Exploiting ecology in drug pulse sequences in favour of population reduction
To quantify pulse efficiency , we primarily study the minimal population size nmin of our two-species system as a proxy for the extinction probability of the population ., Antibiotic stewardship programmes suggest that for some diseases , such as pneumonia , the immune system can clear the residual infection once the bacterial population size is sufficiently reduced 19 , 20 ., Thus , the minimal population size may be a more relevant parameter than the exact extinction probability itself ., Additionally , the general behaviour of the deterministic system and its observable nmin is more robust than the extinction probability in the stochastic model , as in the latter , the precise form of stochastic noise , or the system size , would be important ., The total population minimum nmin can still serve to gauge the latter , which scales as exp ( nmin ) 21 ., Before introducing our model in detail , we jump ahead and summarise the essential result of our work in Fig 1 , which we discuss later in more detail ., Fig 1 shows the value of the minimal total population size in the configuration space of drug concentration profiles , spanned by the width of the pulse ( or duration of its high stress environment ) on the x-axis and the form of the pulse on the y-axis , which will be explained later ., Practically , these two properties of a pulse—its high stress duration and its form—are likely constrained: a very long duration of the high stress environment or stronger drug might be detrimental for patients due to , for example , a destructive impact on the gut microbiome 22 , 23 ., Similarly , some pulse forms , such as those where the highest possible drug concentration suddenly drops to zero at the pathogen location at the end of the pulse ( here denoted by temporal skewness s = 1 ) , may not be realistic for clinical treatments ., However , since we do not want to make any assumptions on which parts of the configuration space should be accessible , we examine our system for all possible combinations of pulse form and durations of the high stress environments ., The colour code ( symbols ) signifies which of the four possible pulse sequences sketched on the right of Fig 1 most effectively reduces the population size ., Fig 1 clearly shows that in our simple model setup , different pulse sequences are favourable in different regions of configuration space ., The aim of this work is to outline phenomenologically which pulse sequence yields the lowest minimal population for which part of configuration space in Fig 1 , and might therefore be most likely to drive the species to extinction ., The best pulse sequence at any one point of Fig 1 tends to be the one that maximally exploits the competition between the more resistant and wild-type species , represented by logistic growth in our model ., The simplicity of our approach makes explicit why some references might argue for more moderate treatments involving e ., g ., shorter or lower drug concentrations , but also what the limitation of models and observables are and hence why such moderate treatments may not work in real setups ., We also examine how the population composition ( a measure of how strongly the more resistant species dominates the population ) evolves , should such a pulse sequence not lead to achieving extinction ., Finally , we highlight the need for microbial experiments in such temporally varying drug gradients , in order to evaluate the applicability of simple models to real systems ., The simplest model that can be used to study the effect of the temporal concentration profile on a heterogeneous population ( n ) consists of two phenotypically different species , a susceptible “wild type” species ( w ) , and a more tolerant or resistant species ( r ) ., Its increased resistance comes at the cost of a reduced fitness in the drug-free environment , which is reflected in a smaller growth rate ., As in previous works 24 , 25 , we assume that the drug is bacteriostatic , that is , it only affects growth , such that growth of each species ceases as soon as its minimum inhibitory concentration ( MIC ) is exceeded ., Thus , in this deterministic population dynamics model for the birth-death process , sketched in the inset of Fig 2 , the growth rate of each species η ∈ {r , w} is given by ϕη ( t , n ( t ) ) = ΘMICη − c ( t ) λη ( 1 − n ( t ) ) , where n ( t ) = w ( t ) + r ( t ) is the total number of species at time t expressed in terms of a carrying capacity , which does not require specification as it serves merely as a unit for the population size ., The Heaviside-Theta step function Θ implies that the growth rate is only non-zero when the drug concentration is lower than the MIC of the corresponding species ., The index η ∈ {r , w} refers to the type of species ( resistant or wild-type ) , and λη is its growth rate ., The more resistant species has a lower basal growth rate in the drug-free environment , i . e . , λr = λw − k≔ λ − k , where k > 0 can be interpreted as a cost that the resistant species incurs for being more resistant ., The logistic growth assumed in this model introduces competition between the wild-type and the resistant species for limited space and/or resources , and places an upper bound on the population size ., We also include a constant death rate δ for both species , meaning that a species decays at rate δ when c ( t ) >MICη: For these higher concentrations , growth of species η is inhibited and , since switching is negligible , the species can only die ., All rates and times in this work are given in units of λ ., The time evolution of the population can be studied in terms of the differential equations, w ˙ ( t ) =  ϕ w ( t , n ( t ) ) - δ - μ w  w ( t ) + μ r r ( t ) r ˙ ( t ) = μ w w ( t ) +  ϕ r ( t , n ( t ) ) - δ - μ r  r ( t ) ( 1 ) since for sufficiently large populations stochastic fluctuations can be neglected ., The two species are coupled via the competition from logistic growth , as well as via the switching rates μw and μr ., Phenotypically more resistant states can be characterised by a reduced growth rate , or complete growth arrest , often known as tolerance or persistence 26–28 ( for a recent review , see Ref . 