- Email: [email protected]
Modelling the interaction between bacteriophages and their bacterial hosts
Mathematical Biosciences 279 (2016) 27–32
Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs
Modelling the interaction between bacteriophages and their bacterial hosts Gabor Beke, Matej Stano, Lubos Klucar∗ Institute of Molecular Biology SAS, Laboratory of Bioinformatics, Dubravska cesta 21, 84551 Bratislava, Slovakia
a r t i c l e
i n f o
Article history: Received 19 October 2015 Revised 17 June 2016 Accepted 21 June 2016 Available online 5 July 2016 Keywords: Bacteriophage Bacteria Phage therapy Model
a b s t r a c t A mathematical model simulating the interaction between bacteriophages and their bacterial hosts has been developed. It is based on other known models describing this type of interaction, enhanced with an ability to model the system influenced by other environmental factor such as pH and temperature. This could be used for numerous estimations of growth rate, when the pH and/or the temperature of the environment are not constant. The change of pH or the temperature greatly affects the specific growth rate which has an effect on the final results of the simulation. Since the model aims on practical application and easy accessibility, an interactive website has been developed where users can run simulations with their own parameters and easily calculate and visualise the result of simulation. The web simulation is accessible at the URL http://www.phisite.org/model. © 2016 Elsevier Inc. All rights reserved.
1. Introduction The raising problem of bacterial resistance against antibiotics pointed out the need of novel alternative solutions to eliminate pathogenic bacteria. As a result, the last years of phage research was predominantly devoted to the search of new phages and their potential application as therapeutic agents, generally known as ‘phage therapy’. Bacteriophages, also called phages, are viruses infecting prokaryotic organisms. Phages are ubiquitous and are the most abundant organisms on the planet [1,2] and represent important components of ecological systems. They can modify microbial culture by lysis, transmission of genetic material and by lysogenic conversion. Phages are very important for basic research, but their practical application is still low. Phages can be used in identification of bacterial pathogens [3] or in detection of bacterial contaminations [4]. They are natural enemies of bacteria and are mostly harmless to the human organism. This makes them a promising agent for elimination of undesired bacteria in food industry, medicine or in agro-biotechnology. Phages can be applied individually as solo species; however phages represents one of the major selecting forces in the evolution of bacteria and can develop resistance against phages in couple of days or hours. To slow down this rapid adaptation process, phages can be applied in combinations as ‘phage cocktails’, they can be cyclised (changed over
∗
Corresponding author. Tel.: +421 259307413. E-mail addresses: [email protected] (G. Beke), [email protected] (M. Stano), [email protected], [email protected] (L. Klucar). http://dx.doi.org/10.1016/j.mbs.2016.06.009 0025-5564/© 2016 Elsevier Inc. All rights reserved.
the time) or they can be applied in combination with antimicrobial drugs, like antibiotics [5]. Another possibility is the application of genetic modification of phages or the development of synthetic bacteriophages. There are still several problems in practical application of phages as therapeutic agents, e.g. difficulties during the transport of phages to the place of infection or the fact that phages can cause release of endotoxins and pyrogens, which are the side products of the bacterial lysis and can harm humans or other treated subjects. For the successful practical application of phages it is necessary to understand the interaction between the phages and their hosts in details [6]. Interaction between phages and bacteria were studied in marine environment [7–9], in rhizosphere [10–13] or in fermentation processes, e.g. in the production of biofuels [14,15]. Mathematical modelling of biological systems is a vital tool in this area and has an important application in ecology or in evolutionary and systems biology [16]. One of the most common mathematical techniques used in mathematical modelling to describe a dynamic behaviour of biological system are differential equations [17]. Several mathematical models were developed to describe the qualitative and quantitative aspects of the interaction between phages and bacteria [18,19]. There are mathematical models that include the influence of antibiotics together with the effect of the host’s immune system and might be applied in phage therapy [20,21]. Other models describe the phage-host interaction from ecological point of view, for example the phage-host interaction in marine environment [7,22]. The interaction between the phage and bacteria depends on several factors, both biological (CRISPRs, bacterial resistance, bacterial fitness) and physical (temperature and
28
G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
Fig. 1. Interaction of bacteria and bacteriophages in a simple bioreactor (chemostat). Source of energy is flowing from the resource reservoir (R0 ) in to the system, where the population of bacteria (N) consumes the resource of energy for their growth. The bacterial growth is a function of maximum specific growth rate (μmax ), growth efficiency (ε ) and of Monod constant (k) – the concentration of resources in medium at which the growth is the half. The maximal specific growth rate is defined as a function of optimal growth rate, temperature and pH [25]. At the same time, bacteriophages (P) are attacking bacteria and adsorbing on their surface with constant adsorption rate δ . Hence, infected bacteria are formed. Bacteria and infected bacteria compete for the resource of energy. Bacteriophages attach also to the surface of infected bacteria, but they don’t infect them resulting in decrease of free phages in the system. After latent period infected bacteria are lysed and new phages are added to the system and their number is defined by the ‘burst size’.
