Probiotics as cheater cells: Parameter space clustering for individualized prescription

Journal of Theoretical Biology 361 (2014) 165–174

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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi

Probiotics as cheater cells: Parameter space clustering for individualized prescription Sanhita Ray, Anjan Kr. Dasgupta n Department of Biochemistry and Center for Excellence in Systems Biology and Biomedical Engineering, University of Calcutta, Kolkata, India

H I G H L I G H T S


Dosage requirement for probiotics using cheater worker interaction.
Cheater therapy design includes choice of cheater cells, antibiotics and slow release strategy for these cells against resistant infection.
Optimal prescription for co-administration of probiotics and antibiotics using clustering method.

art ic l e i nf o

a b s t r a c t

Article history: Received 3 May 2014 Received in revised form 15 July 2014 Accepted 16 July 2014 Available online 24 July 2014

Clinicians often perform infection management administering probiotics along with antibiotics. Such probiotics added to an infecting population showing antibiotic resistance can be compared to a dynamical system composed of cheaters and workers. The presence of cheater strains is known to modulate the fitness of the infecting population. We propose a model where probiotics as cheater strain re-establishes the susceptibility of a resistant population towards an antibiotic. Control parameters must assume optimal values in order to attain minimum worker number within a finite time-scale feasible in a clinical set-up. The problem is made non-trivial by the complicated interplay between parameters. The model is an extension of a logistic framework, where a pay-off function has been included to account for the effect of instantaneous worker number on death rates of each species. The outcomes for a randomized set of parameter values and initial conditions are utilized in partitioning the set and desired clusters were identified. For a test case, one can take random combinations of controllable parameters and combine them with fixed parameters and find out the closeness of the points to the desired cluster centroids. This process leads to the identification of optimum antibiotic versus probiotic dosage range leading to elimination or limited existence of the genetically resistant population. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Worker Pay-off Resistance Prebiotics Optimality

1. Introduction The use of cheater cells (Ch) to destabilize pathogenic infections (Diggle, 2010) has been proposed in lieu of the alarming rates of antibiotic resistance development (Spellberg et al., 2013). Cheater cells (Celiker and Gore, 2013) have the ability to invade cooperative cellular societies and benefit from commonly available resources, without having to pay the metabolic costs (Lenski, 2010) of production. The basic idea is to let a blow-up in the number of cheaters to bring down a cooperatively infecting population. Experimental proof of principle was obtained by using cheats, which benefit from but do not participate in crowd sensing, to lower the virulence of Pseudomonas aeruginosa in an

n

Corresponding author. E-mail addresses: [email protected] (S. Ray), [email protected] (A.Kr. Dasgupta). http://dx.doi.org/10.1016/j.jtbi.2014.07.019 0022-5193/& 2014 Elsevier Ltd. All rights reserved.

animal infection model (Rumbaugh et al., 2009). Cheaters may also be used to revert back the antibiotic susceptibility of resistant microbial populations, provided the resistance is linked at some stage to cooperation among individuals. Examples of cooperative resistance are penicillin resistance due to extra-cellular secretion of β-lactamases (Olsson et al., 1976) or when the enzyme itself is periplasmic but the effects of de-toxification are not localized (Francisco et al., 1992). It has been seen that populations of penicillin resistant microbes may harbour non-producers (Lee et al., 2010). The method proposed in this paper, though not restrictive, explores this specific type of cheat therapy, i.e. a combined use of cheaters and antibiotics to treat resistant infections. Here the antibiotic referred to is the one to which infecting strain shows resistance in standard laboratory MIC (minimal inhibitory concentration) determination (Yurtsev et al., 2013). MIC is the minimum antibiotic concentration at which microbial growth is inhibited, typically determined by the use of strips containing a graded concentration of the antibiotic. It is to be

