1
Running head: Coral microbial communities and disease
2
How microbial community composition regulates coral disease transmission
3 4
†
J. Mao-Jones1, K. B. Ritchie2, L. E. Jones1, , and S.P. Ellner1
5 6
1
Department of Ecology and Evolutionary Biology
7
Cornell University, Ithaca NY 14850, USA
8 9
2
Mote Marine Laboratory
10
1600 Ken Thompson Parkway, Sarasota, Florida 34236, USA
11 12 13 14 15 16
†
Corresponding author. E-mail
[email protected]
17 18
Last save: 4/21/2009 2:35 PM EST
19 20 1
20
Abstract: Coral reef cover is in rapid decline worldwide, in part due to bleaching
21
(expulsion of photosynthetic symbionts), infectious disease outbreaks and consequent elevated
22
mortality. One important factor in both disease transmission and initiation of bleaching, is a shift
23
in the composition of the microbial community in the mucus layer surrounding the coral: the
24
microbes normally dominant in the mucus layer, which produce antibiotic substances limiting
25
pathogen growth, are replaced by pathogenic microbes, often species of Vibrio. In this paper we
26
develop computational models for microbial community dynamics in the mucus layer in order to
27
understand how the surface microbial community responds to changes in environmental
28
conditions and under what circumstances it becomes vulnerable to takeover by pathogens. We
29
find that the pattern of interactions in the surface microbial community facilities the existence of
30
alternate stable states, one dominated by beneficial microbes and the other pathogen-dominated.
31
Consequently, a shift to pathogen dominance under transient stressful conditions, such as a brief
32
warming spell, may persist long after conditions have returned to normal. This prediction is
33
consistent with observational findings that antibiotic activity does not return to measurable levels
34
in coral after temperature reduction. The resulting long-term loss of antibiotic activity eliminates
35
a critical component in coral defense against disease, giving pathogens an extended opportunity
36
to infect and spread within the host, elevating the risk of coral bleaching and mortality.
37 38
Key words: Coral, disease, mucus, microbe, pathogen, Vibrio, symbiosis, global
39
warming, climate change.
40
Abbreviations: SMC, surface microbial community. LHS, Latin Hypercube Sampling.
41 42 2
42
Introduction
43
In recent years, coral reef cover has declined dramatically worldwide, due in part to
44
infectious diseases [1], though many other factors likely contribute as well. What facilitates the
45
spread of infections remains uncertain [1], but many disease outbreaks may be opportunistic
46
infections by environmentally endemic organisms following physiological stress to the coral [2-
47
4] Some researchers propose that various forms of physiological stress, namely increased sea-
48
surface temperatures, are correlated with increased virulence in opportunistic pathogens [1,5-7].
49
However the dynamics of opportunistic organisms within the microbial communities they inhabit
50
remain mysterious and are the subject of this paper.
51
Common to almost all instances of disease outbreaks over the past several decades are
52
bleaching events preceded by temperature stress. For example, a severe thermal anomaly in the
53
Caribbean during 2005 was followed by widespread bleaching and then by significant spread of
54
white plague and yellow blotch, resulting in 26-48% losses in coral cover in some areas [8]. [9]
55
found a significant positive correlation between the frequency of warm thermal anomalies and
56
the occurrence of white syndrome on Australia's Great Barrier Reef. The El Niño weather
57
phenomenon in 1998 was followed by widespread bleaching in the western Indian Ocean and
58
then by 50-60% mortality rates [10,11] These recent mortality events are only a few of many
59
instances that have resulted in significant losses in coral cover.
60
A bleached coral is more vulnerable to disease than a non-bleached coral, which is
61
defended by a mucus layer covering the coral’s outer surface, generated in part by its algal
62
symbionts [12]. Under non-stressful conditions, corals produce a mucus layer which contains a
63
complex microbial community commonly referred to as the surface microbial community
64
(SMC). One important function of the mucus layer is that it provides defense against pathogens
3
65
[12] and other stressors not considered in this paper. Most potential pathogens are endemic to the
66
ecosystem and present at low numbers in the SMC, but a rapid shift to pathogen dominance can
67
occur in the SMC after heat stress and prior to a bleaching event [4,5,7,13-15]. For example,
68
during a late 2005 summer bleaching event Ritchie [15] observed that "visitor" bacteria, or
69
bacterial groups that otherwise were not dominant, were the predominant species in mucus
70
collected from apparently healthy Acropora palmata. During temperature stress Vibrio
71
coralliilyticus is an agent of bleaching in Pocillopora damicornis [6] and Vibrio shiloi becomes
72
an agent of bleaching in the coral Oculina patagonica [16-18]. Rosenberg et al. [3] recently
73
summarize the evidence supporting a "microbial hypothesis of coral bleaching", that bleaching is
74
often initiated by a shift to pathogen dominance in the SMC brought on by heat stress, rather
75
than primarily by direct effects of heat stress on the coral and its symbionts [18,19].
76
The mucus environment is rich in nutrients, and microbial population concentrations are
77
orders of magnitude higher in coral mucus than in the surrounding water column [20]. According
78
to [15], "a component universal to coral mucus, independent of species, location, and season, is
79
capable
80
[Pseudoalteromonadaceae bacterium]" suggesting that coral mucus is capable of regulating the
81
activities of a number of visitor and water column microbial isolates. Some visitor microbes
82
produced antibiotic compounds, but this was rarer in visitors (16% of isolates tested) than in
83
residents (41% of isolates tested [15] ) that resident microbes may be beneficial to corals.
of
inhibiting
pigment
and
antibiotic
production
associated
with
84
While mucus provides nutrient substrates on which the beneficial microbes can grow
85
[12,17,21-24], it also readily serves as a growth medium for pathogenic invaders [12,20,25-28].
86
Thus, altered mucus conditions may shift community composition, facilitating pathogen invasion
87
and opportunistic infection. Indeed, beneficial microbes declined in times of increased water
4
88
temperature, when less than 2% of bacteria isolated from the surface of Acropora palmata
89
displayed antibiotic activity - significantly less than the 28% of antibiotic-producing isolates
90
present in cooler months.
91
It is uncertain what causes the decline of beneficial bacteria, but overgrowth of Vibrio
92
spp. has often been observed to precede disease outbreak. During one summer bleaching event,
93
Ritchie [15] found that Vibrio spp. were the predominant species (85% of species) cultured from
94
mucus collected from apparently healthy Acropora palmata sustained at a mean daytime sea
95
surface temperature of 28° to 30°C (the upper extreme of the standard range of thermal
96
tolerance, 18° to 30°C) for 2 months prior to collection.
97
An understanding of the SMC is therefore crucial to ameliorating the effects of thermal
98
stress on corals, including bleaching and infectious disease spread. A number of qualitative
99
models have been proposed for the causes and dynamics of the loss of antibiotic activity and
100
subsequent overgrowth of Vibrio spp. in the SMC. Ritchie [15] suggested that the antimicrobial
101
properties of the mucus are temperature-sensitive due to antimicrobial sensitivity or resident
102
microbe sensitivity to temperature change, with Vibrio overgrowth following a loss of
103
antimicrobial activity by residents. Another hypothesis proposes that Vibrio spp., which thrive at
104
elevated temperatures, out-compete beneficial bacteria in these conditions, and a loss of
105
antibiotic activity follows [5,13-15]. These models agree with the experiments of Ducklow and
106
Mitchell [20] in which coral mucus bacterial populations increased significantly when the coral
107
was stressed. Foster [29] proposed that the competition for space by invasive microbes divides
108
the beneficial microbes into isolated patches, where genetic diversity is significantly decreased.
109
Common to all hypotheses is that disease susceptibility is positively correlated with a loss of
110
antibiotic activity and an increase in Vibrio densities in the SMC.
5
111
Why take a modeling approach?
112
It appears that antibiotic activity and the competition between beneficial and potentially
113
pathogenic microbes (primarily Vibrio spp.) are key to understanding community dynamics
114
within the SMC. In this paper, we develop models for how these interactions affect the outcome
115
of competition within the SMC, and promote overgrowth by pathogenic microbes. We describe
116
first a spatially well-mixed model, and then turn to a model that includes the spatial gradients in
117
nutrient concentration from coral surface to the surrounding seawater, mucus production by the
118
coral, and ablation of mucus into the surrounding seawater. In both models we assume that
119
interactions can be simplified to a few key players, each representing some set of microbial
120
organisms or substances within the mucus layer. Because many quantitative and qualitative
121
characteristics of the SMC are still uncertain, we explore general properties of the model as
122
parameters are varied, rather than attempting to closely simulate any specific coral-pathogen
123
interaction. We thereby identify the parameters and processes that have the most significant
124
impacts on SMC dynamics and the potential for pathogen outbreaks.
