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Nonlinear Stochastic Model for Bacterial Disinfection: Analytical Solution and Monte Carlo Simulation L. T. Fan* and Andres Argoti Department of Chemical Engineering, Kansas State University, Manhattan, Kansas 66506, United States

Ronaldo G. Maghirang Department of Biological and Agricultural Engineering, Kansas State University, Manhattan, Kansas 66506, United States

Song-Tien Chou Department of Finance and Banking, Kun Shan University, Yung-Kang City, Tainan Hsien, 71003 Taiwan

bS Supporting Information ABSTRACT: This contribution presents a sequel to our previously published nonlinear stochastic model for bacterial disinfection whose intensity function is explicitly proportional to the contact time of the bacteria with the disinfecting agent. In the current model, the intensity function is proportional to the square of the contact time to account for an accelerated rate of a disinfection process. The model gives rise to the process’ master equation whose solution renders it possible to obtain the analytical expressions of the process’ mean, variance (or standard deviation), and coeﬃcient of variation. Moreover, the master equation has been simulated via the Monte Carlo method, thereby yielding the numerical estimates of these quantities. The estimates’ values are compared with those computed via the analytical expressions; they are in excellent accord. They are also compared with the available experimental data as well as with the results obtained from our earlier model.

’ INTRODUCTION This contribution constitutes a sequel to our previous eﬀort on nonlinear stochastic modeling of bacterial disinfection.1 Stochastic analysis and modeling of disinfection of bacterial populations are highly desirable: the discreteness and mesoscopic size of the bacteria as well as their incessant and irregular motility in ﬂuid media give rise to ﬂuctuations in the parameters characterizing them, e.g., their number concentration, thereby rendering the temporal evolution of these parameters increasingly uncertain. This is especially the case at the tail-end of disinfection when the number concentration of the bacterial population is extremely low. Our earlier nonlinear stochastic model for bacterial disinfection was formulated on the basis of an intensity function that is explicitly proportional to the contact time of the bacteria with the disinfecting agent.1 In contrast, the model presented herein incorporates an intensity function that is proportional to the square of the contact time to account for an accelerated rate of a disinfection process. Thus, this model might have its own utility in the mathematical description of bacterial disinfection processes exhibiting increased rates of bacterial inactivation; such is the case of bacterial disinfection eﬀected with novel materials, e.g., nanoparticles of metal oxides.2 The model gives rise to the process’ master equation; its analytical solution has rendered it possible to obtain the expressions for the mean, variance (or standard deviation), and coeﬃcient of variation. In addition, the master equation has been simulated numerically by resorting to the Monte Carlo method. Monte Carlo simulation, which is often regarded as numerical experimentation, is an eﬀective technique r 2011 American Chemical Society

for estimating statistical moments of the random variables of processes involving particulate entities.3 To illustrate, the mean, variance (or standard deviation), and the coeﬃcient of variation have been estimated for the number concentration of bacteria at the tail-end of disinfection. These estimates are compared with those computed from the analytical expressions; they are in excellent accord. They are also compared with the available experimental data obtained with E. coli4 as well as with the numerical values resulting from the corresponding deterministic model. Furthermore, the results are compared with those derived from our earlier nonlinear stochastic model.1

’ MODEL FORMULATION The system under consideration comprises a bacterial population being disinfected or deactivated in a unit volume, i.e., the number concentration of bacteria. At the outset of disinfection, it is presumed that the bacteria cease to reproduce due to their deactivation by the disinfecting agent; consequently, their number decreases continually one at a time as they die. As such, the system can be modeled according to the pure-death process5 (Appendix A in the Supporting Information (SI)), in which the Special Issue: Nigam Issue Received: April 25, 2011 Accepted: August 24, 2011 Revised: August 12, 2011 Published: September 21, 2011 1697

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death event is identiﬁed as the elimination of a single bacterium. The number of live bacteria at time t, N(t), is taken as the process’ random variable, the realization of which is denoted by n. All the possible numbers of bacteria are the states of the process, and the collection of these numbers, {n0, n0 1, ..., 2, 1, 0}, is the state space, where n0 is the initial number of bacteria, i.e., n at t = 0. Master Equation. The probability balance for the pure-death process around state n gives rise to the following master equation of the process (Appendix A);5,6 d p ðtÞ ¼ μnþ1 ðtÞpnþ1 ðtÞ μn ðtÞpn ðtÞ, dt n n ¼ n0 , n0 1, :::, 2, 1, 0