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noted that MIC determination is done with a pure clonal population, hence does not account for the effects that other species occupying a real ecological niche might have on antimicrobial resistance. For the practical use of cheater cells as therapy it would be essential to know the dosage or the type of cheater organism that ideally needs to be administered to obtain a favourable end-result. As with pharmacotherapy, a dynamical model describing the growth and mechanism of action of cheaters becomes essential for designing a treatment regimen. Models have emerged from experimental observations explaining the effects of initial antibiotic concentration and worker numbers on the relative dynamics of the two types (Celiker and Gore, 2012; Yurtsev et al., 2013). They have modelled the death rate as a discontinuous function of antibiotic concentration, assuming high and low values, above and below MIC, respectively. Evolutionary game theory describes the dynamics of the two alternate strategies, cheating and cooperation, in terms of the replicator equation (Hofbauer and Sigmund, 2003) where a probability distribution of initial cheater–worker ratio is usually considered. The fixed form of the replicator equation provides a deterministic population dynamics (Traulsen et al., 2009). Materials produced by microbes benefitting producers and non-producers alike are referred to as public goods. Public goods considered in this context are non-rivalrous, i.e. increase in the number of users does not affect the benefit received by each. Both forms make use of the pay-off function, dependent on the number of workers and defined as the increment (for each species) in growth rate that can be attributed to public goods production by workers. Due to saturation effects, i.e. limitations on the amount of help public goods may confer, nonlinear pay-off functions have been proposed (Archetti and Scheuring, 2011, 2012; Hauert et al., 2006). Using deterministic growth models the problem of dose determination may be solved by taking a control systems design (Geevan et al., 1990) approach similar to what is used for determining optimum dose of pharmaceutical agents. In contrast to the use of drugs though, the use of cheater–antibiotic combination is associated with a huge variability in terms of patient immune status (Way et al., 1998), infection progression and characteristics peculiar to the infecting organism itself. In addition a combination therapy would yield a number of controllable parameters since we can design cheater organisms for our benefit (Brown et al., 2009, Ouwehand et al., 2002). This leads us to the problem of optimization in a multi-dimensional parameter space. Clustering (MacQueen, 1967) is commonly used as a means of exploring any tentative grouping within a multi-variate data space. Various criteria (Caliski and Harabasz, 1974; Mouchet et al., 2010) may be used to find out the correct number of groups that exist in the data space. For k-means clustering (MacQueen, 1967), each group is described in terms of its centroid. The coordinates of the centroid are its components along each dimension, corresponding to each of the variables. The length of each component is the mean value of the variable. The same methods are applicable for a multidimensional parameter vs simulation results space. Probiotics are live microbes, which upon ingestion confer some beneficial or therapeutic effects (Surawicz, 2003). For example, probiotics have been shown to be effective against Clostridium difficile disease that often occurs after a phase of antibiotic treatment and necrotizing enterocolitis in pre-term neonates (Bin-Nun et al., 2005). The mechanism of probiotic action has been modelled (Arciero et al., 2010) in terms of its proven antiinflammatory effects. A second, oft-postulated view, that probiotics act by out-competing pathogens, has been used to explain a large number of experimental results, e.g. pathogen exclusion from biofilms in the presence of probiotics (Woo and Ahn, 2013). Competition among gut microflora has been modelled (Coleman

et al., 1996) but without any clear directives for optimizing probiotic dosage. Usefulness of probiotics in the treatment of antibiotic resistant infections has been proven in the case of clarithromycin resistant Helicobacter pylori infection (Ushiyama et al., 2003). It quickly became apparent to us that due to its ability to out-compete pathogens, probiotics could be used to restore the antibiotic susceptibility of resistant populations in a manner parallel to that of cheater therapy. Prebiotics are food components that are selectively fermented (Roberfroid, 2001) by specific probiotic organisms. Consumption of suitable prebiotics along with probiotics effectively increases (Nazzaro et al., 2012) the carrying capacity of the gut niche for the concerned probiotic species, giving them a competitive advantage (Gibson et al., 1995) against any invading pathogens. We incorporate the combined effect of probiotics, antibiotics and prebiotics in our scheme of cheater therapy. In this paper we propose a tractable model to simulate the growth dynamics of cheater cells within an antibiotic resistant infecting population and show how such growth may render cooperative resistance mechanisms ineffective. The model is widely applicable and independent of the exact mechanism of resistance due to the introduction of some novel features into previously established layouts, specifically the reduction of death rate due to payoff acquired from public goods production. We treat cheat therapy as a control systems design and demonstrate how this and similar models of public goods games may be utilized in finding out values that controllable parameters must assume in order to yield favourable end results within a clinically feasible finite time. It may be recognized that optimality of such values is case specific when a wide variability is expected with regard to both controllable and uncontrollable parameters. We predict individualized prescriptions by applying multi-dimensional clustering in order to segregate simulation end results at a fixed time point, each result corresponding to a random combination of parameter values. We have used parameter value ranges approximating those from the existing literature on probiotics and prebiotics to show how cheater therapy works in this context. Further, we show that parameter sensitivities could be determined from clustering results.

2. Methods 2.1. Model Among the most widely used growth rate expressions are the logistic equation and the Monod kinetics. The Monod model has often been employed to describe the dynamics of substrate, biomass and product. Recently it has been shown that as a first approximation, one can employ logistic model (Verhulst, 1845) to avoid the complexities of Monod model (Monod, 1949), the complexity being almost intractable when one involves multispecies system (Kargi, 2009). To describe the interaction between the workers and the cheaters we have used the logistic framework that has less parameters, and to a first approximation matches with the Monod model. The model simulates the effects of introduction of cheaters (Ch) on growth of a cooperative bacterial species (workers, W ). The cooperation is with respect to resistance to an antibiotic, A, as determined by standard laboratory sensitivity tests (MIC determination for A) and the resistance mechanism involves at least one exo-product (public goods, P). Cooperative nature of W arises from the greater benefit accrued if there are more producers of P. The cheaters are susceptible to A as they do not produce the exoproduct. However, they will show resistance if P is supplied externally. The model variables and parameters are described in Table 1.

