Mathematical Biosciences 239 (2012) 106–116
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Stochastic modeling of Pseudomonas syringae growth in the phyllosphere J. Pérez-Velázquez a,⇑, R. Schlicht b, G. Dulla c, B.A. Hense a, C. Kuttler d, S.E. Lindow e a
Institute of Biomathematics and Biometry, Helmholtz Zentrum München, German Research Center for Environmental Health, Ingolstädter Landstr. 1, 85764 Neuherberg, Germany Universität Greifswald, Institut für Mathematik und Informatik, Walther-Rathenau-Straße 47, 17487 Greifswald, Germany c University of Washington, Department of Civil and Environmental Engineering, Benjamin Hall Bldg., 616 NE Northlake Place, Room 476, Box 355014, Seattle, WA 98195-5014, USA d Centre for Mathematical Science, Technical University Munich, M6 Boltzmannstr. 3, 85747 Garching, Germany e Department of Plant and Microbial Biology, University of California, Berkeley, 331A Koshland Hall, Berkeley, CA 94720, USA b
a r t i c l e
i n f o
Article history: Received 15 December 2010 Received in revised form 20 April 2012 Accepted 30 April 2012 Available online 29 May 2012 Keywords: Pseudomonas syringae Bacterial growth Stochastic models of bacterial growth Birth–death-migration Logistic growth
a b s t r a c t Pseudomonas syringae is a gram-negative bacterium which lives on leaf surfaces. Its growth has been described using epifluorescence microscopy and image analysis; it was found to be growing in aggregates of a wide range of sizes. We develop a stochastic model to describe aggregate distribution and determine the mechanisms generating experimental observations. We found that a logistic birth–death model with migration (time-homogeneous Markov process) provides the best description of the observed data. We discuss how to analyze the joint distribution of the numbers of aggregates of different sizes at a given time and explore how to account for new aggregates being created, that is, the joint distribution of the family size statistics conditional on the total number of aggregates. We compute the first two moments. Through simulations we examine how the model’s parameters affect the aggregate size distribution and successfully explain the quantitative experimental data available. Aggregation formation is thought to be the first step towards pathogenic behavior of this bacterium; understanding aggregate size distribution would prove useful to understand the switch from epiphytic to pathogenic behavior. Ó 2012 Elsevier Inc. All rights reserved.
1. Introduction Epifluorescence microscopy and image analysis techniques at the single cell level offer the possibility to determine directly very specific quantitative information of bacterial behavior on leaves. The aggregation distribution of the plant-pathogenic bacterium Pseudomonas syringae pv. syringae strain B728a on leaf surfaces of greenhouse-grown bean plants has been reported using such techniques [7,31]. These bacterial cells were found to be aggregated in a great variety of sizes. In this work we develop a mathematical model to describe and explain the observed aggregation distribution. There are several reasons to study the biological and ecological significance of aggregate formation by these epiphytic bacteria: the aggregated bacteria show increased fitness under environmental stress [31,33,41] and aggregates promote a switch of an epiphytic lifestyle on the plant’s healthy leaves towards an invasive, highly pathogenic [7,40]. Thus, understanding the dynamics of aggregation and growth will facilitate the development of treatment strategies. ⇑ Corresponding author. Tel.: +49 89 3187 2926; fax: +49 89 3187 3029. E-mail addresses:
[email protected] (J. Pérez-Velázquez),
[email protected] (R. Schlicht),
[email protected] (G. Dulla), burkhard.
[email protected] (B.A. Hense),
[email protected] (C. Kuttler),
[email protected] (S.E. Lindow). 0025-5564/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.mbs.2012.04.009
The aim of this work is to use a mathematical model to help understanding the mechanisms that give rise to the growth (an increase in number of viable cells through multiplication) pattern of this bacterium on leaf surfaces following inoculation with P. syringae pv. syringae strain B728a as reported in Dulla and Lindow [7]. Therein, these bacterial cells were found to be distributed in aggregates of a wide range of sizes, from single cells to aggregates with over 104 cells. The majority of aggregates observed were small (less than 100 cells) and aggregate sizes exhibited a strong right-hand-skewed frequency distribution. Large aggregates were not frequent on a given leaf but they contain the majority of cells present. Rather than fitting a distribution to the experimental data, we opted for a dynamical approach which can provide information on the causes of the observed aggregation distribution and the mechanisms that generate it. Successfully describing the mechanism that cause most of the cells to be in large aggregates is of great relevance as it has important biological repercussions. Large aggregates may be protected from bactericides. A colonization pattern where the majority of the cells are located in a few large aggregates can significantly limit microbial control [31]. Our model not only gives a qualitatively correct description of the behavior of P. syringae growth on plant leaves but is also able to propose two mechanisms which can explain the experimental results: migration and heterogeneity of the nutrients availability. The model results agree with the aggregate size distribution
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observed by Dulla and Lindow [7], furthermore, since the model allows to investigate diverse scenarios, it is also able to describe and reproduce similar quantitative data from similar experiments from Dulla and Lindow [31]. This manuscript is organized as follows, in Section 2 we give a biological background, in Section 3 we develop the model, in Section 4 we discuss the model’s simulations results and compare them to the experimental evidence available, finally in Section 5 we present general conclusions. 2. Biological background 2.1. Pseudomonas syringae Although leaf surfaces constitute a hostile environment (fluctuating temperatures, relative humidity, moisture, and UV irradiation), they are often colonized by a variety of microbes. P. syringae is a gram-negative bacterium which is able to grow on plant leaves (the phyllosphere). This bacterium was first isolated as a pathogen of lilac (Syringa vulgaris), it has been studied since the early 1900s due to its pathogenic effect on crop plants [1,10,14,15,24]. P. syringae is also able to produce frost injury due to its ice nucleation properties [10]. On moist leaves, wild-type cells can move on surfaces from a single inoculation point, presumably by flagellum-mediated swimming and swarming [8]. Sugars such as glucose, fructose, and sucrose are the main carbon sources. Fig. 1 This bacterium is often found in tomato, bean, olive, and tobacco among others where they can live symptomless for prolonged periods. Slight changes in the environment which favor the bacteria may cause a rapid outbreak, creating diseases (see Fig. 2) that can destroy entire crops. 