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Simulation of the Dynamics of Bacterial Quorum Sensing Anastasios I. Psarras and Ioannis G. Karafyllidis* Abstract—Quorum sensing (QS) is a signaling mechanism that pathogenic bacteria use to communicate and synchronize the production of exofactors to attack their hosts. Understanding and controlling QS is an important step towards a possible solution to the growing problem of antibiotic resistance. QS is a cooperative effort of a bacterial population in which some of the bacteria do not participate. This phenomenon is usually studied using game theory and the non-participating bacteria are modeled as cheaters that exploit the production of common goods (exofactors) by other bacteria. Here, we take a different approach to study the QS dynamics of a growing bacterial population. We model the bacterial population as a growing graph and use spectral graph theory to compute the evolution of its synchronizability. We also treat each bacterium as a source of signaling molecules and use the diffusion equation to compute the signaling molecule distribution. We formulate a cost function based on Lagrangian dynamics that combines the time-like synchronization with the space-like diffusion of signaling molecules. Our results show that the presence of non-participating bacteria improves the homogeneity of the signaling molecule distribution preventing thus an early onset of exofactor production and has a positive effect on the optimization of QS signaling and on attack synchronization. Index Terms—Bacterial signaling, quorum sensing, systems biology.
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1 INTRODUCTION
A
NTIBIOTIC resistance is the resistance of a bacterium to an antibiotic to which it used to be sensitive. Antibiotic resistance is a growing problem and is due to the fact that antibiotics usually target the metabolism and reproduction bacterial circuits, triggering thus a reaction that produces bacteria strains on which the antibiotic is ineffective [1], [2], [3]. The increasing antibiotic resistance calls for a different approach to antimicrobial treatments. Bacterial signaling and especially quorum sensing (QS) is a possible target for such treatments that will control and reduce bacterial virulence [4], [5]. QS is a signaling mechanism that bacteria use to communicate during the infection process [6], [7]. During infection, bacteria produce and secret virulence exofactors to cause degradation of host cells [8]. Exofactors are very expensive to produce and if bacteria synthesize and secret them before they reach a critical number, the exofactor concentration will not be sufficient enough to cause degradation of host cells, leading thus to an inefficient attack. To count their number, bacteria produce and secret smaller and less expensive molecules, called autoinducers and monitor their concentration in the environment. When the bacterial population reaches the critical number, the autoinducer concentration reaches a critical value and is sensed by the bacteria, which all start to produce and secrete exofactors simultaneously, forming thus an infection “quorum” [9]. QS process is not related to metabolism or reproduction and offers an alternative target for controlling bacterial virulence using QS inhibitors [4], [5].
In infections, during the bacterial population growth phase, mutations occur in two of the most important genes related to QS. These mutations result in the appearance of signal-blind and signal-negative bacteria[10]. Signal-blind mutants produce autoinducer signaling molecules but are unable to respond to the QS signal and do not produce exofactors. Signal-negative mutants do not produce signaling autoinducers but respond to the QS signal and produce exofactors [11]. If we want to understand the QS mechanism we have to study the effect of these mutations on the dynamics of QS and find out if these mutations can be exploited to control bacterial virulence. The mutation effect on QS has been analysed using game theory techniques with Pseudomonas aeruginosa (P. aeruginosa) as the model organism [10], [11], [12], [13]. P. aeruginosa is an opportunistic human pathogen and is a major source of hospital-acquired infections [14]. In this paper we continue our previous work on QS modeling [15 - 17]. We model the bacterial population as a growing graph and compute the evolving spectrum of the graph Laplacian. Graph spectra is a valuable tool for studying the synchronization of processes that take place on a graph [18], [19]. We also study the evolution of the autoinducer concentration distribution using the diffusion equation. We compare the autoinducer concentrations in cases where mutations are present with cases where no mutations occur during the growth of bacterial populations. We formulate a Lagrangian-type cost function in which the syncronizability is the time-like part and the autoinducer concentration is the space-like part. Simulations showed that this function has at least two maxi———————————————— ma, which we interpret as the combination of mutation Both authors are with the Department of Electrical and Computer Engineering, Democritus University of Thrace,Kimmeria Campus, 67100 Xan- rate and synchronizability, that leads to the most effecthi, Greece * Corresponding author. E-mail: [email protected] 1536-1241 (c) 2015 IEEE. Personalxxxx-xxxx/0x/$xx.00 use is permitted,©but republication/redistribution requires IEEE permission. See 200x IEEE Published by the IEEE http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
1
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tive attack with the minimum possible cost for the bacterial population. We conclude that the presence of mutants in growing bacterial populations results in a more homogenous autoinducer distribution while the synchronizability of the bacterial network is not reduced significantly by the mutants. Our results provide a strong indication that mutations enhance the effectiveness of bacterial attacks on their hosts. Therefore, the infection virulence may be reduced by controlling the number of mutants in a growing bacterial population.
2 QUORUM SENSING CIRCUIT AND NETWORK The structure of the QS signaling circuit is shown in Fig. 1. When the signal production gene gI is expressed the protein PI is produced. PI produces the autoinduction signaling molecule A, which is secreted continuously into the environment. The concentration of the autoinducer molecules, A, reflects the size of the bacterial population.
where a mutation in the signal production gene gI renders it ineffective, the bacterium cannot produce the signaling molecule. These bacteria are called signal-negative mutants. Signal-negative mutants do not produce the QS signal, but respond to the signal and produce exofactors. In the case where a mutation renders the signal response gene gR ineffective, the bacterium produces the signaling molecule with a small rate, because the autoinduction loop is broken. Furthermore, it cannot respond to the QS signal and does not produce exofactors [3], [11], [20]. Signal-negative and signal-blind mutants are found in natural bacterial populations with the signal-blind mutant more common [3], [11].