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Table 1 The table summarizes parameter values used in simulation, their description, their controllability status, the actual range used in the simulation and literature values and references if available. Literature (n) are in c.f.u./g faeces transformed to O.D. units MacFarland Standard values, transformed values expressed in parenthesis. Parameter (units)

Description

Con( þ)/UnCon (  )a

Actual rangeb

Literature values

Reference

Knc (O.D.)

Effective carrying capacity of the niche for cheaters

þ

0.001–0.98

1011–1015 (55.75–5.5  105)

Knw (O.D.)

þ

-do-

þ

-do-

107–1011 (0.0067–55.75) 0.1661–0.333

þ   þ þ 

-do-do-do-do-do-do-

þ 

-do-do-

Chnini (O.D.)

Effective carrying capacity of the niche for workers Specific killing rate for cheatersdue to antibiotic Cost of producing the exo-product Cooperativity index Half-saturation parameter Specific growth rate of cheaters Specific kill rate by A, for workers Rate of mutation of cheaters into workers Slow release rate Fraction of pay-off from public goods available to cheaters Initial number of cheaters

Guo et al. (2012) and Nazzaro et al. (2012) (increase in carrying capacity when prebiotic is introduced) Guo et al. (2012) (total lactobacilli counts in faeces) Potel et al. (1991)

þ

-do-

108.13–109.2 (0.0089–0.8836)

Wnini (O.D.)

Initial number of workers



-do-

102–106.9 (10  8–0.0055)

Ac (h  1) C (N.D.) n (N.D.) knm (O.D.) r c (h  1) Aw (h  1) m (h  1) SRn (O.D. h  1) pfrac (N.D.)

a b

0.482–1.57

0.01–0.54 0.1661–0.333 3.75  107– 0.000375

Unpublished data from our lab NA NA Garcia-Cayuela et al. (2014) Potel et al. (1991) Cocconcelli et al. (2003) NA NA Guo et al., 2012 (increase in faecal lactobacilli count after probiotic treatment) Guo et al., 2012 (faecal Salmonella counts)

Controllable or uncontrollable parameters are represented by þ and  symbols. Ranges taken are in the same unit as representative values from literature, except where mentioned.

The rate of change of number of cheaters was modelled as   dCh Ch þ W ¼ rc 1  ð1Þ Ch  γ  Ch  m  Ch þSR dt Kc The first term in Eq. (1) describes the growth rate of cheaters in the form of the logistic equation (May et al., 1976), where rc is its specific growth rate and Kc is the effective carrying capacity for cheaters. Since the direct basis for the de-stabilization of the worker population is competition between the two species for a common pool of resources, the carrying capacity is expected to be the same for both. However, for therapy purposes we may engineer the ecological niche (the infection site) so that the effective carrying capacity for cheaters is greater than that for workers, e.g. by supplementation with pre-biotic (Svensson and Jacobsen, 2014), a nutrient that only the cheaters are able to utilize. However, since the rest of the resources are open to utilization by both species, the available fractional carrying capacity takes the form Keff ¼ 1 

Ch þ W Kc

ð2Þ

In Eq. (1) the second term involves the term γc representing the effective specific death rate of cheaters. The explicit expression for γc is given by

γ c ¼ Ac  ð1  f c Þ

ð3Þ

In Eq. (3) Ac is the specific death rate of cheaters attributable to a particular applied concentration of antibiotic A, when no antibiotic destroying exo-product was present. Its effective value is brought down by the pay-off, fc, accrued from exo-product, P, externally supplied by the worker species. Spatial structure within host may limit access to public goods for cheater cells (Dugatkin et al., 2008). Only cells in the near vicinity of workers will receive protection from antibiotics. Hence it is likely that non-producer, Ch, receives only a fraction, pfrac, of the pay-off (f ) received by workers. In addition, the same case occurs when antibiotic destroying enzyme itself is sequestered within the cells, protection afforded to cheaters is only a fraction

(pfrac) of that available to workers. Thus pay-off for cheaters is modelled by Eq. (4). It may be noted that the maximum possible pay-off for cheaters is given by pfrac, which will be a fraction having value less than 1: f c ¼ pfrac  f

ð4Þ

Here f is the pay-off obtained by workers. It is described as a Heaviside function (Archetti et al., 2011) expressed in the following equation: f¼

