same
carrying capacity as would apply to a similar three-dimensional simulation with equivalent initial concentrations, i.e.
organism concentrations and encounter rates between organisms would not be expected to be significantly different. It
was argued that the main difference between the two cases
would be that bacteria would witness smaller, more rapidly
eroding DOM patches but at higher frequencies in the threedimensional case. The two-dimensional study illustrates that
the microbial food web functions in about the most desertlike conditions imaginable in an ocean, based on established
rates of diffusion and assuming diffusion limitation on uptake. From the slopes of Figs. 2 and 6A it could be estimated
100
500
1,000
2,000
18,000
0.05
3
5
0.1
6
10
0.15
9
15
0.2
12
20
0.6
36
60
that the doubling times for the autotrophs varied from 2 to
4 d and were as rapid as 2 d for the flagellates (Fig. 6B).
The whole bacteria population could at least divide every
few days (Fig. 7) and could have doubling times as high as
15 h (Fig. 2). This is consistent with observations that bacteria are equipped with uptake systems to cope with varied
nutrient levels (Azam and Hodson 198 1; Nissen et al. 1984).
Low numbers of actively growing bacteria (5 X lo4 ml-l)
as predicted here have been observed in several ecosystems
where only a fraction of counted bacteria actually contain
nuclei (Zweifel and Hagstriim 1995). The remaining contain
no DNA (ghosts) and can be assumed to be dead. Zweifel
et al. (1996) and Blackburn et al. (1996) created a model
based on a multistage flowthrough experimental system
showing how population dynamics can explain these observations under nutrient regenerating conditions. The model
predicted that rates of bacterial infection and cell lysis by
viruses
strongly
influenced
bacterial
growth
dynamics
and
that the majority of phosphorous for bacterial growth was
supplied through cell lysis of the bacterial population itself.
Also important to the population dynamics was the role of
protozoa for ingesting lysed bacterial cells. Virus dynamics
offers an interesting extension to the scenario presented in
the present model and offers another important source of
nutrient regeneration through point-source events.
Population dynamics-Population
dynamics was clearly
affected by the small number of individuals resulting in the
possibility of a sudden extinction, as was the case for the
autotrophs in +ChT (Fig. 6A). The high frequency of encounters between autotrophs and flagellates was also seen
after about 200 h in -ChT (Fig. 6A). This was due to the
autotrophs dividing in synchrony at about 60 h (Fig. 6A)
and points to synchrony as being an important factor in the
dynamics. Synchronization has been observed in nature
(Hagstrom et al. 1988; Vaulot et al. 1995) and is an interesting case for study. The bacteria population became desynchronized (Fig. 5), whereas it was observed that the autotrophs did not. This was due to the environment being
identical to each individual (N was uniformly available). In
nature, the diel pulse of solar energy might have synchronizing
effects and point-source
N production
does result in
patches. Inorganic nutrients disperse 10 times faster than
does the DOM simulated here based on molecular weight,
but evaluation of the effect of rapidly diffusing patches
621
Food web simulations
should take into account correspondingly increased potential
uptake. Bacterial activity on DOM patches can also result in
continuous local nutrient release that would enhance potential uptake by autotrophs in the vicinity.
The spatially explicit simulation resulted in stock levels
resembling the initial values determined by a simulation
based on a homogeneously distributed, infinitely large population (Fig. 2). However, this was due to the outcome of a
highly complex game, where subpopulations at different locations in space were continuously intermingling. The condition of a subpopulation at any time depends on the local
level of food concentration and predation pressure. The degree of heterogeneity in the whole population depends subtly
on the motility of predators and prey (De Roos et al. 1991).
To see this, consider the hypothetical situation where both
flagellates and bacteria have very low motility but still forage with the same efficiency given by the model parameters
without responding to concentration gradients. Holes in the
bacterial population would form in the vicinity of flagellates
while bacteria would tend to concentrate by growth elsewhere. The tendency for holes to form would decrease with
increasing flagellate motility, and at the same time flagellates
would increase their exposure to bacteria. It makes sense that
flagellates should swim. Increasing bacterial motility would
seem to have little effect other than increasing their risk of
predation unless they swim to increase their growth rate by
exposure to patches by chemotaxis. This is because, by
swimming, they would increase their flux toward predators
while remaining homogeneously distributed. Mitchell et al.