2.2. Variability of the leaf surface environment There exist several examples of heterogeneity in microbial behavior on leaves: from population distribution to communication patterns [2]. Several studies have given evidence of the heterogeneity of the leaf surface environment, both among leaves and within a leaf [30,31]. Leveau and Lindow [21] showed a highly heterogeneous availability of fructose to individual cells as they colonize the phyllosphere. McGrath and Andrews [29] give further confirmation of differences in carrying capacity among leaves and show how aggregates form around regions where there may be more nutrients available. Woody et al. [47] reports on how differences in habitat quality are primarily responsible for the variation in (A. pullulans) density among leaves in nature. In [34], the spatial heterogeneity of cell sizes observed on leaves suggests that
Fig. 2. Bacterial lesions caused by Pseudomonas syringae pv. syringae B728a in N. benthamiana, picture provided by Dr. Corina Vlot’s group (Inducible resistance signalling), Institute of Biochemical Plant Pathology, Helmholtz Zentrum München. Pictures display the damage to the leaf in a period of 4 days. Reproduced with permission.
nutrient availability is quite variable on the leaf surface environment. Other studies [46] investigated the cause of variations in fungal population density among leaves. Their experiments showed that natural populations vary greatly in density due to sustained differences in carrying capacities among leaves and suggest that the maintenance of populations close to carrying capacities indicates strong density-dependent processes. Given this evidence, studies on the nutrients status on plants could be potentially used to manipulate microbial populations. 2.3. Relevant previous work Formation of aggregates is a diverse and widespread phenomenon. Several examples have been investigated using mathematical modeling [9,37,44], including bacterial behavior such as colony formation [17] and bacterial communication [6]. Here we briefly review some works of relevance to our problem. Bailey [3] was among the first to use stochastic models to study collections of populations which are distributed spatially, and which are subject to birth, death, and migration to or from
Fig. 1. Scanning electron micrographs of Pseudomonas syringae on bean leaves (left picture) and between epidermal cells (right picture). The opening in the lower right-hand corner of the picture on the left is a plant stomate. P. syringae often uses these openings to invade the plant. Pictures contributed by Prof. Gwyn Beattie, Department of Plant Pathology, Iowa State University. Reproduced with permission.
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neighboring populations. He found explicit expressions for the stochastic means and calculated variances and covariances for some special cases. Renshaw [42] further studied the effect of migration between a finite number of colonies each of which undergoes a simple birth and death process. He computed the first two moments for the general process and found deterministic solutions for several special models. More recently and in connection with ecology, a stochastic birth–death-migration for describing the spread of muskrats in the Netherlands was developed in [27]. The mean value functions of the model were derived. The individual migration and the net birth-rate parameters are estimable from spatio-temporal data. Birth–death-immigration models have also been studied [26]. Lves et al. [18] used a combination of laboratory experiments, simulation models, and analytical techniques to examine the impact of dispersal on the mean densities of patchily distributed populations of A. pullulans on apple leaves. Their model suggests that the effect of dispersal on mean densities is enhanced by density dependence of the dynamics within populations, high environmental variability which affects population growth rates, and lack of synchrony among the fluctuations of populations. Ponciano et al. [39] considered how microbial growth dynamics is influenced by environmental conditions, they used extensive experimental data (growth curves of Escherichia coli). They proposed a stochastic ecological model that effectively explains the uncertainty seen in the recorded growth curves. In [36] the onset of swarming of P. aeruginosa was described using a simplified computational model in which cells in random motion communicate via a diffusible signal as well as diffusible, secreted factors that regulate the intensity of movement and metabolism in a threshold-dependent manner. The model predicts sustained colony growth that can be collapsed by the overconsumption of nutrients. 3. Model development Given the heterogeneous behavior seen in bacterial populations we will use a stochastic model approach. The theory of stochastic processes has proven to be a powerful tool for analyzing collective dynamics, such as clonal growth of cultured cells [16], describing populations which are distributed spatially [3,18,27,28], microbial growth [5,38,39], among others. We want to describe how bacterial aggregates form and how their distribution changes in time. Basically we want to, through developing a mathematical model, elucidate the mechanisms that give rise to the aggregate distribution of this bacterium as previously reported by Dulla and Lindow [7] and Monier and Lindow [31]. Rather than fitting a distribution to the experimental data, we have opted for a dynamic approach that may allow us to elucidate the mechanisms that generate the observed behavior by Dulla and Lindow [7], see Fig. 3. In order to obtain these data, a suspension ( 1 ml/plant) of P. syringae pv. syringae strain B728a was sprayed onto leaves of 14-day-old bean plants (Phaseolus vulgaris cv. Bush Blue Lake 274). The frequency and size of bacterial aggregates on leaf surfaces were determined by image analysis of micrographs obtained by epifluorescence microscopy. Note how the number of small aggregates observed decreases with time. Most of the aggregates observed were small. Because of their large size, the majority of the total population (70%) occurred in aggregates larger than 100 cells.