Fig. 2. Bacterial population is modelled as a growing network. (a) The network of the population at times t1 and t2>t1 when no mutations occur in the QS genes. (b) The same network with signal-blind and signal-negative subpopulations. These subpopulations are represented by growing yellow “holes” in the growing network. Fig. 1. The structure of the QS signaling circuit. The names of the las QS circuit parts of P. aeruginosa are given in parentheses.
The auotoinducer molecule concentration is continuously monitored by the bacteria. Autoinducer molecules, A, enter the bacterial cell and bind to the signal response protein PR, which is produced by the gene gR, forming an active complex, PR+. PR+ regulates a number of genes some of which produce exofactors. PR+ also binds to the promoter region of the signal synthase gene gI closing the autoinduction loop of the circuit. In the case of the las QS circuit of the model organism P. aeruginosa the signal synthase gene and protein are lasI and LasI, the signal response gene and protein are lasR and LasR and the signaling molecule, A, is the 3-oxo-C12-HSL. During the growth phase of the bacterial population and before a quorum is formed, mutations in the genes gI and gR may occur in some of the bacteria. In the case
We model the bacterial population as a growing network. The network grows with the bacterial population, as shown in Fig. 2(a), at two different times t1 and t2>t1. If signal-negative and signal-blind mutants appear in the population, they form growing subpopulations as shown in Fig. 2(b). Since the mutation probability increases with time, it is expected that mutant subpopulations will be larger in the center of the population and their size will decrease as we move from the center to the periphery of the network. These mutants do not participate in the QS signal production or their contribution is almost insignificant and their subpopulations are represented by “holes” in the growing network, as shown in Fig. 2(b).
3 SYNCHRONIZATION OF THE GROWING NETWORK In the QS process the bacteria synchronize their attack by producing and diffusing signaling molecules into the
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3
environment. The network that models the bacterial population has in its nodes diffusively-coupled entities. It is well known that the synchronization of a network of diffusively-coupled systems depends strongly on the topology of the network [21], [22], [23]. The network topology is encoded in the spectrum of its Laplacian [24], [25]. We describe the synchronization of the dynamic diffusivelycoupled bacterial network using the coupled lattice map equation [24], [26]: n ci t ci t a Li , j c j t t j 1
(1)
where, ci(t) is the autoinducer concentration in the vicinity of the ith network node. Usually in eq.(1) only the neighbouring nodes are taken into account and the index j runs over these nodes only. The coupling strength, a, is a real number and because of the network symmetry is the same for all couplings of the ith node with its neighbours. The bacterial network is said to be synchronized if:
ci t c j t 0 as t
(2)
for all i and j. Therefore, the network is synchronized if the concentration of the signaling molecules is the same in the vicinity of all the bacteria and quorum is reached if this concentration is equal to or larger than the critical value. The study of the synchronization of a static network using eqs. (1) and (2) will result in a system of n differential equations. In the case of a dynamical growing network the number of nodes and equations will increase with time. We want to study the effect of signal-blind and signal-negative mutants on the synchronization of the bacterial network. Such bacteria do not participate in the synchronization process and the corresponding nodes are deleted from the network. That is, if a bacterium located at node i at time t, undergoes a mutation in its QS genes at time t+1, the node i is deleted permanently from the network. This bacterium will multiply and new mutants will appear in its neighborhood, forming a growing “hole” in the bacterial network. The continuous addition of new nodes, the deletion of nodes and the growing holes in the network, render the analysis of its synchronization using eqs. (1) and (2) impractical. We will instead turn to the study of the spectrum of the bacterial network Laplacian, which acts as an evolution operator in eq. (1) and provides a very satisfactory description of the network synchronization [24], [25], [26]. The Laplacian of a connected network has n eigenvalues, λi , which are ordered as:
0 1 2 3
n
(3)
The most important eigenvalues are: the smallest nonzero eigenvalue, λmin, which is equal to λ2 and the maximum eigenvalue λmax, which is equal to λn. The network topology changes as the bacterial population grows and the Laplacian and its spectrum are time
dependent. The method we use to study the effect of mutations on the synchronization of the bacterial network is to simulate the growth of a bacterial population as a growing network and compute the Laplacian and its spectrum at each time step. We do these computations for two bacterial populations. The first one, which we call the control population, grows without any QS mutants. The second, the mutated population, has the same growth rate and the same initial conditions as the first one, but mutations in the QS genes occur during population growth. The mutations result in growing holes in the network and, therefore, in a different topology. The spectrums of the Laplacians of these two networks are computed and compared at each growth time step. As explained above mutations result in deletion of nodes and growing holes in the evolving mutation network. Because of this, the number of nodes and the number of eigenvalues of the control and mutated networks differ. To compare the two spectra on a common ground, we convolve their eigenvalues with a common Lorentz kernel [24], [25]:
G x i
Gm x im
i x
2
2
im x 2
(4)
(5)
2
Equations (4) and (5) are the spectral functions of the control and mutated populations respectively. The eigenvalues of the mutated population are marked with an “m” which is a symbol and not an exponent. The variable x is the same in both (4) and (5) and varies in the same range. The parameter ρ is a real number and its value is important in the plot of the spectral functions. Small values of ρ highlight the details of the spectral plot, whereas large values are used to visualize the large-scale form of the spectral plot. In our simulations we choose an intermediate value, ρ = 0.2 . Figures 3(a) and 3(b) show the time evolution of the spectra of the growing control and mutated populations, respectively. The z axis is the spectral function G(x) in Fig. 3(a) and Gm(x) in Fig. 3(b). The x axis in both figures is the x variable of (4) and (5) and the y axis gives the time steps of population growth. For each time step the spectral functions G(x) and Gm(x) are computed and plotted. The surfaces of Fig. 3 are generated by the plots of the spectral functions at all time steps. Figure 4 shows the difference of the two spectral functions. The x and y axis are the same as in Fig. 3, but the z axis is the difference G(x)-Gm(x). We performed a large number of simulations and in all of them, the structure of the spectral plots in both cases was found to be the same. The difference between the two cases, shown in Fig. 4, varies between 5% and 8%. This difference is located between x=7 and x=11.5 and is due to the difference between eigenvalues the values of which are in the middle of the spectrum.