pH affecting adsorption of phages and bacterial growth, concentration of organic acids affects the growth of bacteria). Aim of the work described in this paper was to develop an improved mathematical model based on combinations of existing models describing the phage-host interaction under different conditions and affected by several factors as pH and temperature. It was aimed also at developing an interactive utility simulating the dynamics between the phages and their hosts, which would be easily available for users via a web application.
dM = dt
μmax .R.N k+R
− δ.N.P − ω.N
δ.N.P − δ.N (t − τ ).P (t − τ ).e−ω.τ − ω.M
dP = β . δ.N (t − τ ).P (t − τ ).e−ω.τ − δ.N.P − δ.M.P − ω.P dt
(2)
(3)
(4)
and the mathematical model by Beretta and Kuang [22]:
2. Methods
Using the criterion of relative size and mode of action, the interactions between the virulent phages and their bacteria are usually defined as parasitism [23]. Because replication by most virulent phages necessarily results in bacterial death, some authors describe these interactions as predation and certain interactions could even be termed mutualistic, as some temperate phage encode phenotypic characteristics that are of direct benefit to their hosts [24]. The interactions of the phage-host system in a simple controlled environment (chemostat) are described in Fig. 1.
2.1. Mathematical models describing the phage-bacteria interaction The two, most commonly used mathematical models describing the phage-bacteria interaction are the mathematical model by Schrag and Mittler [25] and their modifications, which describe the phage-host interaction in a chemostat:
dR ε .μmax .R.(N + M ) = ω . ( R0 − R ) − dt k+R
dN = dt
(1)
N+M dN = μmax N 1 − − δ NP dt C
(5)
dM = −μi M + δ NP − e−μi τ δ N (t − τ )P (t − τ ) dt
(6)
dP = b − μ p P − δ NP + β e−μi τ δ N (t − τ )P (t − τ ) dt
(7)
which describes the phage-host interaction in marine environment. These two models differ in a way how they define the increase of bacterial population. The other parts (attack of phages on bacteria and generation of new phages) are almost the same in both models. Several models describing the growth of bacterial population alone were developed. Some of them describe the bacterial growth affected by pH and/or temperature, e.g. the model of Rosso et al. [26]:
μmax = μopt .τ (T ).ρ ( pH )
(8)
G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
29
Table 1 Variables and parameters used for simulations. Symbol R N M P R0
ω ε μmax
K
B
τ
t T Tmin Topt Tmax pH pHmin pHopt pHmax
μopt ∗
τ (T ) =
Definition Concentration of glucose in the system Population density of bacteria Population density of infected bacteria Population density of bacteriophages Glucose concentration in the reservoir Flow rate Growth efficiency Maximal growth rate Glucose concentration at which the bacteria grow at one-half μmax Adsorption rate of bacteriophage on bacteria Burst size Latent period time Run time of the experiment Experimental temperature Minimal temperature Optimal temperature Maximal temperature Experimental pH Minimal pH Optimal pH Maximal pH Optimal growth rate
Source −1
user defined [mg.ml ] user defined [count] user defined [count] user defined [count] 0.5 mg.ml−1 0.2 h−1 2 × 106 mg calculated∗ [h−1 ] 0.0727 mg.ml−1 2 × 107 phage.cell−1 .ml−1 .h−1 98 viruses per bacterial cell infected 0.5 h user defined [h] user defined [°C] 3.06°C 41.10°C 45.06°C user defined 3.88 7.2 12.17 2.635 h−1
[25] [25] [27] [29] [29] [24] [19] [19]
[26] [26] [26] [26] [26] [26] [26]
For formulas 1, 2 the value of 0.7726 h−1 was used [29].
(T − Tmax ).(T − Tmin )2 (Topt − Tmin ).[(Topt − Tmin ).(T − Topt ) − (Topt − Tmax )]
ρ ( pH ) =
Default value
( pH − pH Tmin ).( pH − pH Tmax ) ( pH − pHmin ).( pH − pHmax ) − ( pH − pHopt )2
(9)
(10)
but these physical factors were not included in any known mathematical models of phage-host interactions. All variables, parameters and their values are summarised in Table 1.
2.2. Running the simulations and developing the web application To build and to simulate the mathematical model, R and its additional package deSolve were used. For running the simulations the values of parameters estimated by Schrag and Mittler [25] for Escherichia coli B and Rosso et al. for [26] Escherichia coli O157:H7 were used (Table 1). For developing an interactive web application, which solves the system of delay differential equations and shows the results both graphically and numerically, R packages ggplot2, shiny and shinyIncubator were used.
