Modelling of filamentous phage-induced antibiotic tolerance of P.aeruginosa

Filamentous molecules tend to spontaneously assemble into liquid crystalline droplets with a tactoid morphology in the environments with the high concentration on non-adsorbing molecules. Tactoids of filamentous Pf bacteriophage, such as those produced by Pseudomonas aeruginosa, have been linked with increased antibiotic tolerance. We modelled this system and show that tactoids, composed of filamentous Pf virions, can lead to antibiotic tolerance by acting as an adsorptive diffusion barrier. The continuum model, reminiscent of descriptions of reactive diffusion in porous media, has been solved numerically and good agreement was found with the analytical results, obtained using a homogenisation approach. We find that the formation of tactoids significantly increases antibiotic diffusion times leading to stronger antibiotic resistance.

Polymicrobial biofilms, where microcolonies of bacteria form structured communities 2 surrounded by a self-produced extracellular polymeric substances (EPS), are increasingly 3 linked to infections and to providing an environment that encourages bacterial survival 4 and persistence. Bacterial survival in biofilms is supported by parameters such as lower 5 metabolic rates, stress responses, decreased nutrient diffusion and low oxygen gradients. 6 Recent work [1][2][3][4] on structuring and organisation of such systems has found that 7 biofilms can exhibit order comparable to liquid crystals. The formation of the liquid 8 crystalline order was also detected in other biological materials including cell 9 tissue [5][6][7] and cellulose [8][9][10]. While, in general, the mathematical modelling of 10 phage (Pf4), a phase separation occurs, with phages forming a nematic liquid crystalline 18 phase consisting of droplets with a rugby ball shape, called tactoids (Fig 1), surrounded 19 by an isotropic phase predominantly consisting of the non-adsorbing polymers. 20 Although the tactoid morphology has not been directly observed in biofilms, nematic 21 structure has [1]. Due to their uniform properties such as length, diameter, and charge, 22 filamentous phages (with or without non-adsorbing polymers) can provide an 23 experimental model system to study the physics of liquid crystal formation and 24 depletion attraction, which has been described previously [18][19][20]. 25 The Pf4 phage (genus Inovirus) is approximately 2 µm in length and 6 -7 nm in 26 diameter. It is a long, negatively charged filament with more uniformity than synthetic 27 filamentous nanoparticles. P. aeruginosa itself is an important bacterial pathogen, 28 causing a wide range of infections including wound and burn infections, pneumonia, 29 urinary tract infections, and it is a dominant pathogen in cystic fibrosis airway 30 infections. P. aeruginosa can readily form biofilms, which are often characterized by a 31 highly mucoid EPS containing polysaccharides, proteins, lipids and DNA. Further 32 research has shown that Pf virions encapsulate P. aeruginosa cells [2]. 33 The production of Pf phages by P. aeruginosa is stimulated by high viscosity 34 environments [21], anoxic conditions [22], and oxidative stress [23], conditions that 35 mimic those found at infection sites. Pf bacteriophages are often found in cystic fibrosis 36 sputum samples [24] and have a direct role in biofilm formation [25,26]. Therefore, the 37 behaviour of these phages in the presence of polymers (representing those which could 38 be found in the EPS) offer a model to start understanding how Pf virions in 39 environments that facilitate liquid crystal formation could impact tolerance and 40 antibiotic diffusion. 41 There is growing concern over the increasing prevalence of antibiotic resistance and 42 tolerance. Resistance requires genetic mutation, while tolerance relates to an increased 43 capability to be able to withstand antibiotic exposure temporarily, for example through 44 the effects of decreased diffusion caused by the EPS. In P. aeruginosa biofilms, liquid 45 crystal formation by Pf virions leads to increased antibiotic tolerance [1]. The ability of 46 Pf virions to encapsulate P. aeruginosa cells suggests that a possible cause of the 47 increased tolerance is this protective phage barrier [2]. It has been proposed that 48 binding of cationic antibiotics by anionic Pf virions (observed by Janmey (2014) [27] 49 and Secor et al. (2015) [1]) in a liquid crystalline state plays a central role in mediating 50 antibiotic tolerance [1]. On the other hand, Pf virions do not provide protection from 51 uncharged antibiotics such as ciprofloxacin, which is not bound by Pf4 phage [28].