125
A focal aspect of our modeling is exploring the importance of the spatial dimension -
126
which permits spatial heterogeneity – in this system. A spatially unstructured model (which
127
posits that the mucus layer is "well-mixed”) does not depict the physical processes of mucus
128
production by the coral and its endosymbionts and of mucus loss by sloughing off into the water
129
column. Furthermore, it is possible that gradients in chemical concentrations and population
130
densities within the mucus layer may have considerable effect on the dynamics of the microbial
131
community. Our spatial model considers a gradient of fixed length with endpoints at the coral
132
surface and at the interface between the mucus layer and the water column. By contrasting this
6
133
model with a spatially unstructured. “well-mixed” model, we examine the role of spatial
134
gradients in SMC dynamics and in defense against pathogen invasion.
135 136
Results
137
The well-mixed model. We begin with a simple well-mixed model to explore community
138
dynamics within the SMC, assuming first that there is no pathogen inoculation from external
139
sources, and later relaxing this assumption. This model and its underlying assumptions are
140
presented in detail in the Materials and Methods section. Insight into the behavior of the model
141
without external inputs can be gained by a simple nullcline analysis. The analysis is further
142
simplified if we consider a general rescaled well-mixed model
143
(1)
144
where
145
substrate within beneficials and antibiotics combined. The growth-rate functions
146
increasing functions of s, and
147
concentration that is proportional to b. We assume
148
maximum possible value of s); otherwise the populations die out because there is never enough
149
substrate for population increase. We also assume that
150
neither population can persist in the absence of substrate.
151 152
; p is the total substrate within pathogenic microves, and b represents the total
where
are
is a strictly decreasing function of b, reflecting the antibiotic and
and
are both positive when s=1 (the
are both negative when s=0, so
To see how model (1) behaves, we can study its nullclines in the nullcline is the line
and
is the solution of
7
, i.e., the line
plane. The b
153 154
(1) which has constant slope of
. The p nullcline is the curve
155 156 157 158 159
(2) where
is the solution of
. Because antibiotics are harmful to the pathogens,
is an increasing function of b. Therefore the p nullcline has a negative slope that is always below -1. Consequently, there are only three possible qualitative behaviors (Figure 1). If one
160
nullcline lies completely above the other, then all initial conditions with
lead
161
to competitive exclusion of the species with the lower nullcline, exactly like the Lotka-Volterra
162
model. If the nullclines cross, their intersection is a saddle (locally unstable), so there is
163
competitive exclusion again but with the identity of the winner depending on initial conditions
164
(sometimes called contingent exclusion). Both of the single-species equilibria (on the coordinate
165
axes) are then locally stable.
166
Without the antibiotic-mediated interactions, we would have pure resource competition
167
for a single limiting substrate. The p and b nullclines would then be parallel lines, and the
168
pathogen is the superior competitor if the p nullcline is above the b nullcline (as in Figure 1a).
169
Adding antibiotic effects, if they are sufficiently strong, will make the p nullcline decrease more
170
quickly as b increases, giving the situation shown in Figure 1b. Control of potentially dominant
171
pathogens through antibiotic activity is thus the "recipe" for contingent exclusion. A numerically
172
dominant pathogen population can prevent regrowth of beneficials by keeping substrate
173
concentrations too low for beneficials to increase, while a numerically dominant beneficial
8
174
population can prevent pathogen regrowth by maintaining a high ambient concentration of
175
antibiotics.
176
What if there is a pathogen source (
)? The empirical observation that microbes are
177
orders of magnitude less abundant in seawater than in mucus implies that
in the scaled
178
model. A pathogen source is then a small perturbation to the dynamics shown in Figure 1, so its
179
only effect is to keep pathogens from being eliminated: in cases of pathogen exclusion in Figure
180
1 (panels A and C of Figure 1), the stable equilibria with p=0 and b large are replaced by stable
181
equilibria with p close to 0 and b large.
182 183
Pathogen Invasion. We now explain how the properties of the well-mixed model can lead to a
184
sudden and persistent "takeover" of the SMC by pathogens following a brief period of conditions
185
stressful to the host and to beneficial microbes. To illustrate the process we consider thermal
186
stress, which has been implicated most consistently as the environmental driver linked to
187
pathogen outbreaks.
188
For a healthy coral we can presume that during colder months, which are less favorable to
189
the pathogens, the beneficials are able to exclude the pathogens (Figure 2a) – only the pathogen-
190
exclusion equilibrium is stable. During warmer months (Figure 2b) the higher temperatures may
191
give the pathogens a higher intrinsic growth rate than the beneficials, but pathogen growth is
192
kept in check by the effect of antibiotics, so the pathogen-exclusion equilibrium remains locally
193
stable. However, a thermal anomaly that causes the loss of antibiotic activity (Figure 2c)
194
eliminates the coexistence equilibrium, so the system jumps to being pathogen-dominated.
195
Figure 3 illustrates the corresponding microbial population dynamics during a warm
196
anomaly scenario based on sea surface temperatures at Glover's Reef, Belize (Fig. 3a). Under
9
197
normal seasonal variation in temperature, beneficial microbes are dominant year-round (Figure
198
3b). Regardless of how a simulation is initiated, during winter the beneficials become dominant
199
and they remain dominant through the summer, while pathogens persist at very low levels due to
200
inoculation from the water column. But even a brief thermal anomaly that eliminates antibiotic
201
activity for 14 days (Fig. 3b) allows the pathogens to become dominant and remain so for
202
approximately 3 months, until temperatures drop low enough that the pathogen-dominant
203
equilibrium becomes unstable.
204
The well-mixed model therefore provides a mechanistic explanation for the empirically
205
observed sudden switches to pathogen dominance following a change in conditions, and
206
moreover the model predicts that such sudden switches will occur even if the change in
207
conditions is gradual. Another key prediction is that the return from pathogen-dominance to
208
beneficials-dominance when conditions improve will also be sudden, but it will occur under
209
different conditions: pathogens may remain dominant even after environmental conditions return
210
to those where beneficials were initially dominant, while beneficials may not recover dominance
211
until the environment becomes considerably more favorable for them.
212 213
The spatial model: Qualitative properties. A fundamental question inherent in including
214
spatial heterogeneity in a model is whether spatial dimension allows for a broader range of
215
qualitative outcomes than a well-mixed, or non-spatial model. In particular, spatial variability
216
might allow stable coexistence of pathogenic and beneficial microbes, for example if pathogens
217
segregate away from beneficials and so avoid the effects of antibiotics produced by the
218
beneficials. The well-mixed model's prediction of potentially abrupt changes in community
219
composition in response to gradual changes in environmental conditions might then prove to be
10
220
an artifact of neglecting spatial variability. We therefore expand the model to include spatial
221
dimension; the model and its underlying assumptions are presented in Materials and Methods,
222
while details of numerical analysis and simulation are found in the Appendices.
223
Numerical study of the spatial model shows that the nullcline analysis of nonspatial
224
model (Figure 1) continue to hold. Specifically, the spatial model behaves like a two-
225
dimensional system of differential equations, even though it has an infinite-dimensional state
226
space. This occurs because, apart from a brief transient period, the spatial distribution of the state
227
variables is completely predictable from the total abundance of beneficial and pathogenic
228
microbes. Figure 4 shows an example. Two model runs were initialized by first two different
229
shapes for the population distributions at time t=0, and then using numerical optimization to find
230
total population sizes at t=0 such that the total beneficial and pathogen populations at time t=12
231
would both be 4. As Figure 4 illustrates, when the total beneficial and pathogenic populations are
232
at equal values, everything else (antibiotics, population distributions, etc.) are the same in the
233
two runs. In technical terms, numerical solutions show the model converging onto a two-
234
dimensional inertial manifold [30].
235
So given that there are
total beneficials and
total pathogens in the community
236
at time t, we can compute the complete state of the system and thus compute the instantaneous
237
rates of change dB/dt and dP/dt. This procedure defines a two-dimensional dynamical system for
238
the microbial populations, so we can again determine its qualitative behavior by plotting the
239
nullclines. Figure 5 shows nullclines for the slower "baseline" parameters listed in Table 1,
240
computed by the methods described in Appendix C. These show that the spatial model is in the
241
bistable situation of Figure 1(b), indicating that a healthy population of beneficial microbes can
242
keep pathogens from increasing, but beneficials would be at a competitive disadvantage in a
11
243
community dominated by pathogenic microbes. Given the large uncertainties in our parameter
244
estimates, we cannot regard this property as a prediction about nature. The important feature of
245
Figure 5 is that, as in the well-mixed model, the pathogen nullcline is steeper than the beneficials
246
nullcline, which is the property that precludes stable coexistence of beneficials and pathogens
247
(rather than one or the other being sustained at very low numbers by small levels of input from
248
the surrounding water column). Consequently, the spatial model preserves the key qualitative
249
prediction of the well-mixed model: if temporary extreme conditions allow the community to
250
become dominated by pathogenic microbes, the pathogen takeover may persist even after
251
conditions return to normal and may not terminate until conditions occur (such as winter
252
temperatures) that are highly unfavorable to the pathogens.
