How microbial community composition regulates coral disease development

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Running head: Coral microbial communities and disease

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How microbial community composition regulates coral disease transmission

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†

J. Mao-Jones1, K. B. Ritchie2, L. E. Jones1, , and S.P. Ellner1

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Department of Ecology and Evolutionary Biology

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Cornell University, Ithaca NY 14850, USA

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Mote Marine Laboratory

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1600 Ken Thompson Parkway, Sarasota, Florida 34236, USA

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†

Corresponding author. E-mail [email protected]

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Last save: 4/21/2009 2:35 PM EST

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Abstract: Coral reef cover is in rapid decline worldwide, in part due to bleaching

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(expulsion of photosynthetic symbionts), infectious disease outbreaks and consequent elevated

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mortality. One important factor in both disease transmission and initiation of bleaching, is a shift

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in the composition of the microbial community in the mucus layer surrounding the coral: the

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microbes normally dominant in the mucus layer, which produce antibiotic substances limiting

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pathogen growth, are replaced by pathogenic microbes, often species of Vibrio. In this paper we

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develop computational models for microbial community dynamics in the mucus layer in order to

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understand how the surface microbial community responds to changes in environmental

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conditions and under what circumstances it becomes vulnerable to takeover by pathogens. We

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find that the pattern of interactions in the surface microbial community facilities the existence of

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alternate stable states, one dominated by beneficial microbes and the other pathogen-dominated.

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Consequently, a shift to pathogen dominance under transient stressful conditions, such as a brief

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warming spell, may persist long after conditions have returned to normal. This prediction is

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consistent with observational findings that antibiotic activity does not return to measurable levels

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in coral after temperature reduction. The resulting long-term loss of antibiotic activity eliminates

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a critical component in coral defense against disease, giving pathogens an extended opportunity

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to infect and spread within the host, elevating the risk of coral bleaching and mortality.

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Key words: Coral, disease, mucus, microbe, pathogen, Vibrio, symbiosis, global

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warming, climate change.

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Abbreviations: SMC, surface microbial community. LHS, Latin Hypercube Sampling.

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Introduction

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In recent years, coral reef cover has declined dramatically worldwide, due in part to

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infectious diseases [1], though many other factors likely contribute as well. What facilitates the

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spread of infections remains uncertain [1], but many disease outbreaks may be opportunistic

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infections by environmentally endemic organisms following physiological stress to the coral [2-

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4] Some researchers propose that various forms of physiological stress, namely increased sea-

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surface temperatures, are correlated with increased virulence in opportunistic pathogens [1,5-7].

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However the dynamics of opportunistic organisms within the microbial communities they inhabit

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remain mysterious and are the subject of this paper.

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Common to almost all instances of disease outbreaks over the past several decades are

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bleaching events preceded by temperature stress. For example, a severe thermal anomaly in the

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Caribbean during 2005 was followed by widespread bleaching and then by significant spread of

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white plague and yellow blotch, resulting in 26-48% losses in coral cover in some areas [8]. [9]

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found a significant positive correlation between the frequency of warm thermal anomalies and

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the occurrence of white syndrome on Australia's Great Barrier Reef. The El Niño weather

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phenomenon in 1998 was followed by widespread bleaching in the western Indian Ocean and

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then by 50-60% mortality rates [10,11] These recent mortality events are only a few of many

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instances that have resulted in significant losses in coral cover.

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A bleached coral is more vulnerable to disease than a non-bleached coral, which is

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defended by a mucus layer covering the coral’s outer surface, generated in part by its algal

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symbionts [12]. Under non-stressful conditions, corals produce a mucus layer which contains a

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complex microbial community commonly referred to as the surface microbial community

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(SMC). One important function of the mucus layer is that it provides defense against pathogens

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[12] and other stressors not considered in this paper. Most potential pathogens are endemic to the

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ecosystem and present at low numbers in the SMC, but a rapid shift to pathogen dominance can

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occur in the SMC after heat stress and prior to a bleaching event [4,5,7,13-15]. For example,

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during a late 2005 summer bleaching event Ritchie [15] observed that "visitor" bacteria, or

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bacterial groups that otherwise were not dominant, were the predominant species in mucus

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collected from apparently healthy Acropora palmata. During temperature stress Vibrio

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coralliilyticus is an agent of bleaching in Pocillopora damicornis [6] and Vibrio shiloi becomes

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an agent of bleaching in the coral Oculina patagonica [16-18]. Rosenberg et al. [3] recently

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summarize the evidence supporting a "microbial hypothesis of coral bleaching", that bleaching is

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often initiated by a shift to pathogen dominance in the SMC brought on by heat stress, rather

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than primarily by direct effects of heat stress on the coral and its symbionts [18,19].