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The main focus of our work is to determine whether tactoids can cause antibiotic 53 tolerance by acting as an adsorptive diffusion barrier without any further biological 54 mechanisms. We approach this problem from a physical and mathematical angle by 55 modelling diffusion and adsorption of antibiotics in a tactoid. The model presented here 56 describes the diffusion of antibiotics using continuum equations for diffusion in a 57 domain perforated by a regular lattice of filamentous phages, with adsorption at the 58 phage boundary; this domain represents the tactoid (see Fig 1). It is solved numerically 59 and compared against an analytical approximation, which is derived using the principle 60 of homogenisation [29]. In homogenisation, scale separation and local periodicity at a 61 microscopic scale lead to effective macroscopic equations. The technique is well 62 developed in application to porous media [29][30][31][32][33][34][35][36], where it leads to estimating effective 63 diffusion coefficients and time scales. For instance, homogenisation has been used to 64 model adsorption-induced blockage of porous filters [37] and obstructed diffusion in antibiotics in tactoids [1]. Since this coincides with tactoids forming an effective barrier 70 against such type of antibiotics, we predict that antibiotics will agglomerate in the outer 71 tactoid layer before diffusive equilibrium sets in.

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Diffusion of antibiotics through a single tactoid encapsulating a bacterium was captured 74 in a mathematical model and expressed through a system of dynamical equations.

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These equations were solved numerically in Comsol, to assess the influence of the 76 tactoid on the diffusion time. The mathematical model, and its numerical 77 implementation, are discussed in the following paragraphs.

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Description of the model 79 We describe the diffusion and adsorption of antibiotics by a system of equations analogous to diffusion in porous media, as described by Allaire et al. (2010) [29]:

Numerical implementation 97
Modelling an entire tactoid in three dimensions is computationally expensive. However, 98 it is not necessary to do so, because a simpler modelling domain naturally follows from 99 the geometry of the tactoid. The phages are very long, aligned with the bacterium, and 100 have a diameter much smaller than that of the bacterium. The first two properties 101 imply that antibiotic diffusion is only significant across the phages and not along them, 102 thus reducing the problem from three to two dimensions. The last property allows us 103 the neglect the curvature of the bacterium and to consider only a thin section of the 104 tactoid (shaded in yellow in Fig 1) with periodic horizontal boundary conditions, so that 105 the tactoid slice is effectively one layer in a vertical stack of identical slices. In summary, 106 we can model the tactoid as a rectangular domain of length equal to the tactoid 107 thickness, as shown in Fig 2. We assume that the phages form a regular lattice in the 108 tactoid and choose the height of the rectangle to be the period of this lattice. This is a 109 standard assumption in models of diffusion in porous media, made for numerical 110 convenience [39,40]. The antibiotics come in from the right, with the condition that

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The diffusion of the antibiotic tobramycin through an adsorptive tactoid layer was 118 modelled, and the timescale at which antibiotics diffuse through to the bacterium was 119 estimated as t 90 , the time at which the free antibiotic concentration at the bacterium 120 boundary is 90% of the equilibrium concentration inside the domain. The modelling 121 parameters, summarised in Table 1, were estimated from the experimental conditions 122 in [1] and [2], and the equilibration times were compared with their experimental results. 123 A full account of the parameter estimation is given in S2 Appendix.