253 254
Sensitivity analysis of the spatial model.
We performed a local sensitivity analysis to
255
determine the relative impact of each parameter on system dynamics. Due to the high uncertainty
256
of parameter estimates, parameters were varied up to ±50% from their default values (Table 1)
257
using Latin Hypercube sampling (see Appendices D and E for additional information about our
258
sources for parameter values and the methods used to carry out the sensitivity analysis).
259
We carried out sensitivity analysis under three different scenarios: baseline (the
260
parameter values listed in Table 1), heat stress, and high antibiotic conditions. For the heat stress
261
scenario, beneficials growth rate was reduced, pathogen growth rates increased, and the
262
production of antibiotics was decreased. For the high antibiotic secnario the antibiotic production
263
rate was increased and the efficacy of the antibiotics against the pathogens increased. We also
264
considered both the "slower growth" and "faster growth" values for the microbe growth rate
265
parameters
. Parameter values for these scenarios are listed in Table 2.
12
266
Overall, the results of the sensitivity analysis (Figure 6) indicate that the most important
267
parameters are either (i) the advection and diffusion coefficients, which control the balance
268
between movement towards favorable conditions and mortality through mucus ablation, or (ii)
269
the maximum growth rates, which are important for the direct competitive interactions between
270
the microbial populations. Movement parameters were generally less important in the faster
271
growth scenarios where competitive interactions are stronger. Changes in antibiotic production
272
have the most effect on pathogen success in the faster growth baseline and heat stress scenarios,
273
because the rate of antibiotic production correlates with the beneficials' population growth rate.
274
In the high antibiotic scenario, the antibiotic inhibition of pathogens is so strong that even major
275
changes in the strength of antibiotic inhibition have no effect on the outcome (i.e., pathogen
276
exclusion). The most surprising outcome is that antibiotic conversion rate, bacteriostatic efficacy,
277
and antibiotic degradation rate are never among the most significant parameters, even though the
278
beneficials' ability to produce antibiotics is essential for their persistence. This suggests that (if
279
our parameter ranges are realistic) the role of antibiotics in normal conditions is to tip the balance
280
in a competition between near-equals. To explore this idea further, we modified the
281
baseline/faster growth scenario by holding the ratios
282
Latin Hypercube sample parameter vector we perturbed
283
the values
284
As expected, with these constraints (Fig 6g) the importance of growth rate variation (indicated
285
by r in the axis label) is greatly reduced relative to Fig 6b, and the importance of antibiotic-
286
related parameters is greatly increased. The fact that higher overall growth rates are detrimental
287
to the pathogen also reflects the impact of antibiotics, because of the proportionality between
288
beneficials' growth rate and antibiotic production rate in the model.
and
constant (i.e., for each
and then set the value of
so that
in the sample parameter vector and the default parameter vector were equal).
13
289 290
Discussion
291
Previous studies have shown that a sudden shift to pathogen dominance occurs in the
292
surface microbial communities (SMC) of corals prior to a bleaching event [5,13-15]. It has also
293
been demonstrated that antibiotic activity and antibiotic-producing bacteria in the SMC decline
294
in times of increased water temperature when bleaching is most likely to occur [15]. Disease
295
susceptibility in hard corals is thus positively correlated with a loss of antibiotic activity and an
296
overgrowth of pathogenic bacterial densities in the SMC.
297
In this paper we have developed mathematical models to explore how interactions
298
between resident and invading microbes within the coral SMC affect the health and disease
299
susceptibility of reef-building corals, assuming first a well-mixed mucus layer and then allowing
300
spatial heterogeneity in microbial population densities and nutrient substrate. A surprising but
301
robust finding is the consistency in outcomes between well-mixed and spatially heterogeneous
302
systems. Though spatial heterogeneity might be expected to allow spatial segregation and thus
303
coexistence of pathogenic and beneficial microbe types, in both cases stable coexistence is
304
precluded. Instead, the situation is one of competition and contingent exclusion between two
305
more or less equal competitors, with antibacterial production usually shifting the balance in favor
306
of microbes beneficial to the coral.
307
The well-mixed model. Analysis of the well-mixed model shows that under competition
308
for a single limiting substrate, control of pathogen via antibiotic activity is the key to the
309
contingent exclusion of pathogens by the beneficial bacteria. However, under the empirically
310
supported assumption of reduced antibacterial production during heat stress, the model predicts a
311
rapid switch from dominance by beneficial microbes to dominance by pathogens during thermal
14
312
anomalies. In addition, dominance by beneficials is not restored when temperatures return to the
313
normal conditions under which the beneficials were previously dominant. Instead, conditions
314
must become unusually unfavorable to pathogens before a switch back to dominance by
315
beneficial bacterial can occurs. This is consistent with observational findings that antibiotic
316
activity does not return to measurable levels in coral even after recovery and temperature
317
reduction (Ritchie, unpublished data, 2009).
318
The spatially heterogeneous model. Simulations of the spatial model show that the key
319
qualitative result from the analysis of well-mixed model (contingent exclusion) holds in the
320
heterogeneous case as well. However, because of the level of uncertainty in our parameter
321
assumptions and estimations, we performed sensitivity analyses to better quantify the effects of
322
parameter variation under normal warm-season conditions, heat stress conditions, and finally
323
conditions of high antibiotic production.
324
Under normal conditions, the ability to spread to fresh sources of substrate is critical to
325
the success of a slowly reproducing pathogen. For a more rapidly growing pathogen under these
326
conditions, the ability to compete for nutrients becomes more important. Under conditions
327
causing heat stress, the pathogen uniformly has the upper hand and its success hinges primarily
328
on its potential population growth rate. Results for high antibiotic production are similar to
329
baseline conditions, except antibiotic production benefits the beneficial bacteria across the entire
330
range of parameters considered. Thus the critical parameters overall are those which govern
331
movement (microbial advection and diffusion coefficients, mucus ablation rates) and maximum
332
microbial growth. Finally, a numerical nullcline analysis of the heterogeneous model suggests
333
that it retains the hysteresis observed in the well-mixed model: once temporary extreme heat
15
334
allows pathogens to overgrow, their dominance will persist until conditions become highly
335
unfavorable for pathogenic persistence.
336
Future Questions and Research. Our models have omitted for clarity and simplicity
337
some biological processes which might prove important to microbial community dynamics. For
338
example: why does antibiotic production decline as sea temperatures increase? Are the beneficial
339
bacteria being succeeded by a more temperature-tolerant (and virulent) bacterial type, as
340
suggested by [5], who discovered that a Vibrio species becomes more virulent and invades at
341
high temperatures; and if so, why? Future models could address the cost of antibiotic production
342
by beneficial bacteria (does antibiotic production become costly as temperature rises?),
343
incorporate variability in antibiotic conversion efficiency and allow production of defensive
344
antibiotics by the pathogens themselves, as suggested by [15], who showed that visitor microbes,
345
like the Vibrio inoculated into the SMC at the mucus-seawater interface, also produce antibiotics.
346
The present models focus primarily on the loss of defenses within the surface microbial
347
community, without considering the defenses of the host coral itself once an infection has
348
penetrated through the mucus layer. An earlier paper focused entirely on cellular immune
349
responses by soft corals to an established fungal infection [31] without addressing how
350
vulnerability to infection is modulated by process in the SMC. Future models need to integrate
351
the process of infection with host immune responses, to elucidate how the onset of pathogenic
352
overgrowth in the SMC, and the quality of host cellular response to a successful invasion,
353
interact to determine the impact of pathogen attack.
354
16
355
Conclusions
356
We have presented models that yield insights into a current crisis in our oceans: the
357
decline of coral cover due to increased vulnerability to disease in a warming climate. Our models
358
show that the structure of interactions in the surface microbial community facilitates the
359
existence of alternate stable states, one dominated by beneficial microbes and the other
360
pathogen-dominated. This provides a mechanistic explanation for the empirically observed
361
sudden switches to pathogen dominance following heat stress. The models also predict that
362
sudden switches will occur even if the temperature increase is gradual, and that the switch to
363
pathogen dominance will persist long after thermal stress has ceased, so that a short-term heating
364
event may give pathogens an extended opportunity to establish and spread. These predictions are
365
robust consequences of an interaction between beneficial and pathogenic microbes mediated by
366
beneficials' production of antibiotic substances, rather than depending on any fine details of our
367
models.
368
An important practical implication of our findings is that a shift to pathogen dominance is
369
much more easily prevented than reversed. To prevent potential consequences such as bleaching
370
events and elevated colony mortality, it is advantageous to be able to forecast the location and
371
timing of situations (e.g. heat anomalies) conducive to pathogen dominance, and to intervene
372
before a shift occurs. Reversing the damage once it has occurred will require even greater efforts
373
than rapid, timely, and proactive interventions.
374 375 376 377
17
378
Materials and Methods
379
Ethics statement: N/A.