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The mucus environment is rich in nutrients, and microbial population concentrations are

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orders of magnitude higher in coral mucus than in the surrounding water column [20]. According

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to [15], "a component universal to coral mucus, independent of species, location, and season, is

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capable

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[Pseudoalteromonadaceae bacterium]" suggesting that coral mucus is capable of regulating the

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activities of a number of visitor and water column microbial isolates. Some visitor microbes

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produced antibiotic compounds, but this was rarer in visitors (16% of isolates tested) than in

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residents (41% of isolates tested [15] ) that resident microbes may be beneficial to corals.

of

inhibiting

pigment

and

antibiotic

production

associated

with

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While mucus provides nutrient substrates on which the beneficial microbes can grow

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[12,17,21-24], it also readily serves as a growth medium for pathogenic invaders [12,20,25-28].

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Thus, altered mucus conditions may shift community composition, facilitating pathogen invasion

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and opportunistic infection. Indeed, beneficial microbes declined in times of increased water

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temperature, when less than 2% of bacteria isolated from the surface of Acropora palmata

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displayed antibiotic activity - significantly less than the 28% of antibiotic-producing isolates

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present in cooler months.

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It is uncertain what causes the decline of beneficial bacteria, but overgrowth of Vibrio

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spp. has often been observed to precede disease outbreak. During one summer bleaching event,

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Ritchie [15] found that Vibrio spp. were the predominant species (85% of species) cultured from

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mucus collected from apparently healthy Acropora palmata sustained at a mean daytime sea

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surface temperature of 28° to 30°C (the upper extreme of the standard range of thermal

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tolerance, 18° to 30°C) for 2 months prior to collection.

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An understanding of the SMC is therefore crucial to ameliorating the effects of thermal

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stress on corals, including bleaching and infectious disease spread. A number of qualitative

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models have been proposed for the causes and dynamics of the loss of antibiotic activity and

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subsequent overgrowth of Vibrio spp. in the SMC. Ritchie [15] suggested that the antimicrobial

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properties of the mucus are temperature-sensitive due to antimicrobial sensitivity or resident

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microbe sensitivity to temperature change, with Vibrio overgrowth following a loss of

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antimicrobial activity by residents. Another hypothesis proposes that Vibrio spp., which thrive at

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elevated temperatures, out-compete beneficial bacteria in these conditions, and a loss of

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antibiotic activity follows [5,13-15]. These models agree with the experiments of Ducklow and

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Mitchell [20] in which coral mucus bacterial populations increased significantly when the coral

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was stressed. Foster [29] proposed that the competition for space by invasive microbes divides

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the beneficial microbes into isolated patches, where genetic diversity is significantly decreased.

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Common to all hypotheses is that disease susceptibility is positively correlated with a loss of

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antibiotic activity and an increase in Vibrio densities in the SMC.

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Why take a modeling approach?

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It appears that antibiotic activity and the competition between beneficial and potentially

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pathogenic microbes (primarily Vibrio spp.) are key to understanding community dynamics

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within the SMC. In this paper, we develop models for how these interactions affect the outcome

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of competition within the SMC, and promote overgrowth by pathogenic microbes. We describe

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first a spatially well-mixed model, and then turn to a model that includes the spatial gradients in

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nutrient concentration from coral surface to the surrounding seawater, mucus production by the

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coral, and ablation of mucus into the surrounding seawater. In both models we assume that

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interactions can be simplified to a few key players, each representing some set of microbial

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organisms or substances within the mucus layer. Because many quantitative and qualitative

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characteristics of the SMC are still uncertain, we explore general properties of the model as

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parameters are varied, rather than attempting to closely simulate any specific coral-pathogen

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interaction. We thereby identify the parameters and processes that have the most significant

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impacts on SMC dynamics and the potential for pathogen outbreaks.