120 100
Frequency
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80 60 40 20 0 1−10 11−20 21−30 31−40 41−50 51−60 61−70 71−80 81−90 91−100 100−500 500−1000 1000+
Aggregates size
4
3
2
1
Days after innoculation Fig. 3. Frequency distribution of bacterial aggregates on leaf surfaces following inoculation with P. syringae pv. syringae strain B728a. Experimental data from Dulla and Lindow [7].
in Section 2, there is a high degree of stochasticity involved due to environmental heterogeneity. It is a stochastic model since, given a population of cells which grow by division, we cannot say that a cell will divide in a specific time interval, but only that there is a certain probability that it will do so. This is the model’s stochasticity source. Bacterial population dynamics in time and space will depend on events such as immigration, migration, growth, and death. Given this setting, population dynamics will vary across colonies (not across bacteria). From the experimental evidence, we know that bacteria are dispersed across the leaf forming aggregates. We then consider a continuous-time Markov process consisting of a component NðtÞ, the number of colonies that were formed until time t, and components X 1 ðtÞ; X 2 ðtÞ; . . ., the number of cells in the individual colonies, i.e.
AðtÞ ¼ ðNðtÞ; X 1 ðtÞ; X 2 ðtÞ; . . .Þ; AðtÞ;
tP0
is a continuous-time Markov process with initial state Nð0Þ ¼ 1; X i ð0Þ ¼ 0; i ¼ 1; 2; . . .. We choose in the first instance a simple birth-and-death process; due to its mathematical transparency, flexibility and relevance to fundamental biological processes is a standard mathematical tool in mathematical biology. We have chosen to include death explicitly as reported in [33]. Explicit results are available when colony sizes fluctuate according to a linear birth-and-death process: a Poisson distribution (for a pure birth process, Bailey 1964) or a negative binomial (Cox and Miller, 1965) for the birth–death case. However, neither of these distributions are in qualitative agreement with our reported distributions; considering a simple birth and death process is therefore not a valid framework for our case. Each colony will then be modeled by a logistic birth–death process. We elaborate on this in the following section.
3.1. The mathematical model 3.2. Variability of the leaf surface environment Our model is a stochastic model. We want to stress that we do not use a deterministic approach since we know that all aggregates grow differently and independently of each other and, as discussed
We assume that the colonies follow logistic growth (i. e. not a pure birth). Several instances of microbial populations on leaves
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seem to reach an equilibrium value [11,13,18,46,47], and therefore assuming logistic growth is appropriate. Ives [46] uses logistic growth, and reported that A. pullulans populations seem to be near the leaf carrying capacity most of the time. A logistic growth model can be described in several ways. We use the definition from [4]. Let XðtÞ; t P t 0 be a Markov stochastic process with state space S ¼ 0; 1; . . . ; K, where K > 0 is a large positive integer representing the maximum population size (carrying capacity). Definition 1. Let ki ¼ ik 1 Ki and di ¼ id, where k > 0; d P 0. The Markov process XðtÞ given above is called a logistic birth–death process with birth rate ki and death rate di if and only if the following conditions are satisfied, for j 2 f0; . . . ; Kg and k 2 N n fj 1; j; j þ 1g. (a) PrfXðt þ DtÞ ¼ j þ 1jXðtÞ ¼ jg ¼ kj Dt (b) PrfXðt þ DtÞ ¼ j 1jXðtÞ ¼ jg ¼ dj Dt (c) PrfXðt þ DtÞ ¼ kjXðtÞ ¼ jg ¼ oðDtÞas Dt ! 0.
3.2.1. Colony-individual carrying capacity In order to realistically reproduce the variability of the leaf surface environment, we will further assume that each of the colonies has its own individual carrying capacity. This aims to reproduce the heterogeneous environment found on leaves: each bacterial cell landed in a site with has a specific nutrients availability (carrying capacity). Since we have (initially) a number n of bacterial cells, we will need to choose n carrying capacities, to do this appropriately and realistically we take these carrying capacities to be lognormally distributed following several pieces of evidence [12,21,30,47]. Mercier and Lindow [30] reported that the amount of sugars on a population of apparently identical individual bean leaves before and after microbial colonization exhibited a righthand-skewed distribution and total bacterial population sizes also exhibited a right-hand-skewed distribution. In [12] epiphytic bacterial populations were found to approximate a log normal distribution. [47] reported that A. pullulans population densities among leaves were log-normally distributed. See Sections 2.2, 3.2 and reference [21] for further discussion on nutrient variability of the leaf surface environment. See [22] for a review of the use of lognormal distributions.
0.08
P.d.f. of a log−normal distribution
0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
0
200
400
600
800 1000 Colony Size
1200
1400
1600
Fig. 4. Probability density function of the (log-normal) distribution of the carrying capacities among aggregates, with parameters l ¼ 2:71; r ¼ 1:78, figure generated using the Matlab Statistics Toolbox (MathWorks).
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See in Fig. 4 an example of the probability density function (pdf) of the carrying capacities distribution (log-normal) among aggregates. This type of distribution was chosen from the relevant biological data on the distribution of nutrients (sugars) on the leaf environment [21,30]. To estimate the parameters l ¼ 2:71; r ¼ 1:78 we used the data from [7], used a PDF Estimator (V3.2) for Matlab by Yi Cao and ‘‘A Collection of Fitting Functions’’ also for Matlab by Ohad Gal, which uses a maximum likelihood fit of the log-normal distribution of independent and identically distributed samples (the experimental data).