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Since experiments show that the adsorptive capacity of the phages is different in the 125 isotropic and nematic phases [1], the equilibration time was determined for multiple 126 values of the equilibrium adsorption coefficient α (α I , α N and α 0 in Table 1), 127 corresponding to isotropic, nematic and zero adsorption respectively. In the last case The numerical diffusion times (see Table 2) can be compared to the analytical showing good agreement between the two. The diffusion time, as calculated from the 140 homogenised model, also agrees well with the numerical equilibration time, see Table 2. 141 To assess the agglomeration of antibiotics, the distribution of antibiotics across the 142 tactoid was modelled and plotted after 10 s, before diffusive equilibrium has set in, and 143 after 100 s, when the system has (almost completely) reached equilibrium. These results 144 are shown in Fig 4 and indicate that antibiotics agglomerate in the tactoid: at 145 equilibrium, the concentration in the tactoid is higher than outside of it. Furthermore, 146 antibiotics agglomerate in the outer tactoid layer before equilibrium has set in; this 147 effect is transient and more pronounced for stronger binding equilibrium coefficient.  Values are obtained from numerical simulation of the system of equations Eq (1) (t 90 ) and from the homogenised model (τ H D ).

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The model outlined in this paper is an extremely powerful tool to link experiments that 150 measure phage parameters, e.g. adsorption, to antibiotic resistance. The case discussed 151 in the previous section is just an example, based on parameter values estimated from 152 the results reported in [1] and [2]. However, the power of the model lies in its ability to 153 link changes in the experimental conditions to antibiotic resistance. For example,   Having analysed the effect of the parameters, we can better assess the calculated 185 diffusion times. The results in Table 2 show that encapsulation of bacteria by tactoids 186 composed of Pf virions strongly increases antibiotic diffusion time, indicating that 187 tactoids serve as an effective diffusion barrier. However, the resulting diffusion times, of 188 the order of tens of seconds to minutes, are not sufficiently long to fully explain the 189 observed antibiotic tolerance over timescales of tens of minutes to a few hours [1]. On 190 the other hand, Table 2 shows that adsorption is essential to the efficacy of the tactoids 191 as a diffusion barrier, and this agrees with the observed correlation between adsorption 192 and antibiotic resistance [1,2,28]. Furthermore, the results in Fig 4 are consistent with 193 the observation that antibiotics agglomerate in tactoids [1,2]. These results reinforce 194 the observation that the barrier effect of the tactoid directly impacts antibiotic 195 tolerance [2], and support the hypothesis that diffusion inhibition and adsorption are 196 sufficient for the barrier to be effective. Moreover, they provide a mechanism for 197 predicting the influence and effect of this tactoid barrier.

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The quantitative difference between modelled and observed tolerance timescales may 199 be explained by underestimation of parameter values, the influence of adsorption on 200 global antibiotic concentration, and the combined effect of tactoid encapsulation with 201 other tolerance mechanisms. We will discuss these in turn.

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The analysis of parameter effects earlier in this section shows that, while κ plays 203 little role in antibiotic resistance, underestimating α, the packing density, or the tactoid 204 thickness has a significant effect. We outline in S2 Appendix the procedure we followed 205 to estimate the parameter values used to obtain the data in Table 2. In all cases more 206 targeted experiment will be needed to constrain the parameters further. For example, we 207 may have underestimated the values of α, since these were derived from measurements 208 at a macroscopic scale; further experiments could elucidate how well these results carry 209 over to the scale of a tactoid. Furthermore, the authors are currently developing a 210 model to investigate why phages in the nematic tactoid phase have a stronger binding 211 capacity α than in the isotropic phase, which was observed by Secor et al. (2015) [1].

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However, the effect of α is relatively modest (see Fig 5). The remaining two parameters 213 can potentially have a much larger impact. The phage packing density is dependent on 214 the ionic strength of the environment (due to the anionic phage charge) and the 215 molecular weight of the polymers serving as depletion agents; this yields an opportunity 216 for future experiments to test the effect of packing density on antibiotic tolerance.