380
(1) A model for community dynamics in a well-mixed mucus layer.
381
To frame our studies of the surface microbial community we have developed a simple
382
model for a spatially homogeneous ("well mixed") mucus layer focusing on the key community
383
members. We list and explain each state variable and equation of the model below.
384
Substrate. A nutrient substrate (S) is supplied by the host and consists mostly of organic carbon
385
(we use "host" to mean the coral and its endosymbionts together, and the carbon substrate in the
386
mucus is provided primarily by endosymbionts). We assume that substrate is supplied at a
387
constant rate
388
microbes, so the net supply rate as perceived by the populations of interest is constant. Mucus is
389
assumed to slough off at a constant rate
390
equation is then:
, and is consumed only by the modeled populations of beneficial and pathogenic
(fraction of mucus lost per unit time). The substrate
391
,
(4a)
392
the terms on the right-hand side representing input from the host, uptake by pathogenic and
393
beneficial microbes (discussed below), and mucus loss.
394
Beneficial microbes and Antibiotics.
395
saturates as a function of nutrient concentration, and is described by a Monod equation
396
, where K is the half-saturation constant (in units of substrate concentration). We
397
make the simplifying assumption that the substrate content of microbes is released back into the
398
substrate pool immediately upon death, so that net substrate uptake is proportional to the net
399
population growth rate. Maximum per-capita growth rate (as a function of S) for the beneficial
We assume the growth of beneficial microbes (B)
18
. We assume that a constant fraction α of net substrate uptake by
400
microbes is denoted
401
beneficials is used to produce antibiotic, and the remainder goes towards population growth.
402
Without loss of generality we scale the microbial populations and antibiotic so that there is a 1:1
403
conversion between substrate uptake and population growth or antibiotic production (i.e., we
404
measure microbes and antibiotic in terms of the amount of substrate required to produce them).
405
The equations for beneficial bacteria and antibiotic substances ( A ) are then:
406
(4b)
407
.
(4c)
408
Pathogenic microbes (P), are passively inoculated into the system at rate Ip. Maximum growth
409
rate for the pathogens is
410
growth is described by a Monod equation. We again make the simplifying assumption that the
411
substrate content of these microbes is released back into the substrate pool immediately upon
412
death, so that net substrate uptake is proportional to the net population growth rate;
, which is a decreasing function of antibiotic concentration, and
413
(4d)
414
We now invoke conservation of mass, thus eliminating S and A as state variables without
415
affecting the model’s long-term behavior (see Appendix A for details). Furthermore, scaling the
416
model into dimensionless units gives the rescaled model:
417
(3)
19
418
where
, and rescaled parameters (such as
) are defined in Appendix A.
419
Some biological assumptions of this model should be noted. First, consistent with
420
observations by Ritchie (2009, unpublished data) we assume that the antibiotics produced by
421
beneficial bacteria are bacteriostatic (decreasing the pathogens' ability to reproduce) rather than
422
bacteriocidal (causing pathogen mortality). Second, we assume equal half-saturation constants K
423
for pathogens and beneficials. We will see in the next section that neither of these assumptions
424
affects the qualitative properties of the well-mixed model, so long as the substrate contents of
425
pathogens killed by antibiotics are immediately recycled back into the substrate pool. Third, we
426
do not consider possible effects of oxygen limitation, which might occur at night when the host's
427
photosynthetic endosymbionts are not producing oxygen. Most of the taxa that [15] identified as
428
"resident" are facultatively anaerobic, as are the main potentially pathogenic visitors (Vibrio
429
species). For simplicity we thus assume that the microbes of interest can function equally well
430
under anaerobic conditions.
431 432
(2) A Spatial Model for Interactions between Beneficial and Pathogenic Microbes
433
Spatial structure within the mucus layer is potentially very important, and inevitably
434
present because of the essential differences between host tissue at the base of the layer which is
435
providing mucus and nutrients, and the seawater environment into which mucus and nutrients are
436
lost. We therefore generalize our well-mixed model by allowing spatial variability in the surface
437
microbial community along the gradient from host to water column. In this section we describe
438
the model in some technical detail; readers who wish can omit this section on first reading.
439 440
We consider a one dimensional spatial gradient on the interval surface is at
and the mucus layer meets the water column at
20
, where the coral . Because mucus is
441
provided at the coral surface and lost by ablation into the water column, we conceptualize the
442
mucus layer as a "conveyor belt" moving from coral to seawater at some constant velocity δ. The
443
conveyor belt motion carries along microbes and substrate away from the coral surface (i.e. in
444
the positive x-direction), but this is counteracted (in part) by diffusion and by active chemotactic
445
motion of microbes. Because the relative concentrations of substrate, microbes, and antibiotics
446
can vary from one place to another, we cannot reduce the model from four to two state variables.
447
Thus the model tracks the S, P, B and A as functions of space x and time t. We assume that all
448
particles inside the mucus remain inside the mucus [32] and that no particles diffuse into the
449
water column or through the coral surface. Particles may leave the SMC via mucus sloughing.
450
At any fixed location x within the mucus layer (
), the local interactions are
451
described by the well-mixed model (4-5), but without the supply terms
because these
452
are "active" only at the boundaries. Added to the local interactions are transport terms
453
representing diffusion and advection (directed motion). For the substrate and antibiotics, the
454
transport terms are random Fickian (concentration independent) diffusion and the "conveyer
455
belt" motion at rate
, so we have:
456
(6)
457
where
are the diffusion coefficients for S and A, respectively, and
458
degradation rate.
is the antibiotic
459
The microbes also have diffusion and "conveyor belt" transport terms, and in addition we
460
assume that they are positively attracted to increases in substrate concentration and (for the
461
pathogen) decreases in antibiotic concentration. To represent this mathematically we posit that
21
462
the chemotactic velocity component is linearly proportional to the gradient in reproductive rate
463
W, where W is given in our model by
464 465
.
(7)
The microbial dynamics are then
466
(8)
467
where the chemotaxis coefficients
determine how strongly the microbes respond to
468
gradients in substrate and antibiotic concentration. Flagella are energetically expensive (and
469
often the target of antibody responses), so we assume that microbial motility will be limited, and
470
not much more than the minimum needed to avoid being "swept out to sea" at x=1 through
471
mucus ablation (because substrate is supplied at the coral surface, attraction to substrate
472
automatically favors motion away from x=1).
473
The differential equations (6) and (8) apply for x between 0 and 1, so to complete the
474
model we need to specify what happens at the mucus layer boundaries. Here we give a brief
475
description; see Appendix B for full details and a description of how we numerically solved the
476
spatial model. At the water column boundary x=1 we expect a fairly sharp transition. This can be
477
represented most simply by assuming that anything that reaches the end of the "conveyer belt"
478
falls off it instantly, so the boundary at x=1 is effectively coupled to a void from which nothing
479
returns. We therefore impose the "absorbing" boundary conditions,
480
.
22
(9)
481 482 483
To allow some immigration from the water column we could set
with
. For simplicity we use (9), but recognize that immigration would prevent complete extinction of either beneficial or pathogenic bacteria.
484
Substrate is supplied at the coral surface, which means in our "conveyor belt" model that
485
new mucus has a high substrate concentration determined by the host. The boundary condition
486
for substrate at x=0 is therefore
487
coral surface, so the appropriate boundary condition is that there be zero flux across the
488
boundary. The same is true for the microbial populations, but a simple no-flux condition would
489
lead to microbes piling up at the coral surface to get the most possible substrate. This is not
490
observed, perhaps because there is increased viscosity in newly released mucus that would
491
inhibit mobility and keep the microbes from reaching the coral surface. Schneider and Doetsch
492
(1974) observed the effect of viscosity on motility under experimental conditions, finding that
493
motility decreased at high and low viscosities and was maximized at intermediate viscosity.
494
Therefore, following [31] we made the boundary at x=0 inaccessible to the microbes by having
495
the diffusion and advection coefficients decrease smoothly to zero near the coral surface.
. Antibiotic is neither supplied nor absorbed at the
496 497
Acknowledgements. We thank Alex Vladimirsky (Department of Mathematics, Cornell) for
498
advice on boundary conditions for the spatial model, C. Drew Harvell (EEB, Cornell) for
499
initiating our collaboration, and participants in the EcoTheory Lunch Bunch (Michael Cortez,
500
Ben Dalziel, Matt Holden, Paul Hurtado, Katie Sullivan, Rebecca Tien) for comments on the
501
manuscript. Finally, Parviez Hosseini (Consortium for Conservation Medicine, Wildlife Trust,
502
New York) provided very helpful and timely comments and suggestions.
503
Funding disclosure. This research was supported by National Science Foundation
23
504
grant OCE-0326705 in the NSF/NIH Ecology of Infectious Diseases program. The funders had
505
no role in study design, data collection and analysis, decision to publish, or preparation of the
506
manuscript. The authors do not have any financial, personal, or professional interests that could
507
be construed to have influenced their paper.