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A focal aspect of our modeling is exploring the importance of the spatial dimension -

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which permits spatial heterogeneity – in this system. A spatially unstructured model (which

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posits that the mucus layer is "well-mixed”) does not depict the physical processes of mucus

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production by the coral and its endosymbionts and of mucus loss by sloughing off into the water

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column. Furthermore, it is possible that gradients in chemical concentrations and population

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densities within the mucus layer may have considerable effect on the dynamics of the microbial

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community. Our spatial model considers a gradient of fixed length with endpoints at the coral

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surface and at the interface between the mucus layer and the water column. By contrasting this

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model with a spatially unstructured. “well-mixed” model, we examine the role of spatial

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gradients in SMC dynamics and in defense against pathogen invasion.

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Results

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The well-mixed model. We begin with a simple well-mixed model to explore community

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dynamics within the SMC, assuming first that there is no pathogen inoculation from external

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sources, and later relaxing this assumption. This model and its underlying assumptions are

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presented in detail in the Materials and Methods section. Insight into the behavior of the model

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without external inputs can be gained by a simple nullcline analysis. The analysis is further

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simplified if we consider a general rescaled well-mixed model

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(1)

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where

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substrate within beneficials and antibiotics combined. The growth-rate functions

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increasing functions of s, and

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concentration that is proportional to b. We assume

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maximum possible value of s); otherwise the populations die out because there is never enough

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substrate for population increase. We also assume that

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neither population can persist in the absence of substrate.

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; 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

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, i.e., the line

plane. The b

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(1) which has constant slope of

. The p nullcline is the curve

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(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

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nullcline lies completely above the other, then all initial conditions with

lead

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to competitive exclusion of the species with the lower nullcline, exactly like the Lotka-Volterra

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model. If the nullclines cross, their intersection is a saddle (locally unstable), so there is

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competitive exclusion again but with the identity of the winner depending on initial conditions

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(sometimes called contingent exclusion). Both of the single-species equilibria (on the coordinate

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axes) are then locally stable.

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Without the antibiotic-mediated interactions, we would have pure resource competition

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for a single limiting substrate. The p and b nullclines would then be parallel lines, and the

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pathogen is the superior competitor if the p nullcline is above the b nullcline (as in Figure 1a).

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Adding antibiotic effects, if they are sufficiently strong, will make the p nullcline decrease more

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quickly as b increases, giving the situation shown in Figure 1b. Control of potentially dominant

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pathogens through antibiotic activity is thus the "recipe" for contingent exclusion. A numerically

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dominant pathogen population can prevent regrowth of beneficials by keeping substrate

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concentrations too low for beneficials to increase, while a numerically dominant beneficial

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population can prevent pathogen regrowth by maintaining a high ambient concentration of

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antibiotics.

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What if there is a pathogen source (

)? The empirical observation that microbes are

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orders of magnitude less abundant in seawater than in mucus implies that

in the scaled

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model. A pathogen source is then a small perturbation to the dynamics shown in Figure 1, so its

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only effect is to keep pathogens from being eliminated: in cases of pathogen exclusion in Figure

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1 (panels A and C of Figure 1), the stable equilibria with p=0 and b large are replaced by stable

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equilibria with p close to 0 and b large.

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Pathogen Invasion. We now explain how the properties of the well-mixed model can lead to a

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sudden and persistent "takeover" of the SMC by pathogens following a brief period of conditions

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stressful to the host and to beneficial microbes. To illustrate the process we consider thermal

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stress, which has been implicated most consistently as the environmental driver linked to

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pathogen outbreaks.

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For a healthy coral we can presume that during colder months, which are less favorable to

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the pathogens, the beneficials are able to exclude the pathogens (Figure 2a) – only the pathogen-

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exclusion equilibrium is stable. During warmer months (Figure 2b) the higher temperatures may

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give the pathogens a higher intrinsic growth rate than the beneficials, but pathogen growth is

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kept in check by the effect of antibiotics, so the pathogen-exclusion equilibrium remains locally

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stable. However, a thermal anomaly that causes the loss of antibiotic activity (Figure 2c)

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eliminates the coexistence equilibrium, so the system jumps to being pathogen-dominated.