(1995a) observed that a small fraction of marine bacteria
moved at high velocities, and this has been supported by
direct observations with dark-field microscopy (Blackburn
unpubl. obs.). Apart from the apparent fact mentioned above
that a substantial fraction of bacteria are inactive, the unnecessary increase in predation risk at times between encounters of patches may offer an explanation for remaining
relatively immobile in the absence of concentration gradients.
Two simulations with identical start conditions displayed
different population dynamics (Fig. 6). Although the two
simulations were not identical (+ChT and -ChT), differences in outcome were attributed to a chance clustering of
both flagellates and autotrophs in +ChT at -100 h resulting
in the extinction of the latter.
Turbulence and shear-Eddies in turbulent water have a
certain minimum size and a limited potential for mixing. The
size spectrum of eddies ranges from many meters down to
the Kolmogorov scale of a few millimeters. Most of the mixing energy, however, is contained in eddies of a few centimeters in size (Lazier and Mann 1989). Within eddies there
is always a shear that results in water motion much like a
river, where the current in the middle is faster than the current at the edges. Objects floating a bit away from the middle
will experience other objects closer to the middle floating
past them at higher velocities. The velocity gradient, also
called the shear rate a, is mostly linear and unidirectional
over small distances (Bowen et al. 1993; Bowen and Stolzenback 1992) and is often estimated as
a
x-x-Av
Ax
(e/v) “2
2’
(8)
where E is the turbulent dissipation rate (ranges from 10 2
cm2 s-j in the upper mixed layer to lo-” cm2 s-’ when mixing is low) and v is the kinematic viscosity (-0.02 cm2 s-l).
The shear rate thus varies from -0.5 s I to 0.005 s-l.
Shear creates a mechanism whereby objects can be
brought into contact and result in capture or coagulation. The
larger the objects in question, the larger the relative velocity
over their diameters and the higher the frequency of contact.
However, for all but the largest microbes (100 pm radius or
more) or the slowest diffusing particles (- 1 pm), diffusion
will dominate as a means of transport to organisms (Jumars
et al. 1993). For example, a bacterium of 0.7 pm radius
experiencing a shear of 0.5 s-l and utilizing a substrate with
a diffusion coefficient of lo+ cm2 s-l would only witness
-1% increase in flux due to the flow.
The dynamics of patches can be altered considerably by
the effects of shear that were not taken into account in our
simulations. Shear tends to stretch patches into streamers,
while diffusion continues to account for dispersion perpendicular to the length axis (Bowen and Stolzenback 1992).
The effects of shear can be simulated dynamically in spatially explicit models. A low shear rate of 0.005 s-l over the
distance of one grid cell (0.013 cm) would give a velocity
of 6.5 X 10 5 cm s-l or 0.5% of the extent of the grid cell
per second. The relative velocity in the next grid cell would
be l%, etc. Thus, a flow can be implemented by shifting
grid cells in a fixed direction at intervals corresponding to
their distance from a reference point, e.g. the middle of the
grid.
Assuming a low shear rate of 0.005 s-l and length scales
for dispersion of 0.2 cm, the induced velocity is on the order
of 1 X lo-” cm s-l, resulting in patches stretching -4 cm
h-l. At high shear rates, patches would be stretched within
a few minutes. By using these rough scaling arguments, it
can bc concluded that at low shear rates or in long periods
between bursts of shear, resulting dispersion is limited, At
medium to high shear rates, patches probably become tendril-like structures interlaced with others. While dispersion
of patches is more rapid in shear, the effects should be evaluated taking into account increased availability to the bacterial community and subsequent erosion, as was done in the
purely diffusional case above. More detailed quantitative
analysis must take into account the varying degrees of diffusivity of released organic matter.