3.3. Migration The described framework thus far does not account for new (and therefore small) aggregates seen experimentally even after 4 days post-inoculation. There must be another process involved that is responsible for the reported large numbers of small aggregates. From the experimental evidence, we know that bacteria are dispersed across the leaf at particular sites, sites mainly associated with high nutrients concentration. In addition to the birth and death kinetics of the previously discussed section, P. syringae populations are known to be able to move across the leaf. We therefore include migration into our model. We will assume that a bacterium is able to leave its landing site and start a new colony, this would be the source of the new aggregates seen. The model will therefore be a logistic-birth and death process with migration. Note that this model is not a ‘‘transfer of state’’ model in the sense that when cells move they do so to form a new colony, not to join an existing one [32]. There is no exchange between colonies and therefore this is not a compartmental model or a stepping stone model. We also note that with migration we mean dispersal within the leaf and no new bacteria are introduced after initial inoculation. Note the following further assumptions:
Cells land, attach and then move. Cells can at any point leave their aggregate. Cells leave one by one, not in groups. There are two sources of new colonies: inoculation and migration.
Note that by migration, we mean the actual movement (dispersal) of the bacteria on the leaf. From the experimental setting it is clear that we consider leaves where immigration (arrival of viable propagules on the leaf) and emigration (physical loss or removal of viable propagules from the leaf) are neglectable from the seen dynamics. Migration in the sense of movement from or to the leaf has been studied; see [18,23]. In summary, we develop a mathematical model to account for the microbial population dynamics on leaves as follows: a stochastic logistic birth–death process, i. e. a sequence of (initially) n random variables each with different (randomly chosen, log-normally distributed) carrying capacities to describe n individual colonies, which are also subject to migration events. Note that since we want to avoid two events happening at the same time, we will deal with a continuous model. We then consider a continuous-time Markov process consisting of a component NðtÞ, the number of colonies that were formed until time t, and components X 1 ðtÞ; X 2 ðtÞ; . . ., the number of cells in the individual colonies, i.e. **AðtÞ ¼ ðNðtÞ; X 1 ðtÞ; X 2 ðtÞ; . . .Þ; AðtÞ; t P 0 is a continuous-time Markov process with initial state Nð0Þ ¼ 1; X i ð0Þ ¼ 0; i ¼ 1; 2; . . .. The model’s parameters are described in Table 1, time scale is hours. Note that migration does not depend on the number of cells in the colony, migration events happen randomly at a constant rate.
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J. Pérez-Velázquez et al. / Mathematical Biosciences 239 (2012) 106–116 1 X qx ðtÞ ¼ E 1fi6NðtÞ;X i ðtÞ¼xg
Table 1 Model’s parameters. Time scale is hours. Parameter
Description
a ¼ 0:4 d ¼ 0:1 I
Growth rate [21] Death rate Migration rate Lognormal parameter Lognormal parameter
l r
1 1 X X Eð1fi6NðtÞ;Xi ðtÞ¼xg Þ ¼ Prði i¼1
i¼1
1 X 6 NðtÞ; X i ðtÞ ¼ xÞ ¼ PrðNðtÞ P iÞPrðX i ðtÞ ¼ xjNðtÞ P iÞ: i¼1
3.5. Moments Solutions of the differential equations in (1b) can be obtained recursively for n ¼ 1; 2; . . . (see, for example, [19, p. 122]:
Xi Xi ðtÞki;x1 þ ðdðx þ 1Þ þ IÞpxþ1 ðtÞ ðkx þ dx þ IÞpXx i ðtÞ ð1aÞ p_ Xx i ðtÞ ¼ px1
where ki;x
¼
i¼1
3.3.1. Equations The transition probabilities satisfy:
p_ Nm ðtÞ ¼ pNm1 ðtÞðm 1ÞI pNm ðtÞmI
!
ð1bÞ
¼ xk 1 Kx . Note that each colony has its own carrying i
capacity but all colonies share the same growth/death rate. System (1) holds for x ¼ 1; 2; . . . and for m ¼ 1; 2; . . ., respectively, because the cases x ¼ 0 and m ¼ 0 are handled in (2). Note that Eq. (1b) is a linear birth process, because the rate (intensity) is not deterministic but grows as the number of colonies increases; Eq. (1a) are birth logistic-death-migration processes. Note also that
n1 pNn ðtÞ ¼ eIt 1 eIt : The exponential decay in n implies that for any polynomial f ðnÞ the absolute value of dtd pNn ðtÞf ðnÞ is bounded, in view of (1b), by a funcP1 tion gðnÞ with n¼1 gðnÞ < 1, locally in t. We can therefore exchange summation and differentiation as follows: 1 1 X d dX d N Eðf ðNðtÞÞ ¼ p ðtÞf ðnÞ ¼ EðAf ðNðtÞÞÞ; pNn ðtÞf ðnÞ ¼ dt dt n¼1 dt n n¼1
X X p_ 0 i ðtÞ ¼ ðd þ IÞp1 i ðtÞ
ð2aÞ
where A is the infinitesimal generator of the Markov process, Af ðnÞ ¼ nIf ðnÞ þ nIf ðn þ 1Þ. For the particular functions f ðnÞ ¼ n and f ðnÞ ¼ n2 , we obtain the differential equations dtd EðNðtÞÞ ¼ IEðNðtÞÞ and dtd EðNðtÞ2 Þ ¼ 2IEðNðtÞ2 Þ þ IEðNðtÞÞ with solutions
NðtÞ p_ 0 ðtÞ ¼ IpN0 ðtÞ:
ð2bÞ
EðNðtÞÞ ¼ eIt
3.4. Construction of the process To analyze our model it is essential to assume that migration does not affect the size of the colony i from which migration occurs. Otherwise migration events affect the size X i of colony i and the total number of colonies N simultaneously, which makes independence assumptions impossible. The model is constructed as follows. Let N; Y 1 ; Y 2 ; . . .be independent stochastic processes with the following properties: (1) N is a linear birth process with Nð0Þ ¼ 1 and
d N p ðtÞ ¼ ðn 1ÞIpNn1 ðtÞ nIpNn ðtÞ dt n for pNn ðtÞ :¼ PrðNðtÞ ¼ nÞ. (2) Y i is a birth–death process with Y i ð0Þ ¼ 1 and
d Yi Yi Yi p ðtÞ ¼ ki;x1 px1 ðtÞ þ di;xþ1 pxþ1 ðtÞ ðki;x þ di;x ÞpYx i ðtÞ dt x Y
for px i ðtÞ :¼ PrðY i ðtÞ ¼ xÞ.