ð1Þ

ð2Þ

For the process under consideration, the intensity of death, μn(t), in eq 1 is assumed to be of the form μn ðtÞ ¼

dn ¼ knt 2 dt

γðωÞ ¼ and

t3 k 3

ω¼

ð8Þ

! ð9Þ

ð3Þ

’ ANALYTICAL SOLUTION The analytical solution of the master equation, comprising eqs 4 and 5, can be executed by two approaches. One is based on the probability distribution, pn(t), of random variable N(t), and the other is via the probability generating function of pn(t). The results render it possible to compute the mean and higher moments about the process’ mean, e.g., variance, skewness, and kurtosis, of N(t). Among these higher moments, the second moment about the mean, i.e., the variance, is of utmost importance: it signiﬁes the ﬂuctuations, or scatterings, of the random variable about its mean.79 Probability Distribution. For the process under consideration, pn(t) is obtained by solving the master equation recursively as (Appendix C)

where k is a positive constant given in the unit of t3. Substituting the above equation for μn(t) into eq 1 yields d p ðtÞ ¼ ½kðn þ 1Þt 2 pnþ1 ðtÞ ½knt 2 pn ðtÞ, dt n n ¼ n0 , n0 1, :::, 2, 1, 0

yðωÞ n0

where k is a positive constant given in the unit of t3.

where pn(t) signifies the probability of n live bacteria being present at time t. Naturally, the master equation, eq 1, represents a set of n0 ordinary differential equations. Note that for n = n0, the term, μn+1pn+1(t), in eq 1 lacks any significance; thus, this equation reduces to d p ðtÞ ¼ μn0 pn0 ðtÞ dt n0

where the dimensionless number concentration of bacteria, γ(ω), and the dimensionless time, ω, are given, respectively, by

pn ðtÞ ¼ where

n0 ! pn ð1 pÞðn0 nÞ n!ðn0 nÞ!

"

t3 p ¼ exp k 3

ð4Þ

ð10Þ

!# ð11Þ

Correspondingly, for n = n0, this expression reduces to d p ðtÞ ¼ ½kn0 t 2 pn0 ðtÞ dt n0

ð5Þ

In the above two equations, the intensity function on the righthand side of each equation comprises the product between the realization, n, of N(t) and the square of time t, which is the independent variable, thereby rendering them nonlinear in t; nevertheless, these equations are linear in n. Consequently, they can be solved analytically; moreover, they can also be solved numerically via Monte Carlo simulation. Both approaches are elaborated in the succeeding sections. Deterministic Model. By integrating the intensity of death, μn(t), as given by eq 3, subject to the initial condition, n = n0 at t = 0, we have ! t3 yðtÞ ¼ n0 exp k ð6Þ 3 where y(t) is the deterministic and continuous counterpart of the mean of N(t) (Appendix B), thereby indicating that the intensity of death, μn(t), corresponds to the deterministic expression of the rate law for the process under consideration. In dimensionless form, eq 6 can be rewritten as γðωÞ ¼ expðωÞ

Equation 10 indicates that N(t) obeys a binomial distribution with parameters n0 and p, i.e., N(t) ∼ Binomial(n0, p), where n0 is the total number of successful events that can possibly occur and p is the probability of occurrence of a single successful event.10 For the pure-death of concern, n0 is the initial number of live bacteria, which will eventually die off, and p is the probability of a single bacterium being alive at time t. Note that eq 10 gives rise to the extinction probability, p0(t), as p0 ðtÞ ¼ or

n0 ! p0 ð1 pÞðn0 0Þ ¼ ð1 pÞn0 0!ðn0 0Þ! "

t3 p0 ðtÞ ¼ 1 exp k 3

!#n0

The quantity, p0(t), signifies the probability of the bacterial population being completely eradicated and/or inactivated at any time t.11 Naturally, p0(t) is 0 at t = 0, and it asymptotically approaches 1 as t f ∞. Probability Generating Function. The probability generating function, G(z;t), is defined as5,8,11 Gðz; tÞ ¼

ð7Þ 1698

∑n znpnðtÞ

ð12Þ

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where z is an auxiliary variable. For the process under consideration, G(z;t) is (Appendix C) Gðz; tÞ ¼ ½ð1 pÞ þ zpn0

ð13Þ

where p is given by eq 11. The above expression is identified as the probability generating function of a binomial distribution with parameters n0 and p.7 Mean. In light of the probability distribution, pn(t), as expressed by eqs 10 and 11, the mean, E[N(t)] or m(t), of N(t) is obtained as (Appendix C)

The standard deviation, σ(t), is the square root of the variance; thus, σðtÞ ¼ ½σ 2 ðtÞ1=2 (" ¼