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concentrations observed in real bacterial populations, does not have a significant effect on the QS synchronization process.
Fig. 3. Spectral plots of (a) the control population and (b) the mutated population. Fig. 5. Time variation of λmin/λmax for: (a) the control population and (b) the mutated population.
Another measure of network synchronizability is the ratio λmin/λmax [24], [26], [27]. The network synchronization condition using this measure takes the form:
min max
Fig. 4. The difference of spectral plots of the control and mutated populations: G(x)-Gm(x).
It is well known that the eiganvalues near the minimum and the maximum eigenvalue are the most important in characterizing the synchronizability of a network [24], [27]. Our simulation results showed that the presence of signal-blind and signal-negative mutants, in
(6)
where the constant α depends on the maximum Lyapunov exponent of the network described by eq.(1). Here we want to study the variation of the network syncronizability with time in the case where no QS mutations are present and in the case where QS mutations are present and compare the two cases. To obtain this we compute the functions:
W t
min t max t
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(7)
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m min t Wm t m max t
5
(8)
where t is the time step. As the bacterial population grows, the minimum and maximum eigenvalues are computed at each time step. The function W(t) gives the variation of the ratio λmin/λmax with time and is plotted in Fig. 5(a) for the case of no QS mutations and the function Wm(t), plotted in Fig. 5(b), gives the same time variation in the case where QS mutants are present in the growing population. Fig. 6 shows the difference W(t)-Wm(t). The structure of the λmin/λmax ratios and their differences were found to be the same in a large number of simulations. The time variations of the ratios in both control and mutated populations, shown in Fig. 5, are almost identical. The difference between them, shown in Fig. 6, starts form zero and increases with time after the 11th time step. The fact that the λmin/λmax ratios are almost identical and their difference is negligible shows that the presence of QS mutants in a bacterial population has an insignificant effect on the QS process. The spectral plots and the λmin/λmax ratio variations provide a strong indication that the effect of QS mutants on the QS synchronization process in a growing bacterial population is not significant. This is actually not a surprise because in all successful infections (from the bacteria point of view) signalblind and signal-negative mutants have been found [11].
2 n x, y , t 2 n x , y , t n x, y , t D t x2 y2 R xs , ys n xs , ys , t
(9)
In eq. (9) n(x,y,t) is the autoinducer concentration at a point with coordinates (x,y) at time t and D is the diffusion coefficient of autoinducer molecules. The coordinates of the distributed sources are (xs,ys) and R(xs,ys) is the autoinducer production rate. This rate is zero at points where a bacterium does not exist. R(xs,ys) is also zero at points where mutated bacteria exist.
Fig. 7. Concentration of autoinducer molecules. (a) Control and (b) mutated bacterial populations.
Fig. 6. The difference W(t) - W m(t).
4 EFFECT OF QC MUTATIONS ON THE DIFFUSION AND DISTRIBUTION OF SIGNALING MOLECULES
The autoinducer concentration is affected by the presence of signal-blind and signal-negative mutants. To study this effect we solve a diffusion equation in which the non-mutated bacteria are modeled as distributed autoinducer sources [28]: Fig. 8. Evolution of the autoinducer molecule concentration of the control population (left) and of the mutated population (right) along the x-axis for the same time steps.
In Fig. 7 the autoinducer molecule concentration of the 1536-1241 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Fig. 9. Relative sizes of the control and mutated bacteria populations (X 10,000) after the formation of a quorum.