3. Results We have developed a mathematical model based on the models by Schrag and Mittler [25] (1, 2, 3, 4), modified by Bohannan and Lenski [27,28], and on the mathematical model by Rosso et al. [26] (8, 9, 10). We have combined these two models in order to obtain a mathematical model based on the Schrag and Mittler model [25] – a system of four delay differential equations. The first equation describes the change of glucose in the reservoir (1), the second describes the number of bacteria (2), the third describes infected bacteria (3) and the fourth equation describes the change in the phage population (4). The specific growth rate in the models by Schrag and Mittler [25] (1, 2), Bohannan and Lenski [27,28] is a constant parameter (0.7726 h−1 for Escherichia coli B, [29]). However, in real conditions the growth rate is not a constant value but differs according to the environmental conditions. Hence we took the equations from Rosso et al. [26], which define the actual growth rate as the function dependent on the optimal growth
rate, pH and temperature, and incorporated them into the model by Schrag and Mittler [25]. The advantage of this definition is that the actual growth rate can be numerically easily calculated under different conditions (though this calculation would not necessarily corresponds to the real growth rate, as it depends on many other factors). This could be used for numerous estimations of growth rate during simulations in cases, where the pH and/or the temperature of the environment are not constant, but they are changing during the experiment. It should be noted that safe margins must be chosen for the parameters, since for instance Tmin is not the lowest known temperature where growth is reported – it is the extrapolated temperature at which no growth will occur. Since the growth rates around that temperature are much too small to measure, the actual Tmin , will always be about one degree lower than the minimal reported growth temperature [30]. When T or pH are set close to Tmin /Tmax or pHmin /pHmax respectively, the simulation will run, however the maximal specific growth rate μmax will be close to zero and the modelled system will not exhibit any growth of bacteria and phages. To evaluate new model, the simulations were made with the values of parameters estimated by Schrag and Mittler [25] and Rosso et al. [26] (Table 1). The results of simulation are shown in Fig. 2. Thanks to incorporation of equations by Rosso et al. [26], several simulations under different conditions could be run (e.g. in lower temperatures or under higher pH), where the bacterial growth is not as effective as under optimal conditions. The change of pH or the temperature greatly affects the specific growth rate which has an effect on the final results of the simulation (Fig. 2). Since the model aims on practical application (e.g. estimation of the growth of pathogenic bacteria in food, related to the amount of phage particles added to fight against this infection) and we wanted to make our model to be easily accessible by any user, even with minimal mathematical background, an interactive website has been developed, where users can run simulations with their own parameters and easily calculate and visualise the result of a simulation. Value for each parameter of the model can be modified and the result of simulation is presented both graphically and numerically. Graphical simulation outputs five graphs representing (i) R, resource content (concentration of glucose), (ii) N, population density of uninfected bacteria, (iii) M, population density of infected bacteria, (iv) N+M, population density of all
30
G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
Fig. 2. Running the simulations under five different conditions. The simulations have been run under different conditions to demonstrate the capabilities of designed model. To set up the simulation the values of maximal, minimal and the optimal pH and temperature have to be experimentally stipulated. Then the model can calculate specific growth rate under different pH and temperature. The trial simulations have been run with parameters shown in Table 1. User defined initial values: R = 0.5 mg.ml−1 , N = 10 0 0, M = 0, P = 10 0,0 0 0.
bacteria and (v) P, population density of bacteriophages. Results of numerical simulation can be downloaded as a CSV spreadsheet and used in other applications for further analysis. The web ap-
plication if prefilled with default values to make the first user interaction easier. The web simulation is accessible at the URL http://www.phisite.org/model.
G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
4. Discussion Physical factors influencing the growth of bacterial population were not included in any known mathematical models of phagehost interactions. To eliminate this deficiency we have integrated the function of the maximal growth rate by Rosso et al. [26] into the model of Schrag and Mittler [25]. The evaluating simulations, made with the experimentally measured values by Lenski [24] and Rosso et al. [26], confirmed great influence of these factors on the simulation. The novelty of designed model is the inclusion of environmental factors (pH and temperature) in described interaction between phages and their host. Constructed model of phage-bacteria interaction was not yet experimentally validated and we cannot completely confirm its validity. Also, the pH and temperature values in our model affect only the maximal specific growth rate of bacteria. However, the survival and persistence of bacteriophages are affected also by different external physical and chemical factors, such as temperature, acidity, and ions [31,32]. Studies showed that e.g. temperature can modify the phage absorption, the time of production of new phage particles and the burst size [33–35]. The stability of the phage particles depends also on the pH. For example, the phage cocktail titer (Staphylococcus aureus phages 88 and 35) was reduced 2 log between 4 and 6 h when pH decreased from 6.19 to 5.38 [36]. However, there are no known models describing influence of pH and temperature on bacteriophage dynamics which could be incorporated into the model. The model describes the interaction between bacteriophages and their hosts in open environment, which has a constant flow rate, bringing to the system new resources and washing out metabolites, toxins, and also bacteria and phages. To eliminate pathogenic bacteria, e. g. in a solid food, one has to take into account the fact that bacteriophages must attach to the bacteria, which are in some cases non-moving or form a biofilm. For these conditions this model would not provide precise estimations, since it is based on homogeneous liquid environment in a chemostat. Our aim was to develop a mathematical model describing the phage-host interaction in more accurate way, but this is still only an abstraction of real existing systems, and the simulation may produce results not corresponding to the behaviour of the real biological model. The model is based on system of delay differential equations that are frequently used for this purpose. Since real biological systems are always exposed to influences that are not completely understood or not feasible to model explicitly, it could be advantageous to use stochastic models in place of deterministic ones. These models embrace more complex variations in the dynamics, and a way of modelling these elements is by including stochastic influences or noise [37] and this approach can enhance future development in the area. Availability of the model was important factor in our work. The accessibility was enhanced using the web application, which can help others to make simulations based on their own data, make predictions or validate their own experiments. The web simulation is an effective way for studying relationship between phage and bacteria with an advantage of being very fast and cheap and it can be beneficial also for educational and training purposes. Many different scenarios can be investigated; e.g. comparing calculated results with literature or with own experimental data. One or several parameters can be changed at once and the results are readily available for comparison. This web interface greatly enhances availability and usability of the model. The model describes the phage-host interaction on the level of biological individuals, respectively populations. For future work, the model could be extended on the level of biomacromolecules, gene regulatory networks, genomes (they are based on the information of DNA or RNA and can predict the effect of mutations in
31
the genetic information to the whole system) and environmental factors.