Wn km þ W n

ð5Þ

The parameter n is the cooperativity index describing the extent of 1=n cooperation and km is the worker population at which half of the maximum payoff possible is attained. According to Eq. (5) the maximum value of f is 1, when the effective death rate becomes 0, according to Eq. (2). The function is continuous and with increase in number of workers the total benefit increases. Below a threshold number of workers the function is convex, implying synergism and above that it is concave implying discounting, i.e. competition for substrates for this scenario (Hauert et al., 2006). Usability of cheater cells has been questioned since there is always the possibility that administered strain itself mutates into workers. In case of hospital acquired infections, such transformations take place at a very high rate (Wilharm et al., 2013) by horizontal gene transfer. We have modelled this scenario by adding a mutation term to the net growth rate of workers in Eq. (6). Mutation rate is given by m and the increase in dW /dt is proportional to the number of cheaters Ch (dCh/dt is decreased by the same amount, see Eq. (1)). To deal with the problem of mutation, a slow release dosage of cheater cells was considered. The term SR in Eq. (1) models the addition of cheaters from a slow release gel at a constant rate. The rate of change of number of workers is modelled as   dW Ch þ W ¼ rw 1  W  Aw ð1  f w Þ  W þ m  Ch ð6Þ dt Kw

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where f w ¼ f . Eq. (6) is similar to Eq. (1), except that a metabolic cost C is always associated with production of the exo-product. Hence, specific growth rate of workers, rw, is obligatorily lower than rc (this would provide a mathematical definition of cheaters), and the relation between the two specific growth rates is r w ¼ r c  ð1  CÞ

ð7Þ

2.2. Model application 2.2.1. Partitioning cause-result space 5000 random combinations of values for 11 model parameters and 2 initial species numbers (in the form of the matrix M, where the rows correspond to observations and the columns correspond to dimensions/coordinates) were generated by Latin Hypercube Sampling (Iman, 2008) lhsdesign in MATLAB over the ranges specified in Table 1. Ranges were around those for probiotics (Guo et al., 2012), since it is proposed that probiotics may be utilized as cheaters. Efficacy of probiotics in in vivo studies is usually determined by studying the faecal counts of both probiotic organisms and pathogenic strain, and this gives a representation of gut microbial niche. Hence all organism counts are in terms of c.f. u./g faeces. However these numbers were too large and calculations with them were not possible with our computer system's memory. Hence the values were transformed into O.D. units according to Macfarland's standard, which gives a linear relationship between O.D. and colony forming units per ml (c.f.u./ml). Parameters like carrying capacity, which are in the terms of c.f.u./ ml, have been scaled similarly. Hence it is possible to use the O.D. values in simulating Eqs. (1) and (6). The ranges actually used were wider than those from the cited literature so as to explore with more efficacy. A wide spectrum meta-analysis would reveal the best ranges to be taken on the basis of existing literature. Model simulations were done for each combination in M, for a fixed time (t¼ 100,000 h), using ode23 function in MATLAB (Version 13a, Mathworks USA) over the range 0.001–0.98, with units specified in Table 1. Species numbers at final time point (for each simulation) were combined with corresponding parameter values to yield another matrix, MR. MR represents the cause-result space for the model (within the specified bounds and for the specified time-point). MR was then subjected to partitioning by k-means, after having determined the optimal number of clusters using the Calinski–Harabasz criterion (Caliski and Harabasz, 1974; Mouchet et al., 2010). The mean value of final cheater and worker number for each cluster was used to obtain a semi-quantitative indication of which clusters give desired end-results. This clustering step is a training exercise that segregates parameter value ranges into regions which give desirable results or not. 2.2.2. From test cluster to optimal parameter values Test cluster here implies a cluster of points, each of which comprises fixed values for 5 uncontrollable parameters and random combinations of 8 controllable ones. The distances of every test point from each of the training cluster centroids were determined. A new component was added for each point and its value was the reference cluster number to which the point was nearest (e.g. if a point was nearer to reference cluster centroid NO. 1 then the value of its last component was assigned to be 1). This new matrix was further subjected to k-means partitioning, after determination of ideal cluster number as before. 3. Results Interaction of cheaters within a cooperatively antibiotic resistant microbial population (worker) has been modelled. As detailed in Section 2, Eqs. (1) and (6) represent this interaction and the