24
508
Table 1: Parameters and baseline default values for the model, after rescaling substrate
509
concentration so that the concentration of substrate in fresh mucus supplied by the host is
510
Values of advection and diffusion coefficients are based on assuming that the mucus layer
511
thickness is L=1mm. C in the table denotes units of substrate relative to
512
parameters
513
spatial model, described in Appendix D. The two values of λ are for the spatial and non-spatial
514
models, respectively.
.
. The two values for
are the "slower growth" and "faster growth" baseline parameters for the
515 Parameter δ
Units
Default Value
Range
mm
0.1
×(0.5 − 2)
Unitless
0.05
×(0.5 − 2)
Beneficial maximal growth rate
1/d
0.8, 5
×(0.5 −1.5)
Pathogen maximal growth rate
1/d
1, 6
×(0.5 − 1.5)
Antibiotic efficacy
1/C
5, 44
×(0.5 − 2)
Beneficials' half-saturation constant
C
0.25
×(0.5 − 2)
Pathogens' half-saturation constant
C
0.25
×(0.5 − 2)
0.5/d
0
×(0.5 − 2)
Substrate diffusion coefficient
mm2/d
.1
×(0.5 − 2)
Antibiotic diffusion coefficient
mm2/d
.1
×(0.5 − 2)
Beneficial microbes diffusion coefficient
mm2/d
.01
×(0.5 − 2)
Pathogenic microbes rate of diffusion
mm2/d
.01
×(0.5 − 2)
Beneficial microbes advection coefficient
mm/d
0.05, 0.01
×(0.5 − 2)
Pathogenic microbes advection coefficient
mm/d
0.05, 0.01
×(0.5 − 2)
Biological Meaning Mucus advection rate Fraction of substrate uptake by beneficials that is used to produce antibiotic.
λ
Degradation rate of antibiotic
516 517 518 25
519
Table 2: Default parameter values for the heat stress and high antibiotic scenarios in the
520
sensitivity analysis of the spatial model. A dash ( − ) indicates no change from the baseline
521
parameter values listed in Table 1. For all parameters not listed here, the default values for these
522
scenarios were the same as those for the baseline scenario listed in Table 1.
523 524 Parameter
λ
Heat Stress
Heat Stress
Slower
Faster
Slower
Faster
0.6
4.0
−
−
1.2
7.2
−
−
0.02
0.02
0.1
0.1
−
−
20
20
525 526 527 528 529 530 531
26
High Antibiotic High Antibiotic
531
Figure Legends
532 533
Figure 1. Nullcline analysis of the general well-mixed SMC model. Stable equilibria are shown
534
as filled circles, and unstable equilibria as open circles. Arrows indicate the direction of
535
population change on the nullclines. (a) The P nullcline (solid red line) lies above the B nullcline
536
(blue dashed line), leading to exclusion of beneficials. (b) Because the P nullcline is steeper than
537
the B nullcline, if the nullclines cross their intersection (open circle) is a saddle and therefore
538
unstable. The stable manifold of the saddle divides the interior of the quadrant into the sets of
539
initial points leading to competitive dominance by one type of microbe and competitive
540
exclusion of the other. (c) The B nullcline lies above the P nullcline, leading to exclusion of
541
pathogens.
542 543
Figure 2. Nullclines (solid red=pathogens, dashed blue=beneficials) of the well-mixed SMC
544
model (equation (3)), for the parameter values such that a brief thermal anomaly allows
545
pathogens to become dominant (the slower baseline parameters in Table 1). Panel (a) shows
546
conditions in winter, when the beneficial-dominant equilibrium is stable (solid circle) while the
547
pathogen-dominant equilibrium is unstable (open circle). Panel (b) shows conditions in summer,
548
when the beneficial-dominant and pathogen-dominant equilbria are both stable (solid circles),
549
while the coexistence equilibrium (open circle) is unstable. Panel (c) shows the effect of a small
550
increase in temperature that eliminates antibiotic activity, so that the nullclines of pathogen and
551
beneficial bacteria become parallel. The beneficial-dominated equilibrium (open circle) becomes
552
unstable, so the community converges to the pathogen-dominated equilibrium (solid circle).
553
27
554
Figure 3. Effects of a brief thermal anomaly on microbial population dynamics in the well-
555
mixed SMC model. (a) NOAA sea surface temperature record for Glover's Reef, Belize (from
556
coralreefwatch.noaa.gov/satellite/data_nrt/timeseries/all_Glovers.txt).
557
The open circles show temperatures considered high enough to elevate the risk of coral
558
bleaching; the dashed curve is the fitted seasonal trend (a periodic smoothing spline) used to
559
simulate the model (b) Simulations of the model using the seasonal temperature trend plotted in
560
panel (a), but with a two-week thermal anomaly (indicated by the vertical dashed lines) during
561
which temperature was elevated by 1 degree C, and antibiotic activity by beneficials was
562
eliminated.
563 564
Figure 4. Two runs of the spatial model (left and right columns) starting from different initial
565
conditions at t=0 (top panels), chosen so that at time t=12 the total abundance of beneficial and
566
pathogenic microbes would both equal 4 on the scale of the rescaled spatial model (Appendix B).
567
In the middle and bottom rows, all state variables (beneficials:blue, pathogens:red,
568
substrate:green, antibiotic:purple) have been scaled relative to their maximum concentration).
569
Figure headings give the total abundances at each time point.
570 571
Figure 5. Numerically computed nullclines (beneficials: dashed blue curve, pathogens: solid
572
blue curve) for the spatial model with the slower growth "baseline" parameter values listed in
573
Table 2. The configuration of the nullclines implies stability of both the pathogen-only and the
574
beneficials-only equilibria, with an unstable equilibrium at the intersection of the nullclines.
575
28
576
Figure 6: Results of local sensitivity analysis using Latin Hypercube Sampling and multiple
577
linear regression of log(model response) on log(parameters); see Appendix E for details on
578
methods. Results are shown for 3 scenarios (baseline, heat stress, high antibiotic) and two sets of
579
values for potential growth rates of the microbe populations (slower and faster). Model responses
580
are the total beneficial and pathogenic microbe densities after 30 days (slower parameters) or 20
581
days (faster parameters). Elasticity is the regression coefficient for each parameter. Due to the
582
log-scale fitting, the elasticity value is the average fractional change in the response relative to
583
the fractional change in the parameter, so that an elasticity value of 2 means that a ±10% change
584
in the parameter causes a ±20% average change in the response.
585
29
585
Figure 1
586
587
30
587
Figure 2
589
31
590 591
Figure 3 593
32
594 595 597
Figure 4.
33
598
Figure 5
599
600 601 602 603
34
603
Figure 6
604
35
605 36
605
Appendix A: Simplifying and rescaling the well-mixed model
606
The total amount of substrate
607
, and so
obeys the differential equation
converges to
. For studying the long-term
608
dynamics of the model we can therefore assume that
has converged to the limiting value, so
609
that
610
the same proportional loss rate δ, so the ratio of A:B converges to
611
to write A= cB, leaving only P and B as state variables. The model is then
. Also, A and B are produced in the constant ratio
and have . This allows us
612
613 614
(4)
with To nondimensionalize the model, we rescale the state variables and time as follows:
615
.
(5)
616
Note that in this rescaling, x represents the total amount of substrate (relative to
617
beneficials and antibiotic. By standard calculations, these rescalings convert (4) into equation (3)
618
with scaled parameters
619
.
) in
(6)
620 621 622
Appendix B. Technical details of the spatial model Here we show how the spatial model can be rescaled into nondimensional form, give
623
additional technical details on the boundary conditions and how they were imposed numerically,
624
and describe our methods for numerical solution of the spatial model.
37
625 626
Rescaling. The equations to be non-dimensionalized are
627
(7)
628
where
629
We measure time in days, distance x in mm, and state variables are all measured in units of
630
substrate (e.g., moles Carbon). The units of parameters are given in Table 2. For simplicity, the
631
spatial dependence of advection and diffusion coefficients used in numerical solutions is not
632
shown explicitly in (7).
633
.
It is convenient to choose substrate units so that the concentration of substrate in fresh
634
mucus supplied by the host is
, because this stabilizes the one non-zero boundary condition
635
(i.e,
636
rescaling the spatial variable x, and likewise for numerical studies there is no gain from rescaling
637
time. We therefore proceed much as in the nonspatial model, and define rescaled state variables
). A typical value of mucus layer thickness is L=1mm so there is no gain from
638 639
(the scaling of B is different here than in the well-mixed model, because we cannot tacitly absorb
640
A into B when both quantities are varying across space). By standard calculations, the resulting
38
641
rescaled model is then identical in form to equation (7), except that the values of the parameters
642
and
are both divided by
643 644
Boundary Conditions. At the right-hand boundary, the sharp transition between mucus layer and
645
the
surrounding
water
column
is
represented
by
absorbing
boundary
conditions
646
. In numerical experiments these gaves very similar results to
647
less extreme boundary conditions (e.g., allowing some two-way diffusion between mucus layer
648
and seawater), so we used the absorbing conditions for simplicity and to avoid introducing
649
additional parameters specifying the boundary conditions.