29 ) ., Provided that μw , r ≪ δ , which is the case for both mutation and phenotypic switching , our choice of μw = 10−6 λ and μr = 0 does not qualitatively affect the results ., For this entire work , we used exemplary values of δ = 0 . 1λ and k = 0 . 1λ , where λ ≡ 1 , i ., e ., we used λ as the basic unit of time ., We investigate several other combinations of costs and death rates , in particular combinations with the same death rate , but a smaller and larger cost , in S1 Text ., There , we show that our results and general statements are still valid for these cases ., We chose the values of δ = 0 . 1 and k = 0 . 1 since this combination allowed us to show the complete and most general picture of possible best pulse shapes in Fig, 1 . A smaller ( yet also biologically possible ) fitness cost would not have contained all different scenarios ., We ask the reader to refer to S1 Text for more details ., Since in our model the only relevant information about the antibiotic concentration is whether it is above or below the MIC of the corresponding species , any pulse sequence is fully determined by the temporal arrangement of low-stress ( low ) and high-stress ( high ) environments ., In these ( low ) and ( high ) environments , the antibiotic concentration is low , MICw < c ( t ) < MICr , or high , c ( t ) > MICr , respectively ( sketched for a single pulse in the top panel of Fig 2 ) ., Before the pulse sequence , the system is in the drug-free environment ( free ) , where the concentration of the antibiotic c ( t ) is less than either MIC , c ( t ) < MICw , r ., We assume that the ( free ) environment appears only before , but not during , a pulse sequence ., Thus , the ( free ) environment determines the initial condition of the population , which we take to be at its fixed point , ( w ( t = 0 ) , r ( t = 0 ) ) = ( w ( free ) * , r ( free ) * ) , shown as the purple dot near the w-axis of the phase space panel ( free ) of Fig, 2 . The change in population size and composition in each of these environments is characterised by the flow field in phase space ( w , r ) , shown in the three lower panels of Fig, 2 . In the ( low ) environment , the population flows towards the more resistant species ( high r and low w ) , while in the ( high ) environment , it flows towards the origin , meaning that both species die out exponentially ., Thus , the effect of a single pulse on the population crucially depends on the times spent by the system in the ( low ) and ( high ) environments ., A single pulse involves a single ( connected ) environment of ( high ) antibiotic exposure , with ( low ) environment potentially preceding or succeeding this ( high ) environment ., In reality , the duration of these ( low ) environments will depend on the experimental setup or host ., A pulse sequence is composed of a succession of identical single pulses ., We refer to the total time of the pulse as τ , and the time during which the system is in the ( high ) environment as tr ., The time periods during which the system is in the ( low ) environment ( initially ) before tr and ( finally ) after tr are denoted by t w ( i ) and t w ( f ) , respectively ., As this would overparameterise the pulse , we combine the latter two time scales into a skewness parameter s = ( t w ( i ) - t w ( f ) ) / ( τ - t r ) , signifying how tr is positioned within τ ., Skewness s = − 1 ( s = 1 ) thus denotes a pulse which starts ( ends ) with the ( high ) environment , while skewness s = 0 denotes a symmetric pulse ., We compared pulse sequences of up to N = 4 pulses ( same tr and, s ) for constant treatment time τ for all possible skewnesses s and durations tr ., Thus , a single pulse with τ = 60 and given tr and s is compared with a sequence of N identical pulses , each defined by τ ( N ) = 60/N and t r ( N ) = t r / N and s ., ( Further values of τ are discussed in S1 Text ) ., The retention of the same skewness within a sequence is motivated by the fact that we assume that the rate of increase or decrease in concentration is primarily determined by the host system of the bacteria ., In this comparison , the ‘best’ pulse sequence for given ( tr ,, s ) is defined as the one that yields the lowest population minimum nmin and so has the highest likelihood of eliminating the pathogen ., In situations where the entire configuration space is accessible , the maximal tr yields the overall lowest population minimum , independent of skewness s ., Since practically the maximal duration tr acceptable for treatments may be limited , it is important to know which pulse sequence is best for each ( tr ,, s ) , such that we can provide intuition on any situation and parameter choice that may arise ., The colour ( and corresponding symbols ) in Fig 1 show the best pulse sequences ( i . e . the best N ) , and the shade indicates the value of nmin ( dark denotes high values ) ., We found that a single pulse is most effective over a large range of parameters ( blue in Fig 1 ) ., In particular , for each duration in the ( high ) environment tr , the lowest minimum across all skewnesses is obtained by a single pulse ( blue line ) ., This means that in practical situations which allow all different pulse skewnesses , a single pulse with a skewness on the blue line would give the lowest minimum ., If , however , the possible pulse skewness is limited due to the host setup , a single pulse may not be the best choice ., For ease of comparison , Fig 3 a shows nmin for just a single pulse of constant treatment time τ = 60 , with the white line marking the lowest minimum ( the blue line