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Figure 3 illustrates the corresponding microbial population dynamics during a warm

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anomaly scenario based on sea surface temperatures at Glover's Reef, Belize (Fig. 3a). Under

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normal seasonal variation in temperature, beneficial microbes are dominant year-round (Figure

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3b). Regardless of how a simulation is initiated, during winter the beneficials become dominant

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and they remain dominant through the summer, while pathogens persist at very low levels due to

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inoculation from the water column. But even a brief thermal anomaly that eliminates antibiotic

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activity for 14 days (Fig. 3b) allows the pathogens to become dominant and remain so for

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approximately 3 months, until temperatures drop low enough that the pathogen-dominant

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equilibrium becomes unstable.

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The well-mixed model therefore provides a mechanistic explanation for the empirically

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observed sudden switches to pathogen dominance following a change in conditions, and

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moreover the model predicts that such sudden switches will occur even if the change in

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conditions is gradual. Another key prediction is that the return from pathogen-dominance to

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beneficials-dominance when conditions improve will also be sudden, but it will occur under

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different conditions: pathogens may remain dominant even after environmental conditions return

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to those where beneficials were initially dominant, while beneficials may not recover dominance

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until the environment becomes considerably more favorable for them.

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The spatial model: Qualitative properties. A fundamental question inherent in including

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spatial heterogeneity in a model is whether spatial dimension allows for a broader range of

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qualitative outcomes than a well-mixed, or non-spatial model. In particular, spatial variability

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might allow stable coexistence of pathogenic and beneficial microbes, for example if pathogens

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segregate away from beneficials and so avoid the effects of antibiotics produced by the

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beneficials. The well-mixed model's prediction of potentially abrupt changes in community

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composition in response to gradual changes in environmental conditions might then prove to be

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an artifact of neglecting spatial variability. We therefore expand the model to include spatial

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dimension; the model and its underlying assumptions are presented in Materials and Methods,

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while details of numerical analysis and simulation are found in the Appendices.

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Numerical study of the spatial model shows that the nullcline analysis of nonspatial

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model (Figure 1) continue to hold. Specifically, the spatial model behaves like a two-

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dimensional system of differential equations, even though it has an infinite-dimensional state

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space. This occurs because, apart from a brief transient period, the spatial distribution of the state

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variables is completely predictable from the total abundance of beneficial and pathogenic

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microbes. Figure 4 shows an example. Two model runs were initialized by first two different

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shapes for the population distributions at time t=0, and then using numerical optimization to find

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total population sizes at t=0 such that the total beneficial and pathogen populations at time t=12

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would both be 4. As Figure 4 illustrates, when the total beneficial and pathogenic populations are

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at equal values, everything else (antibiotics, population distributions, etc.) are the same in the

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two runs. In technical terms, numerical solutions show the model converging onto a two-

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dimensional inertial manifold [30].

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So given that there are

total beneficials and

total pathogens in the community

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at time t, we can compute the complete state of the system and thus compute the instantaneous

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rates of change dB/dt and dP/dt. This procedure defines a two-dimensional dynamical system for

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the microbial populations, so we can again determine its qualitative behavior by plotting the

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nullclines. Figure 5 shows nullclines for the slower "baseline" parameters listed in Table 1,

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computed by the methods described in Appendix C. These show that the spatial model is in the

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bistable situation of Figure 1(b), indicating that a healthy population of beneficial microbes can

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keep pathogens from increasing, but beneficials would be at a competitive disadvantage in a

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community dominated by pathogenic microbes. Given the large uncertainties in our parameter

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estimates, we cannot regard this property as a prediction about nature. The important feature of

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Figure 5 is that, as in the well-mixed model, the pathogen nullcline is steeper than the beneficials

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nullcline, which is the property that precludes stable coexistence of beneficials and pathogens

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(rather than one or the other being sustained at very low numbers by small levels of input from

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the surrounding water column). Consequently, the spatial model preserves the key qualitative

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prediction of the well-mixed model: if temporary extreme conditions allow the community to

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become dominated by pathogenic microbes, the pathogen takeover may persist even after

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conditions return to normal and may not terminate until conditions occur (such as winter

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temperatures) that are highly unfavorable to the pathogens.

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Sensitivity analysis of the spatial model.

We performed a local sensitivity analysis to

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determine the relative impact of each parameter on system dynamics. Due to the high uncertainty

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of parameter estimates, parameters were varied up to ±50% from their default values (Table 1)

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using Latin Hypercube sampling (see Appendices D and E for additional information about our

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sources for parameter values and the methods used to carry out the sensitivity analysis).