N describes the evolution of the number of colonies, Y i describes the (time-shifted) evolution of the size of colony i after its formation. Let
T i ¼ minft P 0 : NðtÞ P ig
given Nð0Þ ¼ 1. Hence the expected value of NðtÞ grows exponentially, while the variance of NðtÞ is e2It eIt .
nþk1 The polynomials f ðnÞ ¼ of degree k ¼ 1; 2; . . .are k eigenvectors of A, namely Af ¼ kIf since Af ðnÞ ¼ nIf ðnÞþ f ðnÞ ¼ kIf ðnÞ for all n. Thus, Eðf ðNðtÞÞÞ ¼ ekIt , this presents nI nk n one way of computing higher moments. The processes Y i ðtÞ and the full process ðNðtÞ; X 1 ðtÞ; X 2 ðtÞ; . . .Þ depend on more parameters, hence corresponding results can only be derived in special situations (for example, if ki;x and di;x both depend linearly on x). 3.5.1. Colony-size dependent migration If, on the other hand, we consider colony-size dependent migration, in this case it is necessary to individually track the numbers N i of bacteria that migrate from each colony i. Let ðY i ; N i Þ; i ¼ 1; 2; . . .be independent processes (where Y i and N i are dependent for each i) such that
d Y i ;Ni Y i ;N i Y i ;N i Y i ;Ni p ðtÞ ¼ ki;x1 px1;n ðtÞ þ di;xþ1 pxþ1;n ðtÞ þ ðx þ 1ÞIpxþ1;n1 dt x;n ðki;x þ di;x þ xIÞpYx i ðtÞ Y ;N
for px;ni i ðtÞ :¼ PrðY i ðtÞ ¼ x; N i ðtÞ ¼ nÞ. Then the total number of colonies N and the colony formation times T i can be computed recursively:
the time of formation of colony i. Then the process
X i ðtÞ ¼
0 if t < T i Y i ðt T i Þ if t P T i
(t P 0) describes the evolution of the size of colony i starting at time 0. Let qx ðtÞ ¼ Eðjf1 6 i 6 NðtÞ : X i ðtÞ ¼ xgjÞ be the expected number of colonies of size x at time t. The proportions of the different qx ðtÞ; x ¼ 0; 1; 2; . . .for large t might provide some insight into the asymptotic colony size distribution. We have
and EðNðtÞ2 Þ ¼ 2e2It eIt
NðtÞ ¼ 1 þ
1 X Ni ðt T i Þ and T i ¼ minft P 0 : NðtÞ P ig: i¼1
The processes X i ðtÞ are again given by
( X i ðtÞ ¼
0
if t < T i :
Y i ðt T i Þ if t P T i :
We now proceed to discuss our results.
J. Pérez-Velázquez et al. / Mathematical Biosciences 239 (2012) 106–116
We point out that in both, the simulation and experimental data:
4. Simulations results In this section we describe how the simulations (Monte Carlo simulations) behave, we first show their agreement with the qualitative behavior we want to portray by making a direct comparison to the experimental data available, we then explore different parameter regimes and discuss its biological significance. We carried out Monte Carlo simulations (i.e. we generated realizations of the stochastic process). Simulations were produced using Matlab (MathWorks), carrying capacities distributions were produced using the random number generator randraw (by Alex Bar-Guy) for the lognormal distribution. This distribution was chosen following experimental evidence of distribution of nutrients on leaves surfaces [12,21,30,47] as discussed in Section 3.2. The time scale of all simulations is in hours. The value for the growth rate k has been taken from [21] where they reported 1 an initial average growth rate of 0:4 h . To create a simulation, we run n (independent) realizations corresponding to a n number of cells (which would have arrived to the leaf by spraying), these are the initial colonies. The initial number of cells chosen was dictated by the observations from the experimental data, with 203 the maximum number of colonies ever observed at a given day. In the figures shown we plot the frequency distribution of the total number of colonies (i.e. including the new aggregates formed as a result of migration) corresponding to 4 days. Note that every time we chose I > 0 the total number of colonies and the initial number of simulations differ, as new colonies are being created at all times.
4.1. Comparison to experimental evidence In this section we directly compare our model results to the experimental data available. In particular, we show that the model is able to produce results with the same qualitative features of the reported data by Dulla and Lindow [7]. See in Fig. 3 the experimental data and in Fig. 5 our simulation.