650
The left-hand boundary conditions are constant for substrate,
to represent
651
substrate supply by the host and its symbionts, and zero-flux for the other state variables. For the
652
antibiotic the zero-flux condition is that
653
condition was imposed by finite difference, as detailed below (equation (9)).
at x=0. In numerical solutions this
654
Boundary conditions for the microbe populations at x=0 are more complicated. Our
655
chemotaxis assumptions favor microbes converging onto the coral surface to maximize nutrient
656
uptake. To avoid this behavior, which is not actually observed, following Ellner et al. (2007) we
657
made the boundary at x=0 inaccessible to the microbes by having the diffusion and advection
658
coefficients decrease smoothly to zero as the coral surface is approached:
659 660
(8) and similarly for the pathogens, where
. The resulting boundary conditions are
661
. Numerical experiments show that the variable coefficient approach (8) gives
662
results very similar to implementing a no-flux boundary conditions by finite difference, but is
39
663
more stable against numerical blowups that can occur when all microbes concentrate near the
664
coral surface.
665 666
Numerical methods. Model solutions were obtained by methods very similar to Ellner et al.
667
(2007, Appendix B). Spatial derivatives of state variables were calculated by Chebyshev
668
interpolation [33] with grid points running from 0 to 1, and we used the "method of lines" to
669
solve the model at interior grid points (i.e. all but x=0 and x=1). Method of lines produces a
670
system of ordinary differential equations representing state variable values at the interior grid
671
points. The odesolve package [34] in R [35] was used to numerically solve these differential
672
equations.
673
Values at the boundary grid points are constants specified by the boundary conditions,
674
except for the value of A at x=0. For this we used a finite difference approximation to the no-flux
675
boundary condition:
676
(9)
677
(indices 0 and 1 indicate the left-most grid point and its neighbor to the right), and solved (9) for
678
as a function of
. Although more complex schemes could be used that achieve higher-order
679
theoretical accuracy, we have found that finite difference is much more robust against grid-scale
680
numerical artifacts in the interior that affect higher-order methods such as spectral estimates [31].
681 682 683 684
Appendix C. Computing Nullclines of the Spatial Model Let B and P denote the total beneficial and microbe densities in the spatial model, i.e. the integrals of
and
from x=0 to 1, respectively. To find the nullclines, we first
40
685
computed values of dB/dt and dP/dt for the rescaled spatial model, at a grid of values for B and
686
P, and then used the contour() function in R [35] to approximate the curves in the (B, P) plane on
687
which dB/dt=0 and on which dP/dt=0 .
688
The derivatives dB/dt and dP/dt need to be computed in the asymptotic spatial
689
distribution (such as the bottom row of Figure 4) determined by the specified values of B and P.
690
To compute the asymptotic spatial distribution for given B and P, we modified the spatial model
691
so that the spatial processes in the microbe populations continued to operate, but the total
692
population sizes did not change. Specifically, consider a modified spatial model in which terms
693
have been added to the equations for B and P that are proportional to the current population at
694
each spatial location, giving
695
(10)
696
(the tilde's in (10) indicate that these are state variables in the modified model). Because the new
697
terms are proportional to current population size, their immediate effect (over a short time-step in
698
numerically integrating the model forward in time) is to change the total population sizes but not
699
the shape of the spatial distribution.
700
We now want to choose values of the γ's in (10) so that the total population size remains
701
constant even though all other aspects of the model, including changes in spatial pattern, are
702
continuing to operate "as usual". Integrating equations (10) over (0,1) and using the boundary
703
conditions, we have
41
704
(11)
705
so total population size is held constant by setting the γ's at each time-step of the numerical
706
integration of the modified spatial model such that the right-hand sides of (11) are zero. For this
707
calculation the integrals on the right-hand side of (11) were computed by Clenshaw-Curtis
708
integration [33] and the spatial derivatives at x=1 were computed using the Chebyshev
709
differentiation matrix, in both cases using the Chebyshev spatial grid being used for numerical
710
solution of (10) by method of lines (as described in the main text).
711
Given B and P values, the modified model using (10) for the microbe populations was
712
initialized with initial distributions satisfying the boundary conditions such that the total microbe
713
populations have the specified values. This model was integrated from t=0 to t=12, which was
714
sufficient time for the spatial distributions to converge to a static form, while the total microbe
715
populations were held constant. The original spatial model was then integrated for 0.1 time units,
716
starting from the final state of the modified model. The changes in total microbe density over the
717
0.1 time units, multiplied by 10, were used as the estimates of dB/dt and dP/dt for the original
718
spatial model.
719
42
719
Appendix D: Parameter Baseline Values for Sensitivity Analysis
720 721
Thickness of mucus layer. According to [36], the outer mucus layer can vary in thickness from
722
a
few
tenths
of
a
mm
to
several
mm.
We
therefore
assume
that
mucus layer thickness is
723
about 1 mm.
724
725
Intrinsic Rates of Microbe Population Growth. Under ideal conditions in the lab, in vitro
726
bacterial growth rates can be extremely high, with doubling times well under one hour. For
727
example [36] observed doubling times of ~30 minutes on diluted coral mucus at 30C, by
728
bacterial strains identified as Vibrio species based on 16S rRNA. This corresponds to an intrinsic
729
growth rate of
at 30C. Applying a Q10 of 2.4 (see below), this corresponds
730
at 25C. These values imply growth by over 9 orders of magnitude within a day.
731
Growth rates on living corals in situ are likely to be much lower due to competition with other
732
species and local resource depletion, but to our knowledge there are no empirical estimates
733
available.
734
We therefore considered two different sets of baseline values for microbial potential
735
growth rates:
("slower") and
736
summer temperatures that are non-stressful to the host (≈29C or slightly cooler). Both of these
737
allow rapid changes in microbial populations – roughly thousand-fold growth within a week at
738
the slower parameters, and more than million-fold growth within 3 days at the faster parameters.
739
Both parameter sets represent our biological premises about the interaction between beneficials
740
and pathogens: (i) pathogens have the higher intrinsic growth rate in warmer months but are kept
741
in check by the antibiotics produced by the beneficials; (ii) effects of seasonal temperature
43
("faster"), at typical
742
variations on microbial growth rates (see below) give the beneficials the advantage during cooler
743
months even in the absence of antibiotics. Without property (i), pathogens would never be able
744
to grow at the expense of beneficials. Without property (ii), hosts could never recover from a
745
pathogen outbreak.
746 747
Antibiotic efficacy. Antibiotic activity in coral surface mucus caused a roughly 10-fold
748
reduction in pathogen growth rate at temperatures near 25C (K. Ritchie, unpublished data). In
749
our scaled well-mixed model, beneficials are at a density near 1 during cooler months, so we
750
need
751
densities during cooler months are roughly 10 in the regions near the coral surface where
752
substrate availability is high and microbe population growth occurs, so for the spatial model we
753
need λ smaller by a factor of 10.
, which is
, where
. In the scaled spatial model, beneficial
754 755
Temperature Dependence of Microbe Growth Rates. [37] estimated that Vibrio cholerae
756
population growth rate was proportional to
757
corresponding Q10 value is exp(0.88) ≈ 2.4. Vital et al. (2007) found a ratio of 2.1 between
758
Vibrio cholerae population growth rates at 30C versus 20C, i.e. Q10=2.1. In our simulations that
759
incorporate temperature effects we assumed Q10=2.4 for the pathogenic microbes. It is also
760
known that some Vibrio species produce a photosynthesis inhibitor at elevated temperatures [38].
761
We do not model this effect explicitly, but assume that elevated temperatures can decrease the
762
growth rate of beneficial microbes.
where T is Centigrade temperature; the
763
44
764
Microbial diffusion coefficients. Diffusion coefficients have been estimated for E. coli in
765
several different media but not (to our knowledge) for other bacteria, so we use those as a rough
766
guide. [39] estimated three-dimensional diffusion coefficient D = 5 × 10-6 cm2/sec for wild-type
767
E.coli in water, very close to their theoretically predicted value of 4 × 10-6 cm2/sec. We need to
768
divide this value by 3 to get the diffusion coefficient for the component of motion in one spatial
769
dimension, and multiply by (60×60×24) to convert to our model's time units (days). The result is
770
0.144 cm2/day. The corresponding diffusion distance (root mean square displacement) over a
771
one day time interval is about 5mm, more than the typical thickness of the mucus layer.
772
So as with population growth rates, these in vitro estimates cannot be applied directly to
773
microbes in vivo in coral mucus. There are at least two differences that need to be considered.