model was subjected to test for a wide spectrum of parameters that include growth rate of cheaters and workers, antibiotic concentrations, initial values of the cheater and worker, and possible mutation of the cheater species to worker species. The objective of this simulation was to find whether in this complex interplay of parameters, one can condense the parameter search space such that a desirable regulated level of worker cheater ratio is attained and worker population does not blow-up in a realistic time scale. It may be noted that probiotics may be utilized as cheaters, provided they are susceptible to the antibiotic in question. Hence parameter values used for simulations are around those for probiotics as obtained from the existing literature. The model and methodology would remain the same for other classes of resistance cheaters, even though the exact ranges for parameter values might differ. 3.1. Numerical results The simulation results for cheater worker growth model for a few combinations of parameter values and initial conditions are given in Fig. 1. When only workers are present in antibiotic rich medium, biomass increases with time following a sigmoidal pattern. This (Fig. 1(a)) is a well behaved dynamical behaviour. Cheaters are unable to survive on their own in antibiotic rich medium excepting by the mutation route (which was not considered here). For the combination of parameter values in Fig. 1(b), a criticality was observed regarding initial conditions. Low initial worker–cheater ratio leads to quick collapse of the entire population. High initial worker–cheater ratio supported a rapid spurt in the number of workers. However, with time the number of cheaters increases and reaches a threshold value. After this point, the population can no longer support its load of non-producers and collapses. It may be noted that no conversion of cheaters into workers was considered. The long period of pseudo-stable state attained by the cheater population is explained on the basis of high worker population that is maintained during this time. The particular combination of parameter values specifically m ¼ 0, SR ¼0 and pfrac¼0 may be the combination where a situation similar to the “tragedy of the commons” is observed, i.e. a few cheaters are capable of bringing down a cooperative population (Smith, 2012). It is to be noted that such might not be the case for other parameter value combinations. 3.2. Stability analysis The model described in Eqs. (1) and (6) has been subjected to stability analysis to determine the parameter values for which stable points might exist at zero or sufficiently low worker number. Such stable points would be the ultimate aim of an optimum prescription. The null-clines (Fig. 2(a)) for the growth equations (Eqs. (1) and (5)) for various combinations of parameter values yielded multiple critical points in each case. Linearization was performed for each critical point and their stability determined (see Fig. 2(a) and (b)). Flow vectors (Fig. 2(a)) drawn on the phase plane support these results. Linearization is not suitable for this model since eigenvalues were found to be 0 in many cases. For cases where multiple stablepoints were found, model simulations showed that some were never attainable within the time limit of simulations. Furthermore analytical solutions do not take into account the initial conditions which affect the end-result, i.e. the stable-point which was ultimately reached. Interestingly the stability analyses yielded saddlepoints for a large number of cases. Simulations for these parameter value combinations showed that trajectories (Fig. 2(c)) initially tended towards the saddle-point and were then repelled away

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Population (cheaters, workers)

5.5 5

ONLY WORKER ONLY CHEATER

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

0

2

4

6

8

10

12

14

16

18

20

time

Species numbers (O.D.)

4 case 1

3

2 case 1

1 case 2

0 100

case 2

102

103

104

105

time (h−1) Fig. 1. Model (Eqs. (1) and (5)) simulation results. (a) Growth dynamics of isolated cheater and worker populations in the presence of an antibiotic. Workers secrete exo-product that degrades the antibiotic, but cheaters do not. Parameter values are Kw ¼ Kc ¼5, Ac ¼Aw ¼ 0.99, C¼ 0.001, n ¼2, km ¼ 0.1, m¼0, SR¼ 0, pfrac ¼1, rc ¼1 and either Chini ¼ 0.1 or Wini ¼ 0.1 for respective cases. Units for parameters as in Table 1. (b) Growth dynamics of mixed populations of workers (green) and cheaters (blue) in the presence of same antibiotic, when (case 1) Wini ¼ 0.1 and Chini ¼ 0.3 and (case 2) Wini ¼ 0.1 and Chini ¼0.4. For case 1 (dashed lines), growth dynamics show three phases: an initial attainment of pseudo-stable state, maintenance of such state for a long time and finally the decline of both worker and cheater populations. Case 2 (solid lines): There is rapid decline of the population. Keeping initial worker number fixed, appearance of the pseudo-stable state showed criticality depending on initial cheater percentage. The two points are from two sides of the critical cheater percentage. All other parameters are same as in (a). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

towards either of the stable points. This is the dynamical feature which is ultimately responsible for the presence of pseudo-stable state as in Fig. 1(b). For some cases, such a pseudo-stable state implies a blow-up and maintenance of the genetically resistant species for a long time before final collapse of the entire population. The end-result might be considered to be a desirable outcome but the time scale over which it was obtained makes it unsuitable for clinical applications. Hence the model was further analysed with the numerical approach. Even though the aim of obtaining an optimum prescription was unattainable by stability analysis, the results revealed some important characteristics of the model as to the time-scales that must be dealt with in applications of cheat therapy and the existence of saddle-points.