in Fig 1 ) ., In the next paragraph we focus on a single pulse in order to understand which pulse parameters ( s , tr ) yield this lowest minimum ., In the previous section , we learnt which pulse sequences yield the maximal relative reduction in population size for which regions in ( tr ,, s ) -space ., This minimal population nmin served as a proxy for gauging when extinction would most likely occur in a setting where an immune response can destroy the population when it is already small ., Now , we would like to address a complementary question: in the event that extinction does not occur , whether because nmin was too high , or the population was small for too short , what is the effect of such a ‘failed’ pulse on the bacterial population ?, We already saw that the composition of the population shifts more towards r with each pulse ., In terms of real treatments , it might often be better not to pursue treatments which , if unsuccessful , entail a high risk of creating a fully resistant population ., In order to evaluate the pulsed treatments associated with the most effective population reductions based on Fig 1 , we now focus on the population composition , quantified by the ratio of resistant to wild-type species , r/w , at the end of the best pulse within the best sequence ., Evaluating r/w at the end of the pulse that yields the global minimum is motivated by the fact that the treatment can be stopped after , but not during , an individual pulse ., Fig 5b shows the dependence of r/w on the pulse configuration , which can be best understood by first considering how the population evolves in ( w ,, r ) phase space during the different pulse sequences ., In Fig 5a , we show trajectories for three pulse sequences , consisting of a single , two , or three pulses respectively , with τ = 60 and tr = 10 ., The qualitative behaviour of the phase space trajectory is independent of skewness ( in Fig 5a , s = 0 . 9 , the corresponding trajectories for s = −0 . 5 , s = 0 . 2 and s = 0 . 5 can be found in Fig D in S1 Text ) ., The colour of the trajectory darkens progressively with every pulse in the sequence ., The trajectory starts at the ( free ) fixed point close to the w-axis beyond the limits of Fig 5a , and evolves towards the r-axis ., Within each sequence , r/w steadily increases from pulse to pulse , as r progressively takes over the population during the ( low ) regimes ., Thus , in the top left corner of Fig 5b , where the first pulse of the sequence yields the lowest minimum , r/w is comparatively smaller ( lighter shading ) ., Indeed , the higher the N of the best sequence , the lower the ratio in Fig 5b , provided the global minimum is reached in the first pulse ( such as in the red region in Fig 1 ) ., The region marked with the white line in Fig 1 , where intermediate pulses ( and not the first pulse ) in the sequence yielded the lowest global minimum , also shows up clearly as darker in Fig 5b ., Here , r has grown more than for a single pulse , as more pulses were applied before the population minimum was reached ., Thus , in our model , when both population reduction and composition are considered , pulse sequences where the minimum is attained in the first pulse are generally more effective than a single long pulse: maintaining the ( low ) regime in the first pulse for around to keeps r/w as well as nmin small ., This argument suggests that treating with this first pulse only achieves the best result , and additionally comes with a shorter total treatment duration τ and a shorter tr ., We would like to note that even if the population does not die out during this short treatment , multiple pulses of this form could be added in order to give the immune system more opportunities to eliminate the infection ., These additional pulses would not drastically change r/w compared to the composition obtained after the single long pulse of τ = 60 ., This can be seen also in Fig 5a , where for all pulse sequences the population composition is similar at the end of the entire sequence ., Experiments with microbes can help investigate minimal antibiotic dosages and treatment times in a well-controlled test tube setup , where the impact of certain treatments on the microbial species itself can be studied without interfering effects , for example from the immune system ., Such microbial experiments have , for example , helped suggest drug combinations or treatment regimens which could retard the development of antibiotic resistance 30–33 ., Increasingly , these experiments try to incorporate practically important aspects of heterogeneities in the environment 34 , such as drug concentration gradients ., These gradients can enhance the development of bacterial resistance relative to spatially homogeneous systems 24 , 25 , 35 , 36 , as the more resistant species can successfully compete with a faster growing , but more susceptible wild-type species ., Not enhancing the selective advantage of the more resistant species , in the context of temporal heterogeneities in drug levels , including the duration , frequency and even the concentration profile during a single antibiotic pulse , as studied also in this work , is also important in real treatments 7 , 37 , and is thus within the limits of current experiments ., Our model makes two drastic simplifications compared to real microbial species ., First , we study only two species , instead of a series of possible phenotypically or genotypically different species ., Typically , the evolutionary pathway that leads to a fully resistant species involves a variety of intermediate mutants , even when the mutational paths are constrained 38 ., Since the fitness benefit diminishes with each successive mutation in a series 39 , 40 , we assumed that the strongest effect is conferred by the first mutation , and neglected all higher order mutants ., For phenotypic switches , it is reasonable to consider only two species , corresponding to , for example , the expression or repression of a protein 41 , 42 ., Thus , our model should be applicable to experimental systems , while in real patients , different types of tolerant or persister cells might be involved 43 , or even interact 44 ., The second simplification concerns how these two species are affected by the antibiotic ., In our model , we assume that the antibiotic is bacteriostatic , i . e . only affects the growth of the species 24 , 25 ., We also assume that the growth rate of each species falls abruptly to zero when the antibiotic concentration is higher than their respective minimum inhibitory concentration ( MIC ) ( see e . g . 