Acknowledgements This work was supported by the Slovak Research and Development Agency [grant number APVV-0098-10]. References [1] A. Sulakvelidze, P. Barrow, Phage therapy in animals and agribusiness, in: E. Kutter, A. Sulakvelidze (Eds.), Bacteriophages: biology and applications, CRC Press, Boca Raton, Florida, 2005, pp. 335–380. [2] Ø. Bergh, K.Y. BØrsheim, G. Bratbak, M. Heldal, High abundance of viruses found in aquatic environments, Nature 340 (6233) (1989) 467–468. [3] H. Milch, Phage typing of Escherichia coli, in: T. Bergan, J.R. Norris (Eds.), Methods of Microbiology, Volume, 11, Academic Press, London, 1978, pp. 123–155. pp. 87–122, 122a, 122b. [4] P.V. Scarpino, Bacteriophage indicators, Ann Arbor Science, Ann Arbor, Michigan, 1978, pp. 201–227. [5] E.C. Keen, Phage therapy: concept to cure, Front. Microbiol. 3 (2012) 1–3. [6] T.K. Lu, M.S. Koeris, The next generation of bacteriophage therapy, Curr. Opin. Microbiol. 14 (5) (2011) 524–531. [7] E. Beretta, Y. Kuang, Modeling and analysis of a marine bacteriophage infection, Math. Biosci. 149 (1998) 57–76. [8] M. Middelboe, Bacterial growth rate and marine virus–host dynamics, Microbiol. Ecol. 40 (2) (20 0 0) 114–124. [9] M. Middelboe, A. Hagström, N. Blackburn, B. Sinn, U. Fischer, N.H. Borch, M.G. Lorenz, Effects of bacteriophages on the population dynamics of four strains of pelagic marine bacteria, Microbiol. Ecol. 42 (3) (2001) 395–406. [10] K.E. Ashelford, M.J. Day, M.J. Bailey, A.K. Lilley, J.C. Fry, In situ population dynamics of bacterial viruses in a terrestrial environment, Appl. Environ. Microbiol. 65 (1998) 169–174. [11] K.E. Ashelford, S.J. Norris, J.C. Fry, M.J. Bailey, M.J. Day, Seasonal population dynamics and interactions of competing bacteriophage and their host in the rhizosphere, Appl. Environ. Microbiol. 66 (10) (20 0 0) 4193–4199. [12] N.J. Burroughs, P. Marsh, M.H. Wellington, Mathematical analysis of growth and interaction dynamics of streptomyces and a bacteriophage in soil, Appl. Environ. Microbiol. 66 (9) (20 0 0) 3868–3877. [13] M.P. Caldas, K.E. Duncan, C.A. Istock, Population dynamics of bacteriophage and bacillus subtilis in soil, Ecology 73 (1992) 1888–1902. [14] Z. Lu, F. Breidt, H.P. Fleming, E. Altermann, T.R. Klaenhammer, Isolation and characterization of a lactobacillus plantarum Bacteriophage, JL-1, from a cucumber fermentation, Int. J. Food Microbiol. 2745 (2003) 1–11. [15] S.S. Yoon, J.W. Kim, F. Breidt, H.P. Fleming, Characterization of a lytic lactobacillus plantarum bacteriophage and molecular cloning of a lysin gene in escherichia coli, Int. J. Food Microbiol. 65 (20 0 0) 63–74. [16] H. Kitano, Foundations of Systems Biology, The MIT Press, 2001. [17] S. Dzeroski, L. Todorovski, Equation discovery for systems biology: finding the structure and dynamics of biological networks from time course data, Curr. Opin. Biotechnol. 19 (4) (2008) 360–368. [18] A.M. Campbell, Conditions for the existence of Bacteriophage, Evolution 15 (2) (1961) 153–165. [19] B.R. Levin, F.M. Stewart, L. Chao, Resource-limited growth, competition, and predation: a model and experimental studies with bacteria and bacteriophage, Am. Nat. 111 (977) (1977) 3–24. [20] B.R. Levin, J.J. Bull, Phage therapy revisited: the population biology of a bacterial infection and its treatment with bacteriophage and antibiotics, Am. Nat. 147 (1996) 881–898. [21] J.H. Payne, A.A. Jansen, Understanding bacteriophage therapy as a density dependent kinetic process, J. Theor. Biol. 208 (20 0 0) 37–48. [22] E. Beretta, Y. Kuang, Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear Anal. Real World Appl. 2 (2001) 35–74. [23] A.T. Bull, J.H. Slater, Microbial interactions and community structure, in: A.T. Bull, J.H. Slater (Eds.), Microbial Interactions and Communities, I, Academic Press, London, 1982, pp. 13–44. [24] R.E. Lenski, Dynamics of interactions between bacteria and virulent bacteriophage, in: K.C. Marshall (Ed.), Advances in Microbial Ecology, 10, Plenum Publishing Corporation, New York, 1988, pp. 1–20. [25] S.J. Schrag, J.E. Mittler, Host-parasite coexistence: the role of spatial refuges in stabilizing bacteria-phage interactions, Am. Nat. 148 (2) (1996) 348–377. [26] L. Rosso, J.R. Lobry, S. Bajard, J.P. Flandrois, Convenient model to describe the combined effects of temperature and pH on microbial growth, App. Environ. Microbiol. 