3.3. Reference cluster generation From Table 1, there are 8 controllable and 5 uncontrollable constant value parameters in the model described in Eqs. (1) and (6). The aim here is to determine what combination of values the controllable parameters must assume, for a given set of uncontrollable parameter values, so that the number of workers is minimized in a mixed population. Firstly, a training exercise was performed to

169

determine what overall combinations of parameter values would yield a favourable result. This step involved randomization (Fig. 3(a)) of both controllable and uncontrollable parameters over the range specified in Table 1. The parameter space was partitioned (see Section 2.2.1) based on simulation results for each point in the parameter space. Final species numbers were taken as extra dimensions, along with the constant value parameters. Final species numbers are usually obtained by iterating till a stable point is reached. However, our results are for a fixed number of iterations regardless of where stability is attained, since the aim would be to minimize worker number within a given time limit. Cluster analysis revealed the presence of two clusters (since the value of Calinski–Harabasz index is maximized when the number of clusters is set at 2 (Fig. 3(b))) in the parameter space – one associated with a much higher (Fig. 3(c), inset and Fig. 3(d)) number of workers than the other. Each cluster is identified by its centroid having 15 components corresponding to the 13 constant value parameters and the 2 species. Fig. 3(c), main plot gives a comparison of centroid components for the 2 clusters. The heights of bars represent mean values that parameters must assume in order to belong within one or the other cluster. The clusters (Fig. 3 (c)) obtained on the basis of simulations are used as training sets or reference clusters (RefClust) in the next step to determine an optimum prescription for cheat therapy, for a given set of uncontrollable parameters. It may be noted that RefClust NO. 1 is the desired cluster, corresponding to a lower final worker number. It is evident that model results are more sensitive to changes in some parameter values than others. The sensitivity for a given parameter may be simply calculated in this case by finding out the normalized difference (difference normalized by range) between the average values of this parameter in each of the 2 clusters (Fig. 3(e)). The highest sensitivity was found for parameter number seven that is specific growth rate of cheaters rc, and low for initial number of cheaters. This implies that not the initial dose of cheaters, but what kind of probiotic is applied is more important. Very little sensitivity was found for initial worker number which implies that how far the infection has progressed is not a very important determinant of whether the cheater therapy will work. Another favourable result is lesser sensitivity to pfrac. This parameter accounts for producer cells storing exo-product in its near vicinity or inside its periplasm, thus reducing its availability for cheater species and also the structured nature of microbial niches. Clustering results show that this parameter would have little effects on the proposed destabilization of infections. Some counter-intuitive results were obtained with regard to effective carrying capacity (Kc), specific death rate for cheaters (Ac) and slow release rate (SR). Mean values of Kc and SR turned out to be lesser for the desired cluster, while that of Ac was greater. This implies that limitations on the growth rate of cheaters are necessary for the therapy to be effective. These results are explained by the fact that mutation of cheaters into workers (e.g. by horizontal gene transfer) was considered in the model. A shootup in the number of cheaters would actually boost worker numbers. This prediction corresponds with high sensitivity of simulation outcome to the mutation rate (m). However too low a value for Kc and SR and a high value for Ac would lead to wipe-out of cheater population, hence the need for parameter value optimization. Similarly it may be noted that the mean value for cost (C) for the desired cluster is lower than for undesired cluster – this is because decreased growth of workers would result in lesser exoproduct formation and hence very little contribution towards cheater cell survival. Additionally, parameter sensitivities were found to be different if determined for a different range. It is to be noted that in case one cycle of clustering does not give a low enough mean value of final W, the points in the desired cluster may be subjected to further cycles of partitioning. This was observed to give even finer partitioning with altered parameter sensitivities (data not shown).

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worker (y)

170

5.5 5 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

(w) STABLE

(c) STABLE

UNSTABLE

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

cheater (x)

Fig. 2. (a) Null clines for parameter values: Kw ¼ Kc ¼ 5, Ac ¼Aw ¼ 0.5, C¼ 0.5, n ¼2, km ¼ 0.1, m¼ 0, SR ¼ 0, pfrac ¼ 1, rc ¼ 1 (black, solid line: y-nullcline. Grey, dashed line: x-nullcline). Units for parameters same as in Table 1. Critical points are given by the points of intersection of nullclines. The arrows represent the flow vectors, toward or away from the critical points. Stable points are indicated in each case. Depending on initial cheater/worker proportions (initial strategy), the system moves towards either of the stable points, as indicated by flow vectors. (b) Results of stability analyses for varying values of specific death rate ðAc ¼ Aw ¼ A and CÞ. Other parameter values are the following: Kw ¼ Kc ¼ 2, n ¼5, km ¼ 0.001, m ¼0.001, SR ¼0, pfrac¼ 1, rc ¼ 1. Similar phase space trajectories were obtained for points A and B marked in the figure. Middle layer: saddle-points, top and bottom layers: stable points, circles: stability undeterminable from linearized form. (c) Trajectories in the cheater–worker phase space for point Ac ¼ Aw ¼ 0.4, C¼0.3, in (b). Showing effect of a saddle-point on trajectories. Also demonstrates how initial species numbers might affect the stable point which is ultimately reached. The dotted line represents an approximate boundary: points above/ left of it evolve into a worker only population, points below/right evolve into a cheater only population.