Conclusions
We have outlined a general purpose method by example
for exploring spatially explicit population dynamics and coupled biogeochemistry based on rates of flow between stocks
and adjacent compartments in a grid configuration.
In order to explore population dynamics on a individual
basis, “electronic microorganisms” were released to interact
with one another in an animated game, resulting in a virtual
reality microbial food web. In that scenario, the organisms
were responsible for the continuous creation and erosion of
food patches. Structure in the microenvironment was pre-
622
Blackburn et al.
dieted to influence strongly short-term growth rates of individual bacteria. The existence of an ecological niche was
predicted, where bacterial chemotactic response to pointsource excretion events by predators represented a significant competitive advantage over bacteria unable to respond
to concentration gradients. It makes little sense for bacteria
to swim unless they can respond to concentration gradients
due to an increased predation risk.
Simulations were two-dimensional and, although it was
argued that reduced dispersion relative to the three-dimensional case is partly cancelled by reduced frequency of
events and organism numbers, the difference in dynamics
between the two- and three-dimensional cases is unpredictable. The two-dimensional case can be used in a qualitative
manner to identify mechanisms of importance in a spatially
explicit scenario as it was used here, or to simulate experiments performed in flat chambers.
A goal for this study was to inspire further exploration of
microbial food web ecology focusing on the effects of interactions between individual species of bacteria, autotrophs,
and predators on an individual scale. Monitoring the behavior of individual organisms can give important clues to the
true structure of the microenvironment and can identify specialized responses by different species to, e.g. food patches.
The use of widefield microscopy and real-time image analysis for tracking individual organisms would be powerful
tools in conjunction with microcosm experiments for acquiring related data.
References
ANDERSSON, A., C. LEE, E AZAM, AND A. HAGSTROM.
1985. Release of aminoacids and inorganic nutrients by heterotrophic
marine microflagellates. Mar. Ecol. Prog. Ser. 23: 99-106.
AZAM, E, AND R. E. HODSON. 198 1. Multiphasic kinetics for D-glucose uptake by assemblages of natural marine bacteria. Mar.
Ecol. Prog. Ser. 6: 213-222.
-,
AND D. C. SMITII. 1991. Bacterial influence on the variability in the Ocean’s biogeochemical state: A mechanistic
view. NATO ASI Ser. 627: 213-236.
-G. E STEWARD, AND A. HAGSTROM. 1993. Bacteria-organic matter coupling and its significance for oceanic
carbon cycling. Microb. Ecol. 28: 167-179.
Bu~icl, H. C. 1983. Random walks in biology. Princeton University
Press.
-,
AND D. A. BROWN. 1972. Chemotaxis in Escherichia coli
analysed by three-dimensional tracking. Nature 239: 500-504.
BLACKBURN, N. B., U. ZWEIFBL, AND A. HAGsrRijM. 1996. Cycling
of marine dissolved organic matter. II. A model analysis.
Aquat. Mar. Ecol. 11: 79-90.
BLACKBURN, N. D., AND T. H. BI,ACKBURN. 1993. A reaction diffusion model of C-N-S-O species in a stratified sediment.
FEMS Microbial. Ecol. 102: 207-215.
BOWEN, J. D., AND K. D. STOLZENBACH. 1992. The concentration
distribution near a continuous point source in steady homogeneous shear. J. Fluid Mech. 236: 95-110.
AND S. W. CHISH~LM. 1993. Simulating bacterial
9 -,
clustering around phytoplankton cells in a turbulent ocean.
Limnol. Oceanogr. 38: 36-5 1.
BROWN, D., AND H. BERG. 1974. Temporal stimulation of che-
motaxis in Escherichia coli. Proc. Natl. Acad. Sci. USA 71:
1388-1392.
DE Roes, A. M., E. MCCAULEY, AND W. G. WILSON. 1991. Mobility versus density-limited predator-prey dynamics on different spatial scales. Proc. R. Sot. Lond. B 246: 117-122.