Fig. 9 shows the relative sizes of the control and mutated bacteria populations after the formation of a quorum (activation), i.e. time step 1 corresponds to the time where a quorum is formed in both populations. At the activation the number of bacteria that participate to the quorum in the control population is about 12% of the corresponding bacteria number in the mutated population. In the next time steps the difference between the two populations increases in favor of the mutated population and then becomes almost stable. This means that the mutated population starts the exofactor production with a significantly larger quorum size, which results in a more effective infection. It should be reminded that the mutated population evolution describes real infections and that signal-blind and signal-negative bacteria are found in all infection sites.
participation of more bacteria in the quorum. It is therefore reasonable to assume that the most appropriate time for bacteria to launch their attack on their host is when the syncronizability and the participation in the quorum reach a favorable combination. To study this, we have to construct a cost function and search for extrema. Since the effects of mutations have a time-like component (synchronizability) and a space-like component (distribution of autoinducer concentration), we believe that a cost function analogous to the Lagrangian used in mechanics would be appropriate. Since the Lagrangian is the difference between kinetic and potential energies we constructed the following cost function: Lg t Q1 G xmax , t Gm xmax , t Q2 W t Wm t (10) n x, y, t nm x, y, t Q3 n x, y , t
In eq. (10), xmax is the point in Fig. 4 where the maximum value of G(x)-Gm(x) occurs. The concentrations n(x,y,t) and nm(x,y,t) are the concentrations of control and mutated populations and are normalized so that the function Lg depends only on time. The values of the weight constants Q1, Q2 and Q3, are determined using the spectral characteristics of the bacterial network and the time evolution of the concentrations, which in turn originate from the structure of the quorum sensing circuits. The values of these constants are expected to be different for bacteria types with quorum sensing circuits that differ from the one shown in Fig. 1. The time variation of the cost function Lg(t) is shown in Fig. 10. The time step values of Figs. 3, 4 and 10 are the same. As the bacterial populations grow, Lg(t) increases and reaches the first maximum, which may be a global maximum. After that, the value of Lg(t) decreases and then increases again to reach a second maximum. Our simulations showed that there are two extrema of Lg(t) and that after the occurrence of the second the values of Lg(t) decrease slowly. 45 40 35 30 25
L(t)
control population is compared to the concentration of the mutated population at the same time step. Fig. 8 shows the evolution the autoinducer molecule concentration of the control population (left) and of the mutated population (right) for the same time steps. The orange horizontal line indicates the concentration threshold at which a bacterial quorum is formed. In the control population this concentration is reached early and only a few bacteria located at the population center start to secret exofactors. In contrast, in the mutated population the presence of signal-blind and signal-negative bacteria results in a more uniform distribution of autoinducer molecules. In this case, the critical concentration is reached with delay, and when reached, a significantly larger bacteria number participate to the quorum, rendering the infection more effective.
20 15
5 EFFECT OF QC MUTATIONS ON THE DIFFUSION AND DISTRIBUTION OF SIGNALING MOLECULES
In the previous sections we have showed that mutations reduce the synchronizability of the bacterial network. This reduction is not significant and as shown in Figures 4 and 6, is oscillating during the growth of bacterial population. On the other hand, the presence of mutations results in more effective infections because of the
10 5 0
0
50
100
150
Time steps
Fig. 10. The cost fucntion Lg(t)
The cost function Lg(t) reaches a maximum value when the conditions are favorable for the bacterial popu-
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lation to initiate the exofactor production and, hence, the attack on their host. It seems that the bacterial population has a limited number of opportunities to start an infection and if these attempts are not successful, the infection ability is reduced, giving thus time to the host immune system to respond. We believe that our study provides strong indications that signal-blind and signal-negative mutants make infections more effective. The infection process is energy demanding and bacteria evolved QS communication systems to manage their energy expenditure. During infections bacteria populations have to decide continuously on how to spend their available energy. This decision is binary: use the energy to produce offspring or use the energy to produce exofactors. Our work showed that it is possible to study bacterial infections using energy as one of the most important parameters. We used the well established Lagrangian formulation, in which we correspond the kinetic energy with the time-like synchronizability and the potential energy with the space-like autoinducer concentration.
7
[6] [7]
[8] [9]
[10] [11]
[12]
[13]
6 CONCLUSION We studied the dynamics of bacterial infections as a process in which the energy available to the bacteria population is used either in producing offspring or in producing exofactors. We used one of the most basic physical approaches to this problem: the Lagrangian dynamics. We modeled the bacterial population as a growing network and computed its synchronization parameters, which we corresponded to the kinetic energy. We used the diffusion equation with distributed sources to compute the autoinducer distribution during the population growth, which we corresponded to the potential energy. Our simulations showed that bacterial populations have a limited number of opportunities to attack their host, which correspond to the maxima of the cost function Lg(t). Preventing bacterial populations to reach these maxima at early stages using QS inhibitors, or deceiving bacteria populations to a very early onset of costly exofactor production, by adding autoinducer molecules, may render bacterial infections less effective and the bacterial populations more vulnerable to antibiotics or to the host immune system.
[14]
[15]
[16] [17]
[18] [19]
[20]
[21]
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Anastasios I. Psarras received the Diploma in Electrical and Computer Engineering from the Democritus University of Thrace, Xanthi, Greece, in 2012. He received a Master’s degree in Microelectronics and Information Systems Technology from the same Department in 2013, where he is currently a PhD student. His research interests include biological circuits and low-cost digital integrated circuits, more specifically, in the area of on-chip interconnection networks.
Ioannis G. Karafyllidis received the Dipl. Eng. and Ph.D. degrees in Electrical Engineering from the Aristotle University of Thessaloniki, Greece. In 1992 he joined the Department of Electrical and Computer Engineering, Democritus University of Thrace, Greece, as a faculty member, where he is currently a Professor. His current research emphasis is on quantum computing, modeling and simulation of nanoelectronic devices and circuits, and biological networks modelling. He is a Fellow of the Institute of Nanotechnology, a Founding Member of the American Academy of Nanomedicine and a member of the Technical Chamber of Greece (TEE).