45 ) ., The experimental situation is more complex: cessation of growth is not instantaneous , the space occupied by a dead cell may not immediately become available 42 , and the general use of the MIC as an indicator for slow growth is questionable 46 ., However , an abrupt change in growth rate at MIC has been verified experimentally for E . coli and chloramphenicol ., 25 ., Additionally , our analysis is based on large numbers rather than extinction events which would be model specific; thus , small changes in the model ( such as reduced but non-zero growth rate ) should still give qualitatively similar results ., Evaluating the effect of different pulse sequences should be possible within a microfluidics setup , where , for example , periodically fluctuating environments have already been investigated for E . coli and tetracycline 42 ., We expect that one should be able to observe that the ( low ) environment of drug concentration can be exploited in order to increase extinction probabilty for a ( high ) environment that is present for as short as possible , with the treatment time being constant ., How long this duration of the ( low ) environment is for best exploitation would be sensitive to the growth rate of the more resistant species , which for tetracycline could be generated using a specific promoter , namely the agn43 promotor 42 , 47 , 48 ., Just as shown in Fig 1 , we expect higher N pulse sequences to do better when this duration is optimal for them , but not for the longer pulse ., In addition , further study of E . coli in combination with other antibiotics and more resistant strains should also show this , in addition to being more realistic than our simple model .
A deterministic population dynamics model involving birth and death for a two-species system , comprising a wild-type and more resistant species competing via logistic growth , is subjected to two distinct stress environments designed to mimic those that would typically be induced by temporal variation in the concentration of a drug ( antibiotic or chemotherapeutic ) as it permeates through the population and is progressively degraded ., Different treatment regimes , involving single or periodical doses , are evaluated in terms of the minimal population size ( a measure of the extinction probability ) , and the population composition ( a measure of the selection pressure for resistance or tolerance during the treatment ) ., We show that there exist timescales over which the low-stress regime is as effective as the high-stress regime , due to the competition between the two species ., For multiple periodic treatments , competition can ensure that the minimal population size is attained during the first pulse when the high-stress regime is short , which implies that a single short pulse can be more effective than a more protracted regime ., Our results suggest that when the duration of the high-stress environment is restricted , a treatment with one or multiple shorter pulses can produce better outcomes than a single long treatment ., If ecological competition is to be exploited for treatments , it is crucial to determine these timescales , and estimate for the minimal population threshold that suffices for extinction ., These parameters can be quantified by experiment .
The possibilities of lower antibiotic dosages and treatment times , as demanded by antibiotic stewardship programmes have been investigated with complex mathematical models to account for , for example , the presence of an immune host ., At the same time , microbial experiments are getting better at mimicking real setups , such as those where the drug gradually permeates in and out of the region with the infectious population ., Our work systematically discusses an extremely simple and thus conceptually easy model for an infectious two species system ( one wild-type and one more resistant population ) , interacting via logistic growth , subject to low and high stress environments ., In this model , well-defined timescales exist during which the low stress environment is as efficient in reducing the population as the high stress environment ., We explain which temporal patterns of low and high stress , corresponding to sequences of drug treatments , lead to the best population reduction for a variety of durations of high stress within a constant long low stress environment ., The complexity of the spectrum of best treatments merits further experimental investigation , which could help clarify the relevant timescales ., This could then give useful feedback towards the more complex models of the medical community .
antimicrobials, medicine and health sciences, ecology and environmental sciences, drugs, immunology, microbiology, antibiotic resistance, probability distribution, mathematics, pharmaceutics, antibiotics, pharmacology, population biology, skewness, research and analysis methods, conservation biology, sequence analysis, antimicrobial resistance, bioinformatics, probability theory, immune system, conservation science, population metrics, species extinction, population size, database and informatics methods, microbial control, biology and life sciences, physical sciences, drug therapy, evolutionary biology, evolutionary processes