3.4. Optimized prescription For prescribing cheaters or probiotics along with pharmaceutical agents, to a particular patient, an approximate knowledge of some of the control parameter values is required, specifically those values which are fixed and characteristic for the pathogen or the patient. Let the values in columns 5, 6, 9, 11 and 13 (Fig. 4(a)) represent the values of the 5 uncontrollable parameters for a particular case (patient). A random set of values was considered for the rest 8 controllable parameters (columns 1, 2, 3, 4, 7, 8, 10, 12). The matrix given in Fig. 4 (a) is the test cluster for the given patient. Optimum prescription was then determined on the basis of distance of each test point from previously obtained training cluster centroids. Points which were nearer to centroid of RefClust NO. 1, i.e. desired cluster, were grouped

into one sub-cluster (sub-cluster NO. 2 in this case) and those nearer to RefClust NO. 2 were grouped into another (sub-cluster NO. 1) as in Fig. 4(c). Since uncontrollable parameters are fixed the sub-clustering partitions only the controllable parameter values. Points closer to desired cluster were found to show better end results (this validation was done by performing simulation for the mean values for each subcluster in Fig. 4(c), results not shown). Hence it may be concluded that mean values for sub-cluster NO. 2 represent a preferred prescription for the specific case. Whether the therapy will be useful may be determined by performing simulation for a single set of values, i.e. the preferred prescription, and seeing whether worker number is brought down to a low enough value within a feasible time-limit. It may be noted that simulations were not performed for all the points in the test cluster, only the sub-cluster means. The generation of reference

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Fig. 3. Partitioning of cause result space (a) the cause-result matrix: 13 columns represent parameters and initial numbers, rows represent random values for each. These combinations (range for each column as in Table 1) were used to explore the model through simulations. The two columns in the inset represent final cheater and worker number at the end of time¼ 100 000 h. (b) Calinski–Harabasz index values for various cluster numbers showing maximum value at 2. Hence the optimum cluster number is 2 for the matrix in (a). (c) The first 13 bar groups represent mean values of each parameter for respective clusters. The last two bar groups give mean values of species final numbers for each cluster. Blue/blueoutlined bars are for the cluster one, red/red-outlined bars are for the cluster two. Since the parameter value dataset is a random number set, their partitioning can only be due to the presence of distinct clusters in the outcome set. Mean values represent cluster-centroid components along each of the 15 dimensions. The inset shows frequency distribution of final worker numbers for the 2 obtained clusters. Cluster one corresponding to lower mean value of worker final and left shifted in the frequency distribution is the desired cluster. The arrows indicate parameters whose values can be neither too big nor too small. (d) Clusters in cheater worker phase space at the end of time¼ 100 000, showing respective centroids (black circles). Green dots represent final species numbers for cluster 1 (desired cluster), blue dots represent final species numbers for cluster 2 (undesired cluster). (e) Sensitivity of the model to the various parameters and initial species numbers. Arrows indicate parameters whose values may not be too high or too low. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

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clusters was on the basis of simulation results but the generation of sub-clusters, for a given case, was on the basis of distance from reference cluster centroids. If a list of reference clusters over a range is readily available, sub-cluster generation becomes a quick process. For validation, only 1 more simulation would be subsequently necessary.

4. Discussion We have undertaken a control system's design approach towards obtaining desired outcome with a specialized type of cheater–antibiotic therapy. This involves simulating a growth model, calculating sensitivities and obtaining parameter values