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Simulation of the Dynamics of Bacterial Quorum Sensing Anastasios I. Psarras and Ioannis G. Karafyllidis* Abstract—Quorum sensing (QS) is a signaling mechanism that pathogenic bacteria use to communicate and synchronize the production of exofactors to attack their hosts. Understanding and controlling QS is an important step towards a possible solution to the growing problem of antibiotic resistance. QS is a cooperative effort of a bacterial population in which some of the bacteria do not participate. This phenomenon is usually studied using game theory and the non-participating bacteria are modeled as cheaters that exploit the production of common goods (exofactors) by other bacteria. Here, we take a different approach to study the QS dynamics of a growing bacterial population. We model the bacterial population as a growing graph and use spectral graph theory to compute the evolution of its synchronizability. We also treat each bacterium as a source of signaling molecules and use the diffusion equation to compute the signaling molecule distribution. We formulate a cost function based on Lagrangian dynamics that combines the time-like synchronization with the space-like diffusion of signaling molecules. Our results show that the presence of non-participating bacteria improves the homogeneity of the signaling molecule distribution preventing thus an early onset of exofactor production and has a positive effect on the optimization of QS signaling and on attack synchronization. Index Terms—Bacterial signaling, quorum sensing, systems biology.
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1 INTRODUCTION
A
NTIBIOTIC resistance is the resistance of a bacterium to an antibiotic to which it used to be sensitive. Antibiotic resistance is a growing problem and is due to the fact that antibiotics usually target the metabolism and reproduction bacterial circuits, triggering thus a reaction that produces bacteria strains on which the antibiotic is ineffective [1], [2], [3]. The increasing antibiotic resistance calls for a different approach to antimicrobial treatments. Bacterial signaling and especially quorum sensing (QS) is a possible target for such treatments that will control and reduce bacterial virulence [4], [5]. QS is a signaling mechanism that bacteria use to communicate during the infection process [6], [7]. During infection, bacteria produce and secret virulence exofactors to cause degradation of host cells [8]. Exofactors are very expensive to produce and if bacteria synthesize and secret them before they reach a critical number, the exofactor concentration will not be sufficient enough to cause degradation of host cells, leading thus to an inefficient attack. To count their number, bacteria produce and secret smaller and less expensive molecules, called autoinducers and monitor their concentration in the environment. When the bacterial population reaches the critical number, the autoinducer concentration reaches a critical value and is sensed by the bacteria, which all start to produce and secrete exofactors simultaneously, forming thus an infection “quorum” [9]. QS process is not related to metabolism or reproduction and offers an alternative target for controlling bacterial virulence using QS inhibitors [4], [5].
In infections, during the bacterial population growth phase, mutations occur in two of the most important genes related to QS. These mutations result in the appearance of signal-blind and signal-negative bacteria[10]. Signal-blind mutants produce autoinducer signaling molecules but are unable to respond to the QS signal and do not produce exofactors. Signal-negative mutants do not produce signaling autoinducers but respond to the QS signal and produce exofactors [11]. If we want to understand the QS mechanism we have to study the effect of these mutations on the dynamics of QS and find out if these mutations can be exploited to control bacterial virulence. The mutation effect on QS has been analysed using game theory techniques with Pseudomonas aeruginosa (P. aeruginosa) as the model organism [10], [11], [12], [13]. P. aeruginosa is an opportunistic human pathogen and is a major source of hospital-acquired infections [14]. In this paper we continue our previous work on QS modeling [15 - 17]. We model the bacterial population as a growing graph and compute the evolving spectrum of the graph Laplacian. Graph spectra is a valuable tool for studying the synchronization of processes that take place on a graph [18], [19]. We also study the evolution of the autoinducer concentration distribution using the diffusion equation. We compare the autoinducer concentrations in cases where mutations are present with cases where no mutations occur during the growth of bacterial populations. We formulate a Lagrangian-type cost function in which the syncronizability is the time-like part and the autoinducer concentration is the space-like part. Simulations showed that this function has at least two maxi———————————————— ma, which we interpret as the combination of mutation Both authors are with the Department of Electrical and Computer Engineering, Democritus University of Thrace,Kimmeria Campus, 67100 Xan- rate and synchronizability, that leads to the most effecthi, Greece * Corresponding author. E-mail: [email protected] 1536-1241 (c) 2015 IEEE. Personalxxxx-xxxx/0x/$xx.00 use is permitted,©but republication/redistribution requires IEEE permission. See 200x IEEE Published by the IEEE http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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tive attack with the minimum possible cost for the bacterial population. We conclude that the presence of mutants in growing bacterial populations results in a more homogenous autoinducer distribution while the synchronizability of the bacterial network is not reduced significantly by the mutants. Our results provide a strong indication that mutations enhance the effectiveness of bacterial attacks on their hosts. Therefore, the infection virulence may be reduced by controlling the number of mutants in a growing bacterial population.
2 QUORUM SENSING CIRCUIT AND NETWORK The structure of the QS signaling circuit is shown in Fig. 1. When the signal production gene gI is expressed the protein PI is produced. PI produces the autoinduction signaling molecule A, which is secreted continuously into the environment. The concentration of the autoinducer molecules, A, reflects the size of the bacterial population.
where a mutation in the signal production gene gI renders it ineffective, the bacterium cannot produce the signaling molecule. These bacteria are called signal-negative mutants. Signal-negative mutants do not produce the QS signal, but respond to the signal and produce exofactors. In the case where a mutation renders the signal response gene gR ineffective, the bacterium produces the signaling molecule with a small rate, because the autoinduction loop is broken. Furthermore, it cannot respond to the QS signal and does not produce exofactors [3], [11], [20]. Signal-negative and signal-blind mutants are found in natural bacterial populations with the signal-blind mutant more common [3], [11].