Number of aggregates observed
400
300
200
100
0
1−10 11−20 21−30 31−40 41−50 51−60 61−70 71−80 81−90 91−100 100+
Number of cells per aggregate
111
4
3
2
1
Days after innoculation
Fig. 5. Simulation results: frequency distribution of bacterial aggregates on leaf surfaces following inoculation with P. syringae pv. syringae strain B728a as reported by [7], where I ¼ 0:1. Other parameter values: k ¼ 0:4; d ¼ 0:1. Note that the value for k has been taken from [21] where they reported an initial average growth rate of 1 0:4h .
Aggregate sizes exhibited a strong right-hand-skewed frequency distribution. Large aggregates are not frequent but they account for the majority of cells present, more than 50% of cells are in aggregates containing 100 cells or more (See Fig. 6). The large majority of aggregates are small. Note that stochastic simulations will always generate quantitatively different results. We stress that we are interested in finding the causes of the reported behavior; our model has allowed elucidating two possible mechanisms: migration and heterogeneity of the nutrient distributions. It is also necessary to point out that even though the model is able to reproduce the tendency of large and small aggregate sizes, simulation from Fig. 5 does not reproduce the exact same proportions for aggregates of other sizes (for example 11–20 or the exact time tendency (strictly decreasing) of the percentage of aggregates of size less than 100. Among causes of this, we could mention that the actual growth rate of this bacterium is perhaps lower than what the simulation used. A sensitivity analysis of this parameter (growth rate) was performed and showed that it is indeed possible to achieve such tendency (strictly decreasing) for r ¼ 0:4 providing we start with a larger number of initial colonies (i. e. increase the number of simulations). However, as mentioned before, since we have stochastic simulations, stochasticity is partially responsible for the observed differences. Generally, however, we could say that simulation results of the model agree with the reported distribution, we can conclude that a logistic birth–death model with migration provides a good description of the observed data. We can therefore suggest that it is the heterogeneous distribution of nutrients that produces the skewed distribution of aggregates, however this alone is insufficient to account for large numbers of small aggregates, these are caused by migration (as it makes more sites available). Migration contributed to the skewness of the observed distribution. At this stage we would also like to compare our results to the reported distribution by Monier and Lindow [31]. A similar experimental protocol and setting were used there to report frequency, size and spatial distribution of P. syringae. There, however, the number of small aggregates increases with time. They reported that the majority of aggregates are small (less than 100 cells), and aggregates sizes exhibited a strong right-hand-skewed frequency distribution. Large aggregates account for the majority of cells present. See our own results in Fig. 7. We can in this case too, successfully reproduce the reported results. One can show that (data not presented) aggregate sizes exhibited a strong right-handskewed frequency distribution; large aggregates (though infrequent) account for the majority of cells present; and the large majority of aggregates are small (less than 100 cells) as reported by Dulla and Lindow [7]. In order to obtain a simulation close to Dulla and Lindow’s experimental data we have taken the parameters of the lognormal distribution to be l ¼ 2:5; r ¼ 0:5 and for Monier and Lindow l ¼ 4:2; r ¼ 0:5. Mean for this case is 13.8 and Fig. 4 (mean is 75.5). To explain these results and relate them to the biological framework, it will be useful at this stage to discuss the experimental protocol from Dulla and Lindow [7] and Monier and Lindow [31]. As mentioned before, in Dulla and Lindow’s experiment [7] plants were inoculated by spraying and kept in a humid chamber. The plants did not become overly wet, only humid enough to maintain the moisture sprayed onto the leaf. In contrast Monier and Lindow [31] applied bacteria by submersion of the leafs in a bacterial
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Dulla and Lindow’s experiments [7] are rather fixed to the attachment site and can then grow and reach a large aggregate size. In Monier and Lindow’s experiments [31], there is perhaps more movement of cells and thus, they are able to form aggregates of wider range of sizes. Due to the heterogeneity involved, it is difficult to answer the question of what plays a bigger role, we could only hypothesize that both processes are needed to produce the observed dynamics. These simulations successfully reproduce some features of the qualitative behavior reported, the model however has four parameters and this allows us to explore the model’s behavior at different conditions. We do this in the following sections and discuss their biological relevance.
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Fig. 7. Simulation results: Frequency distribution of bacterial aggregates on leaf surfaces following inoculation with P. syringae pv. syringae strain B728a as reported by [31], here I ¼ 1. Note that the value for k has been taken from [21] where they 1 reported an initial average growth rate of 0:4h .