61 (2) (1995) 610. [27] B.J.M. Bohannan, R.E. Lenski, Effect of resource enrichment on a chemostat community of bacteria and bacteriophage, Ecology 78 (1997) 2303–2315. [28] B.J.M. Bohannan, R.E. Lenski, Linking genetic change to community evolution: insights from studies of bacteria and bacteriophage, Ecol. Lett. 3 (20 0 0) 362–377. [29] F.Travisano Vasi, M. Lenski, R. E., Long-term experimental evolution in Escherichia coli. II. Changes in Life-History traits during adaptation to a seasonal environment, Am. Nat. 144 (3) (1994) 432–456.
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[30] M.H. Zwietering, J.C. De Wit, S. Notermans, Application of predictive microbiology to estimate the number of Bacillus cereus in pasteurised milk at the point of consumption, Int. J. Food Microbiol. 30 (1) (1996) 55–70. ´ ˛ , A. Górski, The influence of external fac[31] E. Jonczyk , M. Kłak, R. Miedzybrodzki tors on bacteriophages–review, Folia Microbiol. (Praha) 56 (3) (2011) 191–200. [32] M.H. Ly-Chatain, The factors affecting effectiveness of treatment in phages therapy, Front Microbiol. 5 (2014) 51. [33] M.E. Sanders, T.R. Klaenhammer, Phage resistance in a phage-insensitive strain of Streptococcus lactis: temperature-dependent phage development and host-controlled phage replication, Appl. Environ. Microbiol. 47 (5) (1984) 979–985.
[34] R. Moldovan, E. Chapman-McQuiston, X.L. Wu, On kinetics of phage adsorption, Biophys. J. 93 (1) (2007) 303–315. [35] J.W. Kim, S. Kathariou, Temperature-dependent phage resistance of Listeria monocytogenes epidemic clone II, Appl. Environ. Microbiol. 75 (8) (2009) 2433–2438. [36] P. García, C. Madera, B. Martínez, A. Rodríguez, Biocontrol of Staphylococcus aureus in curd manufacturing processes using bacteriophages, Int. Dairy J. 17 (10) (2007) 1232–1239. [37] S. Ditlevsen, A. Samson, Introduction to stochastic models in biology, in: M. Bachar, J. Batzel, S. Ditlevsen (Eds.), Stochastic biomathematical models, Springer, Berlin, 2013, pp. 3–35.
Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs
Modelling the interaction between bacteriophages and their bacterial hosts Gabor Beke, Matej Stano, Lubos Klucar∗ Institute of Molecular Biology SAS, Laboratory of Bioinformatics, Dubravska cesta 21, 84551 Bratislava, Slovakia
a r t i c l e
i n f o
Article history: Received 19 October 2015 Revised 17 June 2016 Accepted 21 June 2016 Available online 5 July 2016 Keywords: Bacteriophage Bacteria Phage therapy Model
a b s t r a c t A mathematical model simulating the interaction between bacteriophages and their bacterial hosts has been developed. It is based on other known models describing this type of interaction, enhanced with an ability to model the system influenced by other environmental factor such as pH and temperature. This could be used for numerous estimations of growth rate, when the pH and/or the temperature of the environment are not constant. The change of pH or the temperature greatly affects the specific growth rate which has an effect on the final results of the simulation. Since the model aims on practical application and easy accessibility, an interactive website has been developed where users can run simulations with their own parameters and easily calculate and visualise the result of simulation. The web simulation is accessible at the URL http://www.phisite.org/model. © 2016 Elsevier Inc. All rights reserved.