Fig. 2. Bacterial population is modelled as a growing network. (a) The network of the population at times t1 and t2>t1 when no mutations occur in the QS genes. (b) The same network with signal-blind and signal-negative subpopulations. These subpopulations are represented by growing yellow “holes” in the growing network. Fig. 1. The structure of the QS signaling circuit. The names of the las QS circuit parts of P. aeruginosa are given in parentheses.
The auotoinducer molecule concentration is continuously monitored by the bacteria. Autoinducer molecules, A, enter the bacterial cell and bind to the signal response protein PR, which is produced by the gene gR, forming an active complex, PR+. PR+ regulates a number of genes some of which produce exofactors. PR+ also binds to the promoter region of the signal synthase gene gI closing the autoinduction loop of the circuit. In the case of the las QS circuit of the model organism P. aeruginosa the signal synthase gene and protein are lasI and LasI, the signal response gene and protein are lasR and LasR and the signaling molecule, A, is the 3-oxo-C12-HSL. During the growth phase of the bacterial population and before a quorum is formed, mutations in the genes gI and gR may occur in some of the bacteria. In the case
We model the bacterial population as a growing network. The network grows with the bacterial population, as shown in Fig. 2(a), at two different times t1 and t2>t1. If signal-negative and signal-blind mutants appear in the population, they form growing subpopulations as shown in Fig. 2(b). Since the mutation probability increases with time, it is expected that mutant subpopulations will be larger in the center of the population and their size will decrease as we move from the center to the periphery of the network. These mutants do not participate in the QS signal production or their contribution is almost insignificant and their subpopulations are represented by “holes” in the growing network, as shown in Fig. 2(b).
3 SYNCHRONIZATION OF THE GROWING NETWORK In the QS process the bacteria synchronize their attack by producing and diffusing signaling molecules into the
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3
environment. The network that models the bacterial population has in its nodes diffusively-coupled entities. It is well known that the synchronization of a network of diffusively-coupled systems depends strongly on the topology of the network [21], [22], [23]. The network topology is encoded in the spectrum of its Laplacian [24], [25]. We describe the synchronization of the dynamic diffusivelycoupled bacterial network using the coupled lattice map equation [24], [26]: n ci t ci t a Li , j c j t t j 1
(1)
where, ci(t) is the autoinducer concentration in the vicinity of the ith network node. Usually in eq.(1) only the neighbouring nodes are taken into account and the index j runs over these nodes only. The coupling strength, a, is a real number and because of the network symmetry is the same for all couplings of the ith node with its neighbours. The bacterial network is said to be synchronized if:
ci t c j t 0 as t
(2)
for all i and j. Therefore, the network is synchronized if the concentration of the signaling molecules is the same in the vicinity of all the bacteria and quorum is reached if this concentration is equal to or larger than the critical value. The study of the synchronization of a static network using eqs. (1) and (2) will result in a system of n differential equations. In the case of a dynamical growing network the number of nodes and equations will increase with time. We want to study the effect of signal-blind and signal-negative mutants on the synchronization of the bacterial network. Such bacteria do not participate in the synchronization process and the corresponding nodes are deleted from the network. That is, if a bacterium located at node i at time t, undergoes a mutation in its QS genes at time t+1, the node i is deleted permanently from the network. This bacterium will multiply and new mutants will appear in its neighborhood, forming a growing “hole” in the bacterial network. The continuous addition of new nodes, the deletion of nodes and the growing holes in the network, render the analysis of its synchronization using eqs. (1) and (2) impractical. We will instead turn to the study of the spectrum of the bacterial network Laplacian, which acts as an evolution operator in eq. (1) and provides a very satisfactory description of the network synchronization [24], [25], [26]. The Laplacian of a connected network has n eigenvalues, λi , which are ordered as:
0 1 2 3
n
(3)
The most important eigenvalues are: the smallest nonzero eigenvalue, λmin, which is equal to λ2 and the maximum eigenvalue λmax, which is equal to λn. The network topology changes as the bacterial population grows and the Laplacian and its spectrum are time
dependent. The method we use to study the effect of mutations on the synchronization of the bacterial network is to simulate the growth of a bacterial population as a growing network and compute the Laplacian and its spectrum at each time step. We do these computations for two bacterial populations. The first one, which we call the control population, grows without any QS mutants. The second, the mutated population, has the same growth rate and the same initial conditions as the first one, but mutations in the QS genes occur during population growth. The mutations result in growing holes in the network and, therefore, in a different topology. The spectrums of the Laplacians of these two networks are computed and compared at each growth time step. As explained above mutations result in deletion of nodes and growing holes in the evolving mutation network. Because of this, the number of nodes and the number of eigenvalues of the control and mutated networks differ. To compare the two spectra on a common ground, we convolve their eigenvalues with a common Lorentz kernel [24], [25]:
G x i
Gm x im
i x
2
2
im x 2
(4)
(5)
2
Equations (4) and (5) are the spectral functions of the control and mutated populations respectively. The eigenvalues of the mutated population are marked with an “m” which is a symbol and not an exponent. The variable x is the same in both (4) and (5) and varies in the same range. The parameter ρ is a real number and its value is important in the plot of the spectral functions. Small values of ρ highlight the details of the spectral plot, whereas large values are used to visualize the large-scale form of the spectral plot. In our simulations we choose an intermediate value, ρ = 0.2 . Figures 3(a) and 3(b) show the time evolution of the spectra of the growing control and mutated populations, respectively. The z axis is the spectral function G(x) in Fig. 3(a) and Gm(x) in Fig. 3(b). The x axis in both figures is the x variable of (4) and (5) and the y axis gives the time steps of population growth. For each time step the spectral functions G(x) and Gm(x) are computed and plotted. The surfaces of Fig. 3 are generated by the plots of the spectral functions at all time steps. Figure 4 shows the difference of the two spectral functions. The x and y axis are the same as in Fig. 3, but the z axis is the difference G(x)-Gm(x). We performed a large number of simulations and in all of them, the structure of the spectral plots in both cases was found to be the same. The difference between the two cases, shown in Fig. 4, varies between 5% and 8%. This difference is located between x=7 and x=11.5 and is due to the difference between eigenvalues the values of which are in the middle of the spectrum.