FHNCHEL,T. 1987. Ecology of protozoa. The biology of free-living
phagotrophic protists. Science Tech. Pub].
FUHRMAN, J. 1992. Bacterioplankton roles in cycling of organic
matter: The microbial food web, p. 361-383. Zn I? G. Falkowski and A. D. Woodhead [cds.], Primary productivity and
biogeochemical cycles in the sea. Plenum Press.
HAGSTROM, A., E AZAM, A. ANDERSSON, J. WIKNER, AND E RASSOULZADEGAN. 1988. Microbial loop in an oligotrophic pclagic marine ecosystem: Possible roles of cyanobacteria and nanoflagellates in the organic fluxes. Mar. Ecol. Prog. Ser. 49: 171-
178.
JACKSON, G. A. 1987. Simulating chemoscnsory responses of ma-
-.
rine microorganisms. Limnol. Oceanogr. 32: 1253-1266.
1989. Simulation of bacterial attraction and adhesion to
falling particles in an aquatic environment. Limnol. Oceanogr.
34: 514-530.
JUMARS, l? A., J. W. DEMING, F? S. HILL, L. KARP-BOSS, I? L. YACXR, ANII W. B. DADE. 1993. Physical constraints on marine
osmotrophy in an optimal foraging context. Mar. Microb. Food
Webs 7: 121-159.
-,
D. L. PENKY, J. A. BAROSS, M. J. PERRY, AND B. W. FROST.
1989. Closing the microbial loop: Dissolved carbon pathway
to heterotrophic bacteria from incomplete ingestion, digestion
and absorption in animals. Deep-Sea Res. 36: 483-495.
LARSSON, U., AND A. HAGSTROM. 1979. Phytoplankton exudate
release as an energy source for the growth of pelagic bacteria.
Mar. Biol. 52: 199-206.
LAZIER, J. R. N., AND K. H. MANN. 1989. Turbulence and the
diffusive layers around small organisms. Deep-Sea Res. 36:
1721-1733.
MITCMELL, J. G. 1991. The influence of cell size on marine bacterial motility and energetics. Microb. Ecol. 22: 227-238.
-,
L. PEARSON, A. BONA~INGA, S. DILLON, H. KHOURI, AND
R. PAXI~NOS. 1995~. Long lag times and high velocities in
the motility of natural assemblages of marine bacteria. Appl.
Environ, Microbial. 61: 877-882.
--,
S. DII,I,ON, AND K. KANTALIS.
1995b. Natural
assemblages of marine bacteria exhibiting high-speed motility
and large accelerations. Appl. Environ. Microbial. 61: 4436-
4440.
T, AND D. L. KIKCIIMAN.
1992. Release of dissolved
organic matter by heterotrohic protozoa: Implications for microbial food webs. Arch. Hydrobiol. Bcih. 35: 99-109.
NISSEN, H., I? NISSEN, AND E AZAM. 1984. Multiphasic uptake of
D-glucose by an oligotrophic marine bacterium. Mar. Ecol.
Prog. Ser. 16: 155-160.
SI-IIMETA, J. 1993. Diffusional encounter of submicrometer particles and small cells by suspension feeders. Limnol. Oceanogr.
NAGATA,
38: 456-465.
VAULOT, D., D. MARIE, R. G. OLSON, AND S. W. CHISIIOLM.
1995.
Growth of Prochlorococcus, a photosynthetic prokaryote, in
the equatoral Pacific Ocean. Science 268: 1480-1482.
ZWEIFEI,, U., N. B. BLACKBURN, AND A. HAGS’IX~M. 1996. Cycling
of marine dissolved organic matter. I. An experimental system.
Aquat. Mar. Ecol. 11: 65-77.
-,
AND A. HAGSTROM. 1995. Total counts of marine bacteria
include a large fraction of non-nucleoid containing bacteria
(ghosts). Appl. Environ. Microbial. 61: 2180-2185.
Received: 25 October 1995
Accepted: 14 October 1996