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It is also possible that adsorption lowers the global antibiotic concentration enough 218 for it to become non-lethal. However, to what extent this concentration is lowered 219 depends on the number of tactoids present and is therefore not straightforward to  Table 2 show that tactoids cause a slow arrival of 227 low concentrations of antibiotics, which may trigger other tolerance and resistance 228 mechanisms such as efflux, stress responses such as the SOS response to DNA damage, 229 etc. that stop bacterial growth, protecting them from antibiotics that kill replicating 230 bacteria. The slow arrival of antibiotics can also afford the bacteria temporary 231 protection while these other mechanisms are not yet active. Furthermore, one should 232 consider that in biofilms, not only the phages, but also the bacteria will aggregate, and 233 microcolonies will form [41]. This means that the bacteria in the inner layers of the 234 biofilm have more protection against antibiotics than one encapsulating tactoid can 235 afford; the effect of tactoids around surrounding bacteria may be cumulative. exists [28]. It would be of interest to verify whether it is the slow diffusion of antibiotics 241 through tactoids (causing gradual exposure of bacteria to antibiotics) which induces this 242 antibiotic resistance.
243 Furthermore, the model presented in this paper can potentially lead towards a 244 further understanding of the complex role of the biofilm EPS matrix. In applying our 245 results to in vivo biofilms, it is important to consider the experimental system on which 246 our model has been based; a mix of Pf4 phages and non-adsorbing polymers which form 247 tactoids have not been observed directly in biofilms. However, the biofilm EPS is likely 248 to contain high amounts of similar non-adsorbing polymeric substances, and indeed 249 there are additional host polymers in the system such as mucins or hyaluronan; 250 therefore, we can assume that the entire biofilm matrix is a nematic liquid crystal (as 251 suggested by Secor et al. (2015) [1]). Hence, the experimental system modelled in this 252 paper offers a simplified model that can then be used to model the complex interactions 253 between phage, bacteria, and polymers in polymicrobial biofilms where the role of the 254 EPS matrix is only beginning to be understood.

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In summary, our results indicate that tactoids can form a significant barrier against 256 antibiotics, merely through adsorption and inhibition of diffusion. This result 257 reproduces the trend observed experimentally and provides important insight into the 258 physical mechanisms. In the next step of our work, we will consider the microscopic 259 tactoid structure in a route towards more comprehensive, mathematical description of 260 liquid crystalline biofilms. 286 S1 Appendix. Homogenisation. An analytically solvable effective model was 287 developed using homogenisation. This is a mathematical technique used to describe 288 systems in which scale separation occurs, i.e. where the physics at a macroscopic scale 289 and a much smaller microscopic scale can be disentangled. Due to the microscopic 290 structure, the system generally cannot be solved analytically, and numerical modelling 291 is computationally expensive. However, in certain cases one can describe such a system 292 system with effective equations on the macroscopic scale. If the microscopic details of  [29]. The liquid crystalline alignment 299 of phages leads to a regular structure which can be considered a lattice of identical unit 300 cells: the microscopic structure to be averaged out. This process is illustrated in Fig S1 301 Fig. Homogenisation leads to the following effective equation: where D (eff) is the effective diffusion coefficient, given by: HereD is a rescaled diffusion coefficient that takes into account of the physical barrier 304 on diffusion caused by phages. The denominator, instead, represents the effect of 305 adsorption. |C| and |Γ| are the free volume between phages and phage surface in a unit 306 cell, respectively.

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This expression can be simplified in the limit of strong adsorption, i.e.

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(α|Γ|/|C|) 1, to thus highlighting that the effective diffusion coefficient is inversely proportional to the 310 equilibrium adsorption coefficient. inside a tactoid [2]. 317 We take the diffusing antibiotics to be tobramycin, for which the greatest amount of 318 data is available; its diffusion coefficient in an aqueous medium is D = 15 µm 2 /s.
which, assuming that τ D and τ κ are of the same order, means κ 1/τ D = 1.7 × 10 6 s −1 . 324 Since κ is the adsorption rate, its value has no influence on the diffusion time, as long as 325 κ 1s −1 . This is confirmed by the fact that κ does not appear in equation (3). Hence, 326 any large value for κ should give identical results.

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A total antibiotic concentration of 200 µg/ml is used in the experiments of Secor et 328 al. (2015) [1] which are comparable to the model presented in this paper. However, 329 since only the relative antibiotic concentration is measured with respect to the average, 330 u and v can be kept dimensionless. Furthermore, we assume that antibiotic adsorption 331 has a negligible effect on the antibiotic concentration in the extracellular matrix 332 surrounding the tactoid. Hence, we take the free antibiotic concentration at the outer 333 tactoid edge to be u = 1.