774
First, mucus is more viscous than water. Second, wild-type E. coli are actively propelling
775
themselves with flagella, and we expect this to be much less important for bacteria in the mucus
776
layer, which often lose their flagella because of their energetic cost.
777
A more relevant estimate may be Berg’s estimate for dead or paralyzed E. coli in water
778
at room temperature: D = 2 × 10-9 cm2/sec ([40]. This translates to 6 × 10-5 cm2/day in a single
779
spatial dimension, a root mean square displacement in 1 day of 0.1mm. If we assume a mucus
780
layer thickness of 1mm, then on the model's length scale (i.e. L=1 being 1mm), this estimate
781
becomes D=0.006.
782
The higher viscosity of mucus cuts both ways: it would slow down the diffusion of a
783
completely passive microbe, but speed up active propulsion because the thrust generated by a
784
flagellar rotor is proportional to the viscosity of the medium [40]. Assuming that microbes in the
785
mucus layer should be viewed as minimally active swimmers, we have used D=0.01 as a default
786
value for the microbes. This is small enough that even substantial changes should not have much
45
787
of an impact, because it is in a range where effects of microbial diffusion are dominated by other
788
transport processes: advection up nutrient gradients, and the "conveyor belt" motion of the
789
mucus medium.
790 791 792
Diffusion coefficients for substrate and antibiotic This is also speculative because we lack good information on the viscosity of coral
793
mucus. A variety of organic compounds, such as sugars and amino acids, have diffusion
794
coefficients in water at 25C on the order of 10-5 cm2/sec, which converts to roughly 30 mm2/d
795
(CRC Handbook of Chemistry and Physics, 2007-08 edition at www.hbcpnetbase.com). The
796
viscosity of water is approximately 1 centipoise, while that of human gut mucus at low shear
797
(probably typical of coral mucus) is higher by a factor of about 3000 according to [41]. The
798
diffusion coefficient is inversely proportional to viscosity, so we have the estimate D=0.01
799
mm2/day. Dog gastric mucus, according to [42] has about 50 times the viscosity of water. Like
800
coral mucus it is a mix of carbohydrates and proteins, so this may be a reasonable viscosity
801
estimate for coral mucus, giving the estimate D= 0.6 mm2/day. Given this wide range we have
802
used D=0.1 as our default value for both substrate and antibiotics.
803 804
Advection coefficients for microbes. Microbes in the mucus layer would be expected to expend
805
the minimum effort needed to keep themselves in preferable regions of the mucus layer, close to
806
the nutrient source (the coral host) rather than the relatively hostile seawater environment. In
807
some host-microbe interactions, once they have colonized the host from the water column,
808
bacteria no longer produce flagella [43] presumably to avoid the metabolic costs of synthesizing
809
flagella and flagellar motion (in addition, flagella can be used as antigenic targets by the immune
46
810
system, but this may not be relevant with an invertebrate host). We therefore set the advection
811
coefficients to values just high enough so that microbe populations at baseline values of the other
812
parameter established a clear gradient in density concentrated near the coral host, but not so large
813
that microbes were able to concentrate exclusively in a narrow band very near the host.
814 815
Substrate limitation of microbe growth. Values of the half-saturation coefficients were chosen
816
so that fresh mucus (substrate level S=1 in our scaled models) would put the microbes near
817
saturation, but 90% substrate depletion would have a substantial effect. Given the sharp substrate
818
gradients that are established in the spatial model, going quickly from S=1 to S near zero within
819
a small distance from the coral host, the value of the half-saturation constant should not be very
820
critically important within a broad range.
821 822 823 824
Appendix E: Sensitivity Analysis Methods and Discussion Here we give additional methodological details for our sensitivity analysis of the spatial model and more extensive discussion of the results.
825
Methods: For each scenario, 500 perturbed parameter sets sets were generated by Latin
826
Hypercube Sampling [44,45]. For each of these we ran the model to compute the total beneficial
827
and pathogenic microbial populations after 30 days (slower growth) or 20 days (faster growth).
828
The relative importance of each parameter was measured by using multiple linear regression to
829
compute the elasticity of the final microbial populations with respect to each parameter.
830
Elasticity is the proportional change in the response relative to the proportional change in the
831
parameter, averaged over the parameter range being considered. So an elasticity value of 2
832
means that a 5% change in the parameter causes a 10% change in the response variable (total
47
833
beneficials or pathogens). Elasticity values were estimated by multiple linear regression of log-
834
transformed final microbial abundances on log-transformed parameters, and the slope coefficient
835
for each parameter is the estimated elasticity.
836
Initial conditions: Different initial conditions were used for each scenario. Baseline
837
scenario initial conditions were chosen to test, for a healthy coral in normal warm-month
838
circumstances, which parameters affected the effectiveness of defense against pathogen invasion
839
and in particular the potential for competitive exclusion of pathogens. The initial conditions
840
therefore had significant concentrations of both beneficials and pathogens to allow opportunity
841
for competitive exclusion by either. Heat stress scenario initial conditions set pathogen initial
842
concentrations just above zero, to test which parameters affected the coral's degree of
843
vulnerability to pathogen invasion under heat stress. High antibiotic scenario initial conditions
844
set pathogens significantly more abundant than beneficials to test the efficacy of the antibiotics
845
as a mechanism for invasion by beneficials.
846
Results: Baseline Scenario: In the baseline/slower growth scenario (Figure 6a) the most
847
significant parameters are the chemotaxis coefficients
, which regulate the rate at which
848
beneficials and pathogens spread to higher concentrations of nutrients. Success of either the
849
beneficials or pathogens is determined by their ability to overcome the potential mortality due to
850
mucus ablation, and all other parameters are insignificant. With faster microbial growth (Figure
851
6b), chemotaxis is much less important than parameters affecting the competitive interactions
852
between pathogenic and beneficial microbes. The directions of parameter effects are all as one
853
would expect, for example a higher value of
854
pathogens, while a higher value of
increases the beneficials and decreases the
is detrimental to both. The efficacy of antibiotics
48
855
(determined by
and
) is drowned out by the dramatic changes in antibiotic production
856
related to changes in the beneficials growth rate.
857
Results: Heat Stress Scenario: In the heat stress/slower growth scenario (Fig. 6c)
858
pathogen success correlates most strongly with pathogen maximum growth rate. Beneficials
859
maximum growth rate and half saturation constant, and the chemotaxis parameters are of
860
secondary importance, and all other parameters are insignificant for pathogen success.
861
Beneficials success correlates most with its advection and potential growth rates and with the
862
mucus ablation rate. Because antibiotic production and efficacy are greatly reduced in this
863
scenario, the corresponding parameters (
864
affecting the resource competition between beneficials and pathogens. Results for the heat
865
stress/faster growth scenario (Fig. 6d) are very similar, except that higher beneficials' growth rate
866
implies higher antibiotic production, so the importance of
867
success correlates most with growth rates and beneficials' half saturation constant, and
868
beneficials success correlates most with advection and growth rates, with all other parameters
869
being less important or insignificant.
) are unimportant relative to parameters
and
is increased. Pathogen
870
Results: High Antibiotic Scenario: Results for the high antibiotic/slower growth scenario
871
(Fig. 6e) are very similar to the baseline/faster growth scenario. Pathogen success is highly and
872
negatively correlated with the beneficials growth rate. Also significant are advection, pathogen
873
growth rate, beneficials half saturation constant, beneficials chemotaxis, and pathogen diffusion.
874
Beneficials success is highly correlated with advection and beneficials growth rate. In the high
875
antibiotic/faster growth scenario (Fig 6f) the combination of high growth rate, high antibiotic
876
production and high antibiotic efficacy gives the beneficials the upper hand across the entire
877
range of parameters considered for the sensitivity analysis. Conditions for the pathogens are bad
49
878
everywhere (either due to low resource levels or high antibiotic concentrations) so they cannot
879
escape through chemotaxis. The only factor affecting their success is the balance between
880
mortality through mucus ablation and diffusion away from the ablation boundary.
881 882
Literature cited
883
1. Harvell C, Jordán-Dalhgren E, Merkel S, Rosenberg E, Raymundo L, et al. (2007) Coral
884
disease, environmental drivers and the balance between coral and microbial associates.
885
Oceanography 20: 172-195.
886
2. Lesser MP, Bythell JC, Gates RD, Johnstone RW, Hoegh-Guldberg O (2007) Are infectious
887
diseases really killing corals? Alternative interpretations of the experimental and
888
ecological data. Journal of Experimental Marine Biology and Ecology 346: 36-44.
889
3. Rosenberg E, Kushmaro A, Kramarsky-Winter E, Banin E, Yossi L (2009) The role of
890
microorganisms in coral bleaching. International Society for Microbial Ecology (ISME)
891
Journal 3: 139-146.
892
4. Bourne D, Iida Y, Uthicke S, Smith-Keune C (2008) Changes in coral-associated microbial
893
communities during a bleaching event. International Society for Microbial Ecology
894
(ISME) Journal 2: 350-363.