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concentrations observed in real bacterial populations, does not have a significant effect on the QS synchronization process.
Fig. 3. Spectral plots of (a) the control population and (b) the mutated population. Fig. 5. Time variation of λmin/λmax for: (a) the control population and (b) the mutated population.
Another measure of network synchronizability is the ratio λmin/λmax [24], [26], [27]. The network synchronization condition using this measure takes the form:
min max
Fig. 4. The difference of spectral plots of the control and mutated populations: G(x)-Gm(x).
It is well known that the eiganvalues near the minimum and the maximum eigenvalue are the most important in characterizing the synchronizability of a network [24], [27]. Our simulation results showed that the presence of signal-blind and signal-negative mutants, in
(6)
where the constant α depends on the maximum Lyapunov exponent of the network described by eq.(1). Here we want to study the variation of the network syncronizability with time in the case where no QS mutations are present and in the case where QS mutations are present and compare the two cases. To obtain this we compute the functions:
W t
min t max t
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(7)
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m min t Wm t m max t
5
(8)
where t is the time step. As the bacterial population grows, the minimum and maximum eigenvalues are computed at each time step. The function W(t) gives the variation of the ratio λmin/λmax with time and is plotted in Fig. 5(a) for the case of no QS mutations and the function Wm(t), plotted in Fig. 5(b), gives the same time variation in the case where QS mutants are present in the growing population. Fig. 6 shows the difference W(t)-Wm(t). The structure of the λmin/λmax ratios and their differences were found to be the same in a large number of simulations. The time variations of the ratios in both control and mutated populations, shown in Fig. 5, are almost identical. The difference between them, shown in Fig. 6, starts form zero and increases with time after the 11th time step. The fact that the λmin/λmax ratios are almost identical and their difference is negligible shows that the presence of QS mutants in a bacterial population has an insignificant effect on the QS process. The spectral plots and the λmin/λmax ratio variations provide a strong indication that the effect of QS mutants on the QS synchronization process in a growing bacterial population is not significant. This is actually not a surprise because in all successful infections (from the bacteria point of view) signalblind and signal-negative mutants have been found [11].
2 n x, y , t 2 n x , y , t n x, y , t D t x2 y2 R xs , ys n xs , ys , t
(9)
In eq. (9) n(x,y,t) is the autoinducer concentration at a point with coordinates (x,y) at time t and D is the diffusion coefficient of autoinducer molecules. The coordinates of the distributed sources are (xs,ys) and R(xs,ys) is the autoinducer production rate. This rate is zero at points where a bacterium does not exist. R(xs,ys) is also zero at points where mutated bacteria exist.
Fig. 7. Concentration of autoinducer molecules. (a) Control and (b) mutated bacterial populations.
Fig. 6. The difference W(t) - W m(t).
4 EFFECT OF QC MUTATIONS ON THE DIFFUSION AND DISTRIBUTION OF SIGNALING MOLECULES
The autoinducer concentration is affected by the presence of signal-blind and signal-negative mutants. To study this effect we solve a diffusion equation in which the non-mutated bacteria are modeled as distributed autoinducer sources [28]: Fig. 8. Evolution of the autoinducer molecule concentration of the control population (left) and of the mutated population (right) along the x-axis for the same time steps.
In Fig. 7 the autoinducer molecule concentration of the 1536-1241 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
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Fig. 9. Relative sizes of the control and mutated bacteria populations (X 10,000) after the formation of a quorum.