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We carried out sensitivity analysis under three different scenarios: baseline (the

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parameter values listed in Table 1), heat stress, and high antibiotic conditions. For the heat stress

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scenario, beneficials growth rate was reduced, pathogen growth rates increased, and the

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production of antibiotics was decreased. For the high antibiotic secnario the antibiotic production

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rate was increased and the efficacy of the antibiotics against the pathogens increased. We also

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considered both the "slower growth" and "faster growth" values for the microbe growth rate

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parameters

. Parameter values for these scenarios are listed in Table 2.

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Overall, the results of the sensitivity analysis (Figure 6) indicate that the most important

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parameters are either (i) the advection and diffusion coefficients, which control the balance

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between movement towards favorable conditions and mortality through mucus ablation, or (ii)

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the maximum growth rates, which are important for the direct competitive interactions between

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the microbial populations. Movement parameters were generally less important in the faster

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growth scenarios where competitive interactions are stronger. Changes in antibiotic production

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have the most effect on pathogen success in the faster growth baseline and heat stress scenarios,

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because the rate of antibiotic production correlates with the beneficials' population growth rate.

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In the high antibiotic scenario, the antibiotic inhibition of pathogens is so strong that even major

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changes in the strength of antibiotic inhibition have no effect on the outcome (i.e., pathogen

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exclusion). The most surprising outcome is that antibiotic conversion rate, bacteriostatic efficacy,

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and antibiotic degradation rate are never among the most significant parameters, even though the

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beneficials' ability to produce antibiotics is essential for their persistence. This suggests that (if

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our parameter ranges are realistic) the role of antibiotics in normal conditions is to tip the balance

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in a competition between near-equals. To explore this idea further, we modified the

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baseline/faster growth scenario by holding the ratios

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Latin Hypercube sample parameter vector we perturbed

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the values

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As expected, with these constraints (Fig 6g) the importance of growth rate variation (indicated

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by r in the axis label) is greatly reduced relative to Fig 6b, and the importance of antibiotic-

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related parameters is greatly increased. The fact that higher overall growth rates are detrimental

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to the pathogen also reflects the impact of antibiotics, because of the proportionality between

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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).

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Discussion

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Previous studies have shown that a sudden shift to pathogen dominance occurs in the

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surface microbial communities (SMC) of corals prior to a bleaching event [5,13-15]. It has also

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been demonstrated that antibiotic activity and antibiotic-producing bacteria in the SMC decline

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in times of increased water temperature when bleaching is most likely to occur [15]. Disease

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susceptibility in hard corals is thus positively correlated with a loss of antibiotic activity and an

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overgrowth of pathogenic bacterial densities in the SMC.

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In this paper we have developed mathematical models to explore how interactions

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between resident and invading microbes within the coral SMC affect the health and disease

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susceptibility of reef-building corals, assuming first a well-mixed mucus layer and then allowing

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spatial heterogeneity in microbial population densities and nutrient substrate. A surprising but

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robust finding is the consistency in outcomes between well-mixed and spatially heterogeneous

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systems. Though spatial heterogeneity might be expected to allow spatial segregation and thus

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coexistence of pathogenic and beneficial microbe types, in both cases stable coexistence is

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precluded. Instead, the situation is one of competition and contingent exclusion between two

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more or less equal competitors, with antibacterial production usually shifting the balance in favor

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of microbes beneficial to the coral.

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The well-mixed model. Analysis of the well-mixed model shows that under competition

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for a single limiting substrate, control of pathogen via antibiotic activity is the key to the

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contingent exclusion of pathogens by the beneficial bacteria. However, under the empirically

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supported assumption of reduced antibacterial production during heat stress, the model predicts a

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rapid switch from dominance by beneficial microbes to dominance by pathogens during thermal

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anomalies. In addition, dominance by beneficials is not restored when temperatures return to the

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normal conditions under which the beneficials were previously dominant. Instead, conditions

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must become unusually unfavorable to pathogens before a switch back to dominance by

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beneficial bacterial can occurs. This is consistent with observational findings that antibiotic

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activity does not return to measurable levels in coral even after recovery and temperature

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reduction (Ritchie, unpublished data, 2009).