895 896
5. Rosenberg E, Ben-Haim Y (2002) Microbial diseases of corals and global warming. Environmental Microbiology 4: 318-326.
897
6. Ben-Haim Y, Thompson FL, Thompson CC, Cnockaert MC, Hoste B, et al. (2003) Vibrio
898
coralliilyticus sp nov., a temperature-dependent pathogen of the coral Pocillopora
899
damicornis. International Journal of Systematic and Evolutionary Microbiology 53: 309-
900
315.
50
901
7. Ben-Haim Y, Zicherman-Keren M, Rosenberg E (2003) Temperature-regulated bleaching and
902
lysis of the coral Pocillopora damicornis by the novel pathogen Vibrio coralliilyticus.
903
Applied and Environmental Microbiology 69: 4236-4242.
904 905 906 907 908 909
8. Miller J, Waara R, Muller E, Rogers C (2006) Coral bleaching and disease combine to cause extensive mortality on reefs in US Virgin Islands. Coral Reefs 25: 418-418. 9. Bruno JF, Selig ER, Casey KS, Page CA, Willis BL, et al. (2007) Thermal stress and coral cover as drivers of coral disease outbreaks. Plos Biology 5: 1220-1227. 10. Goreau T, McClanahan T, Hayes R, Strong A (2000) Conservation of coral reefs after the 1998 global bleaching event. Conservation Biology 14: 5-15.
910
11. McClanahan T (2004) Coral bleaching, diseases, and mortality in the western Indian Ocean.
911
In: Rosenberg E, Loya Y, editors. Coral Health and Disease New York: Springer-Verlag.
912
pp. 157-176.
913 914 915 916
12. Brown BE, Bythell JC (2005) Perspectives on mucus secretion in reef corals. Marine Ecology-Progress Series 296: 291-309. 13. Lipp EK, Huq A, Colwell RR (2002) Effects of global climate on infectious disease: the cholera model. Clinical Microbiology Reviews 15: 757-770.
917
14. Thompson FL, Thompson CC, Naser S, Hoste B, Vandemeulebroecke K, et al. (2005)
918
Photobacterium rosenbergii sp nov and Enterovibrio coralii sp nov., vibrios associated
919
with coral bleaching. International Journal of Systematic and Evolutionary Microbiology
920
55: 913-917.
921 922
15. Ritchie KB (2006) Regulation of microbial populations by coral surface mucus and mucusassociated bacteria. Marine Ecology-Progress Series 322: 1-14.
51
923 924
16. Kushmaro A, Rosenberg E, Fine M, Loya Y (1997) Bleaching of the coral Oculina patagonica by Vibrio AK-1. Marine Ecology-Progress Series 147: 159-165.
925
17. Kushmaro A, Rosenberg E, Fine M, Ben Haim Y, Loya Y (1998) Effect of temperature on
926
bleaching of the coral Oculina patagonica by Vibrio AK-1. Marine Ecology-Progress
927
Series 171: 131-137.
928
18. Kushmaro A, Banin E, Loya Y, Stackebrandt E, Rosenberg E (2001) Vibrio shiloi sp nov.,
929
the causative agent of bleaching of the coral Oculina patagonica. International Journal of
930
Systematic and Evolutionary Microbiology 51: 1383-1388.
931 932 933 934 935 936
19. Teplitski M, Ritchie KB (2009) How feasible is the biological control of coral diseases. Ecology Letters: (in press) 20. Ducklow HW, Mitchell R (1979) Composition of Mucus Released by Coral-Reef Coelenterates. Limnology and Oceanography 24: 706-714. 21. Peters E (1997) Diseases of coral reef organisms. In: Birkeland C, editor. Life and death of coral reefs. New York: Chapman & Hall. pp. 114-139.
937
22. Santavy D, Peters E (1997) Microbial pests: coral disease research in the western Atlantic.
938
In: Lessios HA, Macintyre. IG, editors. Proceedings of the 8th International Coral Reef
939
Symposium, Panama, June 24-29,1996. Balboa, Panama: Smithsonian Tropical Research
940
Institute. pp. 607-612.
941 942 943 944
23. Hayes RL, Goreau NI (1998) The significance of emerging diseases in the tropical coral reef ecosystem. Revista De Biologia Tropical 46: 173-185. 24. Sutherland KP, Porter JW, Torres C (2004) Disease and immunity in Caribbean and IndoPacific zooxanthellate corals. Marine Ecology-Progress Series 266: 273-302.
52
945
25. Rublee PA, Lasker HR, Gottfried M, Roman MR (1980) Production and Bacterial-
946
Colonization of Mucus from the Soft Coral Briarium-Asbestinum. Bulletin of Marine
947
Science 30: 888-893.
948
26. Toren A, Landau L, Kushmaro A, Loya Y, Rosenberg E (1998) Effect of temperature on
949
adhesion of Vibrio strain AK-1 to Oculina patagonica and on coral bleaching. Applied
950
and Environmental Microbiology 64: 1379-1384.
951
27. Banin E, Israely T, Fine M, Loya Y, Rosenberg E (2001) Role of endosymbiotic
952
zooxanthellae and coral mucus in the adhesion of the coral-bleaching pathogen Vibrio
953
shiloi to its host. Fems Microbiology Letters 199: 33-37.
954
28. Lipp EK, Jarrell JL, Griffin DW, Lukasik J, Jaculiewicz J, et al. (2002) Preliminary evidence
955
for human fecal contamination of corals in the Florida keys, USA. Marine Pollution
956
Bulletin 44: 666-670.
957 958 959 960
29. Foster KR (2005) Biomedicine - Hamiltonian medicine: Why the social lives of pathogens matter. Science 308: 1269-1270. 30. Hale J (1988) Asymptotic behavior of dissipative systems. Providence RI American Mathematical Society
961
31. Ellner SP, Jones LE, Mydlarz LD, Harvell CD (2007) Within-host disease ecology in the sea
962
fan Gorgonia ventalina: Modeling the spatial immunodynamics of a coral-pathogen
963
interaction. American Naturalist 170: E143-E161.
964 965 966 967
32. Wild C, Huettel M, Klueter A, Kremb SG, Rasheed MYM, et al. (2004) Coral mucus functions as an energy carrier and particle trap in the reef ecosystem. Nature 428: 66-70. 33. Trefethen L (2000) Spectral Methods in Matlab. Philadelphia, PA: Society for Industrial and Applied Mathematics.
53
968 969
34. Setzer R (2007) odesolve: Solvers for Ordinary Differential Equations, R package version 0.5-19. Vienna, Austria: R Foundation for Statistical Computing.
970
35. R_Development_Core_Team (2008) R: A language and environment for statistical
971
computing. 2.70 ed. Vienna, Austria: R Foundation for Statistical Computing,
972
http://www.R-project.org.
973 974
36. Sharon G, Rosenberg E (2008) Bacterial growth on coral mucus. Current Microbiology 56: 481-488.
975
37. Kirschner AKT, Schlesinger J, Farnleitner AH, Hornek R, Suss B, et al. (2008) Rapid growth
976
of planktonic Vibrio cholerae Non-O1/Non-O139 strains in a large alkaline lake in
977
Austria: Dependence on temperature and dissolved organic carbon quality. Applied and
978
Environmental Microbiology 74: 2004-2015.
979
38. Ben-Haim Y, Banim E, Kushmaro A, Loya Y, Rosenberg E (1999) Inhibition of
980
photosynthesis and bleaching of zooxanthellae by the coral pathogen Vibrio shiloi.
981
Environmental Microbiology 1: 223-229.
982
39. Berg HC, Turner L (1990) Chemotaxis of Bacteria in Glass-Capillary Arrays - Escherichia-
983
Coli, Motility, Microchannel Plate, and Light-Scattering. Biophysical Journal 58: 919-
984
930.
985
40. Berg HC (1983) Random Walks In Biology. Princeton, NJ: Princeton University Press.
986
41. Markesich DC, Anand BS, Lew GM, Graham DY (1995) Helicobacter-Pylori Infection Does
987
Not Reduce the Viscosity of Human Gastric Mucus Gel. Gut 36: 327-329.
988
42. Slomiany BL, Sarosiek J, Slomiany A (1987) Role of Carbohydrates in the Viscosity and
989
Permeability of Gastric Mucin to Hydrogen-Ion. Biochemical and Biophysical Research
990
Communications 142: 783-790.
54
991 992
43. Ottemann KM, Miller JF (1997) Roles for motility in bacterial-host interactions. Molecular Microbiology 24: 1109-1117.
993
44. McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting
994
values of input variables in the analysis of output from a computer code. Technometrics
995
21: 239-245.
996
45. Blower SM, Dowlatabadi H (1994) Sensitivity and Uncertainty Analysis of Complex-Models
997
of Disease Transmission - an Hiv Model, as an Example. International Statistical Review
998
62: 229-243.
999 1000
55