culture and then they were incubated in plastic bags. Note that Monier and Lindow [31], observed that the number of aggregates (particularly small ones) increase with time, whereas in Dulla and Lindow’s experiments [7] small aggregates decrease with time. We hypothesize that cells in the Monier and Lindow’s experiment may have had higher migration possibilities and therefore create new aggregates. In Dulla and Lindow’s experiment plants were left with lower chances to move about the leaf. Presumably submersion of the leaves may have also affected nutrients distribution. Increased moisture would also allow for more leaching of nutrients (increasing the amount of cell growth) and movement of the nutrients across the leaf surface (increasing the small aggregates frequency). Overall, Monier and Lindow’s conditions were very conducive to growth whereas in Dulla and Lindow’s are somewhat stressed. It is clear that nutrients distribution may have been dissimilar in each case. The experimental protocol affected both migration possibilities and nutrients distribution. Note that, in terms of the model, to produce the corresponding simulations both, migration rate and nutrients distribution, are different for each case. Another difference between the results of these two experiments is that in Monier and Lindow’s case, large aggregates are not as frequent as in Dulla and Lindow’s. This may again be due to migration possibilities. Cells in experimental conditions from
In this section we explore how the model’s behavior changes at different conditions of the leaf surface environment (varying nutrients distribution). In Fig. 8 we present two simulations to show the effect on varying the lognormal distribution parameters, leaving the rest of the parameters fixed. In the first simulation, the mean carrying capacity is not very large (13.8) and the aggregate distribution agglomerates around this value, very large aggregates (larger than 500) are not seen. A nutrients distribution with a mean carrying capacity which is low, represents a leaf with most patches poor in nutrients. Changing the lognormal parameters so that we increase the mean of the lognormal to 56.3 produces an agglomeration of aggregate sizes around 300; this distribution is less skewed and does not result in large frequency of small (non solitary cells) aggregates. Note that in both cases (results not shown) small aggregates frequency decreased through time. Monier and Lindow [31] reported a great deal of variation in the degree of aggregation between leaves, leaf segments and within individual leaf segments. Our results have shown that environmental conditions may indeed play an important role. 4.3. Migration In this section we explore the effect of changing the level of migration (see in Fig. 9). The migration rates chosen for the study are 0:0; 0:1; 0:5 and 1, to give some perspective on these numbers, I ¼ 1 may be regarded as strong migration. For I ¼ 0 or low migration, the number of small aggregates decreases with time (results not shown), increasing the migration rate causes the number of small colonies to increase with time. This is an interesting result as the role of migration in a logistic-growth has not been extensively studied. We know that for the immigration case (this would be equivalent to more cells arriving to the leaf), as the level of
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immigration increases, the mean value rises but the variance and the skewness of both the transient and the equilibrium distributions are reduced [25]. Note that low migration allows the formation of aggregates of larger sizes and large migration produces a wider distribution of aggregate sizes. 4.4. Starvation driven migration Under the current framework, cells migrate with the same probability regardless of the condition surrounding them. We now assume that cells only migrate if the colony size approaches the carrying capacity, i.e. nutrient limitation occurs. This could even be compared to an invasion model proposed by Dulla et al. [8]: at arrival, due to the heterogeneity of the leaf surface, the likelihood of P. syringae cells landing near abundant nutrients is low. Thus, cells move about on leaves to nutrient-rich sites. A cell that finds such a site would proliferate and form an aggregate. Although they may have intended to describe a chemotactic process, our model allows us to explore the starvation driven migration
scenario. Beyond the biological model, our results suggest that migration occurs not only at landing but afterwards, when they can leave an already established colony. We implemented this numerically into the model (results not shown). We set up the minimum carrying capacity that allows migration to be 50, that is, cells are able to migrate only until the colony has at least 50 cells. We found that in this case, the number of new colonies being formed neither increase nor decrease with time. Most of the aggregates have a size close to the chosen fixed carrying capacity. Increasing the value of this fixed carrying capacity has the effect of allowing colonies of bigger sizes being created but the number of small colonies, though high, still does not change greatly with time. This scenario allows us to directly compare migration versus heterogeneity of the leaf environment. The simulations results suggest that when comparing these two scenarios, migration is responsible for the skewness and may therefore play an important role. Note that here we are assuming that cells are able to estimate, from the first moment of their first attachment, the carrying
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Fig. 9. A set of four simulations showing the effect of changing the migration rate: in (a) I ¼ 0; (b) I ¼ 0:1; (c) I ¼ 0:5; and, (d) I ¼ 1. All other parameters are fixed to a ¼ 0:4; d ¼ 0:1; l ¼ 3:5; r ¼ 0:6.
capacity. A more realistic description would be to assume that they measure to some degree how close the colony size is to the carrying capacity (e.g. by sensing the amount of available nutrient or the actual growth rate, which is limited by the capacity). Finally, we looked at the dynamics for longer times. Experimentally, carrying out analysis for periods of time longer than 4 days is difficult as the leaves often die after 4 days. This is one case where mathematical modeling allows exploring the dynamics where experimental procedures may not. In this case, it is relevant to show whether new colonies are still being created after 4 days and if even larger colonies can be observed as they potentially could invade the plant. It is known that the presence of large clusters of bacteria on leaves can increase the potential for metabolic and genetic exchange [31]. Results are not shown but we found that the proportion of the total number of cells located in larger aggregates increased with incubation time, in agreement with Monier and Lindow [31].