1. Introduction The raising problem of bacterial resistance against antibiotics pointed out the need of novel alternative solutions to eliminate pathogenic bacteria. As a result, the last years of phage research was predominantly devoted to the search of new phages and their potential application as therapeutic agents, generally known as ‘phage therapy’. Bacteriophages, also called phages, are viruses infecting prokaryotic organisms. Phages are ubiquitous and are the most abundant organisms on the planet [1,2] and represent important components of ecological systems. They can modify microbial culture by lysis, transmission of genetic material and by lysogenic conversion. Phages are very important for basic research, but their practical application is still low. Phages can be used in identification of bacterial pathogens [3] or in detection of bacterial contaminations [4]. They are natural enemies of bacteria and are mostly harmless to the human organism. This makes them a promising agent for elimination of undesired bacteria in food industry, medicine or in agro-biotechnology. Phages can be applied individually as solo species; however phages represents one of the major selecting forces in the evolution of bacteria and can develop resistance against phages in couple of days or hours. To slow down this rapid adaptation process, phages can be applied in combinations as ‘phage cocktails’, they can be cyclised (changed over
∗
Corresponding author. Tel.: +421 259307413. E-mail addresses: [email protected] (G. Beke), [email protected] (M. Stano), [email protected], [email protected] (L. Klucar). http://dx.doi.org/10.1016/j.mbs.2016.06.009 0025-5564/© 2016 Elsevier Inc. All rights reserved.
the time) or they can be applied in combination with antimicrobial drugs, like antibiotics [5]. Another possibility is the application of genetic modification of phages or the development of synthetic bacteriophages. There are still several problems in practical application of phages as therapeutic agents, e.g. difficulties during the transport of phages to the place of infection or the fact that phages can cause release of endotoxins and pyrogens, which are the side products of the bacterial lysis and can harm humans or other treated subjects. For the successful practical application of phages it is necessary to understand the interaction between the phages and their hosts in details [6]. Interaction between phages and bacteria were studied in marine environment [7–9], in rhizosphere [10–13] or in fermentation processes, e.g. in the production of biofuels [14,15]. Mathematical modelling of biological systems is a vital tool in this area and has an important application in ecology or in evolutionary and systems biology [16]. One of the most common mathematical techniques used in mathematical modelling to describe a dynamic behaviour of biological system are differential equations [17]. Several mathematical models were developed to describe the qualitative and quantitative aspects of the interaction between phages and bacteria [18,19]. There are mathematical models that include the influence of antibiotics together with the effect of the host’s immune system and might be applied in phage therapy [20,21]. Other models describe the phage-host interaction from ecological point of view, for example the phage-host interaction in marine environment [7,22]. The interaction between the phage and bacteria depends on several factors, both biological (CRISPRs, bacterial resistance, bacterial fitness) and physical (temperature and
28
G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
Fig. 1. Interaction of bacteria and bacteriophages in a simple bioreactor (chemostat). Source of energy is flowing from the resource reservoir (R0 ) in to the system, where the population of bacteria (N) consumes the resource of energy for their growth. The bacterial growth is a function of maximum specific growth rate (μmax ), growth efficiency (ε ) and of Monod constant (k) – the concentration of resources in medium at which the growth is the half. The maximal specific growth rate is defined as a function of optimal growth rate, temperature and pH [25]. At the same time, bacteriophages (P) are attacking bacteria and adsorbing on their surface with constant adsorption rate δ . Hence, infected bacteria are formed. Bacteria and infected bacteria compete for the resource of energy. Bacteriophages attach also to the surface of infected bacteria, but they don’t infect them resulting in decrease of free phages in the system. After latent period infected bacteria are lysed and new phages are added to the system and their number is defined by the ‘burst size’.
pH affecting adsorption of phages and bacterial growth, concentration of organic acids affects the growth of bacteria). Aim of the work described in this paper was to develop an improved mathematical model based on combinations of existing models describing the phage-host interaction under different conditions and affected by several factors as pH and temperature. It was aimed also at developing an interactive utility simulating the dynamics between the phages and their hosts, which would be easily available for users via a web application.
dM = dt
μmax .R.N k+R
− δ.N.P − ω.N
δ.N.P − δ.N (t − τ ).P (t − τ ).e−ω.τ − ω.M
dP = β . δ.N (t − τ ).P (t − τ ).e−ω.τ − δ.N.P − δ.M.P − ω.P dt
(2)
(3)
(4)
and the mathematical model by Beretta and Kuang [22]:
2. Methods
Using the criterion of relative size and mode of action, the interactions between the virulent phages and their bacteria are usually defined as parasitism [23]. Because replication by most virulent phages necessarily results in bacterial death, some authors describe these interactions as predation and certain interactions could even be termed mutualistic, as some temperate phage encode phenotypic characteristics that are of direct benefit to their hosts [24]. The interactions of the phage-host system in a simple controlled environment (chemostat) are described in Fig. 1.