giving an objective outcome. Stability analyses are generally utilized in control systems design for this purpose. Our approach is novel in its use of parameter value clustering for predicting outcome. The method is based on simulation results at a fixed time-point, regardless of where stability is attained. This is because of feasible time limit within which therapy goals must be attained and made possible by the fact that once the burden of infection is lowered below a certain threshold the body immune system is capable of eliminating it (Way et al., 1998). In control systems design, sensitivity is given by derivative of rates with respect to each parameter keeping values of others constant. The obtained sensitivities may be biased by where those values are fixed. We used clustering solutions to determine sensitivities, where all parameter value combinations were allowed. In addition to its already established uses in preventing collateral damage from antibiotics, probiotics may be used as cheaters to break down cooperative antibiotic resistance, given that the probiotic organism is itself susceptible to the antibiotic in question. This would be probiotic use in direct facilitation of antibiotic action. A probiotic–antibiotic combination therapy is made possible by the ability of probiotics to competitively exclude pathogens from its habitat, e.g. by preventing pathogen adhesion. Probiotics are by no means the only classes of cells that may be used as cheaters. The model itself is general enough to incorporate all types of cheater cells that may be used for antibiotic resistance reversal. However it is to be noted that compared to engineered cheater species, naturally occurring probiotic species are more widely available and extensively studied at present. In addition the selective growth boosting action of prebiotics, which is also widely studied, may be used to give probiotic organisms an obvious competitive advantage. Hence parameter values used for simulations of the model were approximately around those for probiotics, as obtained from the existing literature. A more extensive meta-analytical review would reveal the exact ranges to be covered. The only limitation that must be mentioned is that the use of probiotics as cheaters is only possible for infection sites that are topologically on the exterior and naturally harbor microbiota, e.g. mouth, gut, vaginal tract, and skin. The proposed model describes a wide class of cheater–worker– antibiotic interactions, but it is not meant as a stand-alone. Rather it is viewed as a general birth–death–help framework consisting of modular components which may be altered, depending on the case. It must be mentioned that this model would be valid for a wider class of producers with only a small modification in the birth–death terms, e.g. if the pathogen produces virulence factors (Diggle, 2010; Brown et al., 2009) that help in its growth, the pay-off would directly add to the specific growth rate instead of reducing the deathrate. The payoff function may be some other type of saturation function, e.g. a distorted step function which attains its maximum value at a particular worker number. Our choice, the continuous Heaviside function is normally employed for describing co-operative behaviour with a saturation level indicating the maximum benefit that may be derived, e.g. from cooperation public sector enterprises. The function has been used in different fields, one significant application being in co-operative enzyme behaviour (Qian, 2012). The analogy model in this case may be appropriate since co-operativity is very much contextual in the cheater worker model, phenomenon like quorum sensing often being active in such processes (quorum sensing itself shows a sigmoidal response to microbial density). The specific death rates may be expressed as a time-dependent parameter if the bioavailability of antibiotic at the infection site is time dependent (e.g. daily bolus doses). It is to be noted that parameter values in a real environment are often associated with temporal fluctuations. Hence some parameters may be expressed in the Langevin form. We have modelled the rate of mutation of cheaters into workers as proportional to the number of cheaters only. This would be

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the case for spontaneous mutations or when resistance genes are picked up from DNA in the environment. However for gene transfer by conjugative processes from nearby workers (Nogueira et al., 2009; Mc Ginty et al., 2013), the rate of cheater conversion would be proportional to numbers of both worker and cheater. The effect of such processes may be explored by similar model simulation and clustering. Selection pressures on non-social traits have been shown to reduce the success of cheater infiltration into cooperative worker population (Morgan et al., 2012). This effect has been explained in terms of low cheater numbers. In this case we are at liberty to control the number of cheaters supplied in cheat therapy. Hence by including the effects of sweeping selection (e.g. the flushing of urinary tract by urine at frequent intervals may be modelled as a discontinuous constant negative term added to rate equations for both species) and consequent adaptive mutations into the basic growth equations, followed by optimization as before, we may obtain the cheater doses to be applied. Otherwise cheats may be supplied which are already engineered to withstand the effects of any sweeping selection. Regulatory control of cooperative gene expression has been shown to decrease the efficiency of cheat infiltration into bacterial societies (Kummerli and Brown, 2010; Xavier et al., 2011). Regulatory control may not be an issue here since therapy is planned alongside concomitant use of the positive regulatory signal, i.e. antibiotics. However regulation may play a role when there are fluctuations in antibiotic levels. Its effects will be to decrease cost, C, when antibiotic levels are below a threshold. Thus in these cases cost should be expressed as a function of antibiotic concentration. Optimization by clustering may be done after incorporating such a cost function into the model. The use of clustering in two steps ensures that time taking simulation step is avoided while actually determining the prescription. This would make the system amenable for development into a readily useable toolbox, e.g. in the form of a smartphone App. The second step of clustering is performed instead of simply choosing the single point that is nearest to the desired reference cluster centroid because the nearest point might not be feasible in a clinical scenario (e.g. too high antibiotic dose is not possible if the antibiotic has toxic side effects). A defined cluster would instead provide the clinician a range of choices to decide from. Even though the paper is developed for a clinical scenario, it may be applicable to various other situations for example in biological pest control or other situations involving a cooperative invader. On the other hand the modelling and clustering approach may also be important for microbial chelation of metals from ores or industrial reclamation where the active microbe species is cooperative, i.e. the worker–cheater ratio has to be increased. To conclude, even though the use of cheater cells for therapy has been questioned because of inefficient cheat infiltration under various conditions, the problems may be overcome by treatment regimen optimization. An already available form of cheater cells may be probiotics, provided an informed choice is made regarding their prescription.

Acknowledgement Ms. Sanhita Ray was provided CSIR-India (NET) fellowship Sanction No.: 09/028(0875).

Appendix A. Supplementary data Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.jtbi.2014.07.019.

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