5. Conclusions The description of the spatial distribution of the bacteria at the single cell level provides biological and ecological information about the processes leading to successful leaf colonization. Therefore, the identification and description of growth patterns and how these patterns change (in time or space) is a prerequisite for the development of efficient biological control agents. Our aim in this work was to identify the specific population processes that generate the dynamics reported by Dulla and Lindow [7]. Our model successfully gives information about the dynamics of colony sizes and proposes two mechanism possibly involved in generating the behavior previously reported through experimental data: nutrients distribution and migration. We found that not only local (nutrients) or solely global (migration) effects affect population dynamics, instead both local and global processes interact. It is the heterogeneous distribution of
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nutrients that produces the skewed distribution observed in aggregates, however this alone is insufficient to account for large numbers of small aggregates, these are caused by migration (as it makes more sites available). The model is able to provide quantitative and qualitative information on the frequency and size of aggregates; it successfully reproduces the observation that bacteria in large aggregates account for the majority of the cells present on a leaf. Successfully describing the mechanism that originated most of the cells to be in large aggregates is important as it has great biological relevance. Large aggregates may be protected from bactericides applied to plants. A colonization pattern with a majority of the cells located in a few large aggregates can significantly limit microbial interactions to a few sites and, at the same time, it provides them with refuges, this may be linked to the lack of efficiency of biological control agents applied to leaves [31]. Aggregate formation among epiphytes has significant ecological implications that must be considered when designing strategies to control plant-pathogenic bacteria [31]. Depending on the parameter values, it is possible to show with our model that between 30 and 70% of the total bacterial population is located in aggregates containing 100 cells or more. Our model is able to describe that bacterial populations are highly variable and supports the assumptions that the variability observed seems to be driven by the presence or absence of microsites conducive to bacterial growth, reflecting the spatial heterogeneity of nutrients available on leaf surfaces. This agrees with Monier and Lindow [31] where large variation between leaves were reported. We have examined different migration regimes and their effect on the underlying distribution. In Monier and Lindow’s experiments [31], the number of observed aggregates (particularly small ones) increase with time, whereas in Dulla and Lindow’s experiments [7] small aggregates decrease with time. Throughout our results we are able to hypothesize that cells in the Monier and Lindow’s experiment have a higher dispersal rate and can therefore create new aggregates. Increased moisture would also allow for more leaching of nutrients (increasing the amount of cell growth) and movement of the nutrients across the leaf surface (increasing the small aggregate frequency). Our results suggest that considering a model with migration is essential to explain the observed skewness. A regime with extreme migrations can produce normal-like distributions. The model however does not precisely reproduce the same relation between aggregate sizes observed experimentally, this may be due to growth rate influencing aggregate size. We have assumed the growth rate is the same for all colonies at any time, there is the possibility of growth rate changing when a colony size is reached. Extension of the model can include considering time dependent growth rate of even density dependent growth rate. Indeed, when exploring the scenario of all colonies having the same carrying capacity, this is equivalent to assume that the growth rate is time dependent. Ultimately the model can be used to generate predictions accurate enough to be practical, for example as strategies to control pathogens [31]. Here we give some possibilities: in terms of migration, by influencing water film on the leafs we may affect migration possibilities; with regards to nutrients availability, competing (less aggressive) organisms could be introduced so that resources are limited and growth is halted. In [20] by studying motile and nonmotile strains of P. fluorescens it was shown that bacterial motility conveyed a selective advantage during surface colonization. It is relevant to note that we have assumed in our model that cells can move at any point, however this needs to be revised as a cell in the middle of a large aggregate may find it difficult to migrate out of the mass. Cells on the outer surface of the aggregate
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may have better possibilities of moving. Thus the average migration rate probably declines with growing colony size. Inversely, it cannot be ruled out that nutrient limitation in large colonies may promote migration. Large aggregates may also be encased in exopolysaccharide matrix that would hamper movement away from the aggregate. Continuous-time Markov processes are useful models in population biology. From the mathematical point of view, we have presented a way to construct the process and compute the first moments but further formal analysis needs to be carried out. The distribution of N will ultimately be influenced by the distribution of the carrying capacities and the migration rate. For a fixed set of colonies with the same carrying capacities and same migration rate (low), the whole process (sizes of colonies at time t) will have a frequency distribution which resembles the individual distribution. The process will reach an asymptotic population size governed by the dynamics of the logistic. For a fixed migration rate, the aggregate size distribution will be governed by the distribution of the carrying capacities. Presumably, migration will have an effect on the skewness. Analytically this represents two questions of interest: the effect of varying rates of migration on the exact distributions and cumulants and the impact of large migration on the accuracy of the cumulant approximations. 5.1. Future work The model has analyzed one P. syringae strain, experimental evidence from Monier and Lindow [35] exists to describe co-existence and exclusion with other strain and other leaf microbes. Moreover, studying competition between bacterial strains and how this affects the patterns observed may be important. Wilson and Lindow [45] reported that the level of coexistence between epiphytes was proportional to the ability to utilize carbon sources not utilized by a competing strain. An important part of this study is its eventual relation with pathogenicity onset. There are suggestions [43] relating pathogen population size on individual leaves to disease incidence. Extensions of our model include considering quorum sensing (QS), a gene regulation system that controls traits involved in epiphytic fitness and virulence of P. syringae. QS regulated genes themselves can influence development of aggregates, e.g. via control of migration behavior [40]. As QS is critical for epiphytic fitness and emergence of pathogenecity of P. syringae, it is a promising target for treatment strategies as e.g. biocontrol [8,32]. The ultimate aim will be prediction of pathogenecity and recommendations for adequate treatment strategies. A number of other epiphytic bacterial species use QS regulation to control gene expression [8]. Thus the general outcomes of our analysis may be of broad relevance. Acknowledgements We would like to thank several people who have been involved in preparing this manuscript. We are very much grateful to Prof. Johannes Müller (Centre for Mathematical Science, Technical University Munich) for all his help and all insightful discussions. We want to thank the kind help of Prof. Gwyn A. Beattie (Iowa State University, Department of Plant Pathology) and Dr. Corina Vlot (Institute of Biochemical Plant Pathology, Helmholtz Zentrum München) for providing pictures. Finally we also want to thank Dr. J. Monier for helpful comments and discussions. J. P. V. wants to acknowledge the project ‘‘Systems Ecology of Oligotrophic Biofilms’’ from the Helmholtz Zentrum München for providing funding to carry out this research.
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