Fig. 9 shows the relative sizes of the control and mutated bacteria populations after the formation of a quorum (activation), i.e. time step 1 corresponds to the time where a quorum is formed in both populations. At the activation the number of bacteria that participate to the quorum in the control population is about 12% of the corresponding bacteria number in the mutated population. In the next time steps the difference between the two populations increases in favor of the mutated population and then becomes almost stable. This means that the mutated population starts the exofactor production with a significantly larger quorum size, which results in a more effective infection. It should be reminded that the mutated population evolution describes real infections and that signal-blind and signal-negative bacteria are found in all infection sites.
participation of more bacteria in the quorum. It is therefore reasonable to assume that the most appropriate time for bacteria to launch their attack on their host is when the syncronizability and the participation in the quorum reach a favorable combination. To study this, we have to construct a cost function and search for extrema. Since the effects of mutations have a time-like component (synchronizability) and a space-like component (distribution of autoinducer concentration), we believe that a cost function analogous to the Lagrangian used in mechanics would be appropriate. Since the Lagrangian is the difference between kinetic and potential energies we constructed the following cost function: Lg t Q1 G xmax , t Gm xmax , t Q2 W t Wm t (10) n x, y, t nm x, y, t Q3 n x, y , t
In eq. (10), xmax is the point in Fig. 4 where the maximum value of G(x)-Gm(x) occurs. The concentrations n(x,y,t) and nm(x,y,t) are the concentrations of control and mutated populations and are normalized so that the function Lg depends only on time. The values of the weight constants Q1, Q2 and Q3, are determined using the spectral characteristics of the bacterial network and the time evolution of the concentrations, which in turn originate from the structure of the quorum sensing circuits. The values of these constants are expected to be different for bacteria types with quorum sensing circuits that differ from the one shown in Fig. 1. The time variation of the cost function Lg(t) is shown in Fig. 10. The time step values of Figs. 3, 4 and 10 are the same. As the bacterial populations grow, Lg(t) increases and reaches the first maximum, which may be a global maximum. After that, the value of Lg(t) decreases and then increases again to reach a second maximum. Our simulations showed that there are two extrema of Lg(t) and that after the occurrence of the second the values of Lg(t) decrease slowly. 45 40 35 30 25
L(t)
control population is compared to the concentration of the mutated population at the same time step. Fig. 8 shows the evolution the autoinducer molecule concentration of the control population (left) and of the mutated population (right) for the same time steps. The orange horizontal line indicates the concentration threshold at which a bacterial quorum is formed. In the control population this concentration is reached early and only a few bacteria located at the population center start to secret exofactors. In contrast, in the mutated population the presence of signal-blind and signal-negative bacteria results in a more uniform distribution of autoinducer molecules. In this case, the critical concentration is reached with delay, and when reached, a significantly larger bacteria number participate to the quorum, rendering the infection more effective.
20 15
5 EFFECT OF QC MUTATIONS ON THE DIFFUSION AND DISTRIBUTION OF SIGNALING MOLECULES
In the previous sections we have showed that mutations reduce the synchronizability of the bacterial network. This reduction is not significant and as shown in Figures 4 and 6, is oscillating during the growth of bacterial population. On the other hand, the presence of mutations results in more effective infections because of the
10 5 0
0
50
100
150
Time steps
Fig. 10. The cost fucntion Lg(t)
The cost function Lg(t) reaches a maximum value when the conditions are favorable for the bacterial popu-
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lation to initiate the exofactor production and, hence, the attack on their host. It seems that the bacterial population has a limited number of opportunities to start an infection and if these attempts are not successful, the infection ability is reduced, giving thus time to the host immune system to respond. We believe that our study provides strong indications that signal-blind and signal-negative mutants make infections more effective. The infection process is energy demanding and bacteria evolved QS communication systems to manage their energy expenditure. During infections bacteria populations have to decide continuously on how to spend their available energy. This decision is binary: use the energy to produce offspring or use the energy to produce exofactors. Our work showed that it is possible to study bacterial infections using energy as one of the most important parameters. We used the well established Lagrangian formulation, in which we correspond the kinetic energy with the time-like synchronizability and the potential energy with the space-like autoinducer concentration.
7
[6] [7]
[8] [9]
[10] [11]
[12]
[13]
6 CONCLUSION We studied the dynamics of bacterial infections as a process in which the energy available to the bacteria population is used either in producing offspring or in producing exofactors. We used one of the most basic physical approaches to this problem: the Lagrangian dynamics. We modeled the bacterial population as a growing network and computed its synchronization parameters, which we corresponded to the kinetic energy. We used the diffusion equation with distributed sources to compute the autoinducer distribution during the population growth, which we corresponded to the potential energy. Our simulations showed that bacterial populations have a limited number of opportunities to attack their host, which correspond to the maxima of the cost function Lg(t). Preventing bacterial populations to reach these maxima at early stages using QS inhibitors, or deceiving bacteria populations to a very early onset of costly exofactor production, by adding autoinducer molecules, may render bacterial infections less effective and the bacterial populations more vulnerable to antibiotics or to the host immune system.
[14]
[15]
[16] [17]
[18] [19]
[20]
[21]
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Anastasios I. Psarras received the Diploma in Electrical and Computer Engineering from the Democritus University of Thrace, Xanthi, Greece, in 2012. He received a Master’s degree in Microelectronics and Information Systems Technology from the same Department in 2013, where he is currently a PhD student. His research interests include biological circuits and low-cost digital integrated circuits, more specifically, in the area of on-chip interconnection networks.
Ioannis G. Karafyllidis received the Dipl. Eng. and Ph.D. degrees in Electrical Engineering from the Aristotle University of Thessaloniki, Greece. In 1992 he joined the Department of Electrical and Computer Engineering, Democritus University of Thrace, Greece, as a faculty member, where he is currently a Professor. His current research emphasis is on quantum computing, modeling and simulation of nanoelectronic devices and circuits, and biological networks modelling. He is a Fellow of the Institute of Nanotechnology, a Founding Member of the American Academy of Nanomedicine and a member of the Technical Chamber of Greece (TEE).
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