2.1. Mathematical models describing the phage-bacteria interaction The two, most commonly used mathematical models describing the phage-bacteria interaction are the mathematical model by Schrag and Mittler [25] and their modifications, which describe the phage-host interaction in a chemostat:
dR ε .μmax .R.(N + M ) = ω . ( R0 − R ) − dt k+R
dN = dt
(1)
N+M dN = μmax N 1 − − δ NP dt C
(5)
dM = −μi M + δ NP − e−μi τ δ N (t − τ )P (t − τ ) dt
(6)
dP = b − μ p P − δ NP + β e−μi τ δ N (t − τ )P (t − τ ) dt
(7)
which describes the phage-host interaction in marine environment. These two models differ in a way how they define the increase of bacterial population. The other parts (attack of phages on bacteria and generation of new phages) are almost the same in both models. Several models describing the growth of bacterial population alone were developed. Some of them describe the bacterial growth affected by pH and/or temperature, e.g. the model of Rosso et al. [26]:
μmax = μopt .τ (T ).ρ ( pH )
(8)
G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
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Table 1 Variables and parameters used for simulations. Symbol R N M P R0
ω ε μmax
K
B
τ
t T Tmin Topt Tmax pH pHmin pHopt pHmax
μopt ∗
τ (T ) =
Definition Concentration of glucose in the system Population density of bacteria Population density of infected bacteria Population density of bacteriophages Glucose concentration in the reservoir Flow rate Growth efficiency Maximal growth rate Glucose concentration at which the bacteria grow at one-half μmax Adsorption rate of bacteriophage on bacteria Burst size Latent period time Run time of the experiment Experimental temperature Minimal temperature Optimal temperature Maximal temperature Experimental pH Minimal pH Optimal pH Maximal pH Optimal growth rate
Source −1
user defined [mg.ml ] user defined [count] user defined [count] user defined [count] 0.5 mg.ml−1 0.2 h−1 2 × 106 mg calculated∗ [h−1 ] 0.0727 mg.ml−1 2 × 107 phage.cell−1 .ml−1 .h−1 98 viruses per bacterial cell infected 0.5 h user defined [h] user defined [°C] 3.06°C 41.10°C 45.06°C user defined 3.88 7.2 12.17 2.635 h−1
[25] [25] [27] [29] [29] [24] [19] [19]
[26] [26] [26] [26] [26] [26] [26]
For formulas 1, 2 the value of 0.7726 h−1 was used [29].
(T − Tmax ).(T − Tmin )2 (Topt − Tmin ).[(Topt − Tmin ).(T − Topt ) − (Topt − Tmax )]
ρ ( pH ) =
Default value
( pH − pH Tmin ).( pH − pH Tmax ) ( pH − pHmin ).( pH − pHmax ) − ( pH − pHopt )2
(9)
(10)
but these physical factors were not included in any known mathematical models of phage-host interactions. All variables, parameters and their values are summarised in Table 1.
2.2. Running the simulations and developing the web application To build and to simulate the mathematical model, R and its additional package deSolve were used. For running the simulations the values of parameters estimated by Schrag and Mittler [25] for Escherichia coli B and Rosso et al. for [26] Escherichia coli O157:H7 were used (Table 1). For developing an interactive web application, which solves the system of delay differential equations and shows the results both graphically and numerically, R packages ggplot2, shiny and shinyIncubator were used.
3. Results We have developed a mathematical model based on the models by Schrag and Mittler [25] (1, 2, 3, 4), modified by Bohannan and Lenski [27,28], and on the mathematical model by Rosso et al. [26] (8, 9, 10). We have combined these two models in order to obtain a mathematical model based on the Schrag and Mittler model [25] – a system of four delay differential equations. The first equation describes the change of glucose in the reservoir (1), the second describes the number of bacteria (2), the third describes infected bacteria (3) and the fourth equation describes the change in the phage population (4). The specific growth rate in the models by Schrag and Mittler [25] (1, 2), Bohannan and Lenski [27,28] is a constant parameter (0.7726 h−1 for Escherichia coli B, [29]). However, in real conditions the growth rate is not a constant value but differs according to the environmental conditions. Hence we took the equations from Rosso et al. [26], which define the actual growth rate as the function dependent on the optimal growth
rate, pH and temperature, and incorporated them into the model by Schrag and Mittler [25]. The advantage of this definition is that the actual growth rate can be numerically easily calculated under different conditions (though this calculation would not necessarily corresponds to the real growth rate, as it depends on many other factors). This could be used for numerous estimations of growth rate during simulations in cases, where the pH and/or the temperature of the environment are not constant, but they are changing during the experiment. It should be noted that safe margins must be chosen for the parameters, since for instance Tmin is not the lowest known temperature where growth is reported – it is the extrapolated temperature at which no growth will occur. Since the growth rates around that temperature are much too small to measure, the actual Tmin , will always be about one degree lower than the minimal reported growth temperature [30]. When T or pH are set close to Tmin /Tmax or pHmin /pHmax respectively, the simulation will run, however the maximal specific growth rate μmax will be close to zero and the modelled system will not exhibit any growth of bacteria and phages. To evaluate new model, the simulations were made with the values of parameters estimated by Schrag and Mittler [25] and Rosso et al. [26] (Table 1). The results of simulation are shown in Fig. 2. Thanks to incorporation of equations by Rosso et al. [26], several simulations under different conditions could be run (e.g. in lower temperatures or under higher pH), where the bacterial growth is not as effective as under optimal conditions. The change of pH or the temperature greatly affects the specific growth rate which has an effect on the final results of the simulation (Fig. 2). Since the model aims on practical application (e.g. estimation of the growth of pathogenic bacteria in food, related to the amount of phage particles added to fight against this infection) and we wanted to make our model to be easily accessible by any user, even with minimal mathematical background, an interactive website has been developed, where users can run simulations with their own parameters and easily calculate and visualise the result of a simulation. Value for each parameter of the model can be modified and the result of simulation is presented both graphically and numerically. Graphical simulation outputs five graphs representing (i) R, resource content (concentration of glucose), (ii) N, population density of uninfected bacteria, (iii) M, population density of infected bacteria, (iv) N+M, population density of all