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The spatially heterogeneous model. Simulations of the spatial model show that the key

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qualitative result from the analysis of well-mixed model (contingent exclusion) holds in the

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heterogeneous case as well. However, because of the level of uncertainty in our parameter

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assumptions and estimations, we performed sensitivity analyses to better quantify the effects of

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parameter variation under normal warm-season conditions, heat stress conditions, and finally

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conditions of high antibiotic production.

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Under normal conditions, the ability to spread to fresh sources of substrate is critical to

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the success of a slowly reproducing pathogen. For a more rapidly growing pathogen under these

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conditions, the ability to compete for nutrients becomes more important. Under conditions

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causing heat stress, the pathogen uniformly has the upper hand and its success hinges primarily

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on its potential population growth rate. Results for high antibiotic production are similar to

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baseline conditions, except antibiotic production benefits the beneficial bacteria across the entire

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range of parameters considered. Thus the critical parameters overall are those which govern

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movement (microbial advection and diffusion coefficients, mucus ablation rates) and maximum

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microbial growth. Finally, a numerical nullcline analysis of the heterogeneous model suggests

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that it retains the hysteresis observed in the well-mixed model: once temporary extreme heat

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allows pathogens to overgrow, their dominance will persist until conditions become highly

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unfavorable for pathogenic persistence.

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Future Questions and Research. Our models have omitted for clarity and simplicity

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some biological processes which might prove important to microbial community dynamics. For

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example: why does antibiotic production decline as sea temperatures increase? Are the beneficial

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bacteria being succeeded by a more temperature-tolerant (and virulent) bacterial type, as

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suggested by [5], who discovered that a Vibrio species becomes more virulent and invades at

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high temperatures; and if so, why? Future models could address the cost of antibiotic production

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by beneficial bacteria (does antibiotic production become costly as temperature rises?),

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incorporate variability in antibiotic conversion efficiency and allow production of defensive

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antibiotics by the pathogens themselves, as suggested by [15], who showed that visitor microbes,

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like the Vibrio inoculated into the SMC at the mucus-seawater interface, also produce antibiotics.

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The present models focus primarily on the loss of defenses within the surface microbial

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community, without considering the defenses of the host coral itself once an infection has

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penetrated through the mucus layer. An earlier paper focused entirely on cellular immune

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responses by soft corals to an established fungal infection [31] without addressing how

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vulnerability to infection is modulated by process in the SMC. Future models need to integrate

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the process of infection with host immune responses, to elucidate how the onset of pathogenic

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overgrowth in the SMC, and the quality of host cellular response to a successful invasion,

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interact to determine the impact of pathogen attack.

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Conclusions

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We have presented models that yield insights into a current crisis in our oceans: the

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decline of coral cover due to increased vulnerability to disease in a warming climate. Our models

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show that the structure of interactions in the surface microbial community facilitates the

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existence of alternate stable states, one dominated by beneficial microbes and the other

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pathogen-dominated. This provides a mechanistic explanation for the empirically observed

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sudden switches to pathogen dominance following heat stress. The models also predict that

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sudden switches will occur even if the temperature increase is gradual, and that the switch to

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pathogen dominance will persist long after thermal stress has ceased, so that a short-term heating

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event may give pathogens an extended opportunity to establish and spread. These predictions are

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robust consequences of an interaction between beneficial and pathogenic microbes mediated by

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beneficials' production of antibiotic substances, rather than depending on any fine details of our

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models.

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An important practical implication of our findings is that a shift to pathogen dominance is

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much more easily prevented than reversed. To prevent potential consequences such as bleaching

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events and elevated colony mortality, it is advantageous to be able to forecast the location and

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timing of situations (e.g. heat anomalies) conducive to pathogen dominance, and to intervene

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before a shift occurs. Reversing the damage once it has occurred will require even greater efforts

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than rapid, timely, and proactive interventions.

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Materials and Methods

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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

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model for a spatially homogeneous ("well mixed") mucus layer focusing on the key community

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members. We list and explain each state variable and equation of the model below.

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Substrate. A nutrient substrate (S) is supplied by the host and consists mostly of organic carbon

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(we use "host" to mean the coral and its endosymbionts together, and the carbon substrate in the

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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

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beneficial microbes (discussed below), and mucus loss.

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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

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make the simplifying assumption that the substrate content of microbes is released back into the

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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

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beneficials is used to produce antibiotic, and the remainder goes towards population growth.

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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

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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

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883

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