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G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
Fig. 2. Running the simulations under five different conditions. The simulations have been run under different conditions to demonstrate the capabilities of designed model. To set up the simulation the values of maximal, minimal and the optimal pH and temperature have to be experimentally stipulated. Then the model can calculate specific growth rate under different pH and temperature. The trial simulations have been run with parameters shown in Table 1. User defined initial values: R = 0.5 mg.ml−1 , N = 10 0 0, M = 0, P = 10 0,0 0 0.
bacteria and (v) P, population density of bacteriophages. Results of numerical simulation can be downloaded as a CSV spreadsheet and used in other applications for further analysis. The web ap-
plication if prefilled with default values to make the first user interaction easier. The web simulation is accessible at the URL http://www.phisite.org/model.
G. Beke et al. / Mathematical Biosciences 279 (2016) 27–32
4. Discussion Physical factors influencing the growth of bacterial population were not included in any known mathematical models of phagehost interactions. To eliminate this deficiency we have integrated the function of the maximal growth rate by Rosso et al. [26] into the model of Schrag and Mittler [25]. The evaluating simulations, made with the experimentally measured values by Lenski [24] and Rosso et al. [26], confirmed great influence of these factors on the simulation. The novelty of designed model is the inclusion of environmental factors (pH and temperature) in described interaction between phages and their host. Constructed model of phage-bacteria interaction was not yet experimentally validated and we cannot completely confirm its validity. Also, the pH and temperature values in our model affect only the maximal specific growth rate of bacteria. However, the survival and persistence of bacteriophages are affected also by different external physical and chemical factors, such as temperature, acidity, and ions [31,32]. Studies showed that e.g. temperature can modify the phage absorption, the time of production of new phage particles and the burst size [33–35]. The stability of the phage particles depends also on the pH. For example, the phage cocktail titer (Staphylococcus aureus phages 88 and 35) was reduced 2 log between 4 and 6 h when pH decreased from 6.19 to 5.38 [36]. However, there are no known models describing influence of pH and temperature on bacteriophage dynamics which could be incorporated into the model. The model describes the interaction between bacteriophages and their hosts in open environment, which has a constant flow rate, bringing to the system new resources and washing out metabolites, toxins, and also bacteria and phages. To eliminate pathogenic bacteria, e. g. in a solid food, one has to take into account the fact that bacteriophages must attach to the bacteria, which are in some cases non-moving or form a biofilm. For these conditions this model would not provide precise estimations, since it is based on homogeneous liquid environment in a chemostat. Our aim was to develop a mathematical model describing the phage-host interaction in more accurate way, but this is still only an abstraction of real existing systems, and the simulation may produce results not corresponding to the behaviour of the real biological model. The model is based on system of delay differential equations that are frequently used for this purpose. Since real biological systems are always exposed to influences that are not completely understood or not feasible to model explicitly, it could be advantageous to use stochastic models in place of deterministic ones. These models embrace more complex variations in the dynamics, and a way of modelling these elements is by including stochastic influences or noise [37] and this approach can enhance future development in the area. Availability of the model was important factor in our work. The accessibility was enhanced using the web application, which can help others to make simulations based on their own data, make predictions or validate their own experiments. The web simulation is an effective way for studying relationship between phage and bacteria with an advantage of being very fast and cheap and it can be beneficial also for educational and training purposes. Many different scenarios can be investigated; e.g. comparing calculated results with literature or with own experimental data. One or several parameters can be changed at once and the results are readily available for comparison. This web interface greatly enhances availability and usability of the model. The model describes the phage-host interaction on the level of biological individuals, respectively populations. For future work, the model could be extended on the level of biomacromolecules, gene regulatory networks, genomes (they are based on the information of DNA or RNA and can predict the effect of mutations in
31
the genetic information to the whole system) and environmental factors.
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