1=2 n0

t3 exp k 3

!#"

t3 1 exp k 3

!#)1=2

ð22Þ In dimensionless form, this expression can be rewritten as 1=2

ζðωÞ ¼ n0

mðtÞ ¼ n0 p

f½expðωÞ½1 expðωÞg1=2

ð23Þ

where the dimensionless standard deviation, ζ(ω), is defined as

or

"

t3 mðtÞ ¼ n0 exp k 3

!#

ζðωÞ ¼

ð14Þ

Clearly, this expression is identical to that of y(t) given by eq 6, thereby confirming that y(t) is the deterministic and continuous counterpart of the mean, m(t), as claimed in the preceding section. In dimensionless form, eq 14 becomes πðωÞ ¼ expðωÞ

σðtÞ CVðtÞ ¼ mðtÞ (" !#" !#)1=2 t3 t3 1=2 n0 exp k 1 exp k 3 3 " !# ¼ t3 n0 exp k 3

ð15Þ

mðωÞ n0

ð16Þ

or

Note that eq 15 is identical to eq 7 for γ(ω). From the probability generating function, G(z;t), as expressed by eq 13, m(t) is obtained as (Appendix C) mðtÞ ¼

∂ Gðz; tÞjz ¼ 1 ∂z

ð17Þ

Hence, mðtÞ ¼ n0 ½ð1 pÞ þ zpn0 1 pjz ¼ 1

ð18Þ

or ð19Þ

mðtÞ ¼ n0 p In view of eq 11, "

t3 mðtÞ ¼ n0 exp k 3

!# ð20Þ

which is identical to eq 14. Variance, Standard Deviation, and Coefficient of Variation. From pn(t) as expressed by eqs 10 and 11, the variance, Var[N(t)] or σ2(t), of N(t) is obtained as (Appendix C)

"

t3 σ ðtÞ ¼ n0 exp k 3 2

8" !#91=2 > t3 > > > > > > = < 1 exp k 3 > 1=2 " !# CVðtÞ ¼ n0 > > t3 > > > > > > ; : exp k 3

In terms of dimensionless time ω, 1=2 1=2 ½1 expðωÞ CVðωÞ ¼ n0 ½expðωÞ

ð25Þ

ð26Þ

Note that CV(t) signiﬁes the relative ﬂuctuations of a random variable about its mean;9 it is a dimensionless quantity by deﬁnition. From G(z;t) as expressed by eq 13, σ2(t) can be computed as (Appendix C) " # 2 ∂ σ 2 ðtÞ ¼ Gðz; tÞjz ¼ 1 þ mðtÞ ½mðtÞ2 ð27Þ ∂z2 Hence, σ 2 ðtÞ ¼ n0 ðn0 1Þ½ð1 pÞ þ zpn0 2 p2 jz ¼ 1 þ n0 p ðn0 pÞ2

ð28Þ Consequently,

σ2 ðtÞ ¼ n0 pð1 pÞ or

ð24Þ

Moreover, the coeﬃcient of variation, CV(t), i.e., the ratio between the standard deviation, σ(t), and the mean, m(t), is obtained from eqs 14 and 22 as

where the dimensionless time, ω, is given by eq 9, and the dimensionless mean, π(ω), is given by πðωÞ ¼

σðωÞ n0

! #"

t3 1 exp k 3

σ 2 ðtÞ ¼ n0 ðn0 1Þp2 þ n0 p ðn0 pÞ2

!# ð21Þ

or σ 2 ðtÞ ¼ n0 pð1 pÞ 1699

ð29Þ

dx.doi.org/10.1021/ie200890p |Ind. Eng. Chem. Res. 2012, 51, 1697–1702

Industrial & Engineering Chemistry Research In view of eq 11, "

t3 σ ðtÞ ¼ n0 exp k 3 2

! #"

t3 1 exp k 3

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!# ð30Þ

which is identical to eq 21. As a result, G(z;t) yields the same expression for CV(t) as that resulting from pn(t).

’ MONTE CARLO SIMULATION The master equation of the process, as given by eqs 4 and 5, is simulated via the Monte Carlo method with the event-driven approach as proposed by Gillespie12,13 and implemented by various authors.1416 Naturally, the simulation could be performed by means of improved and/or modiﬁed versions of the original method,12 such as those developed by Rao and Arkin,17 Cao et al.,18 or Ramaswamy et al.19 The event-driven approach advances the simulation clock of the process’s temporal evolution by a random time τ signifying the waiting time between successive death events.3,10 For the pure-death process under consideration, τ is obtained as (Appendices D and E) 1=3 3 τ ¼ t þ t 3 lnð1 uÞ kn

Figure 1. Temporal evolution of the dimensionless mean, π(ω), dimensionless sample mean, m(ω)/n0, and envelopes of standard deviation, [π(ω) ( ζ(ω)] and [m(ω) ( s(ω)]/n0, of random variable N(ω) in the termination period of photoelectrochemical disinfection of E. coli4 at two experimental conditions. Symbol (b) represents the dimensionless experimental data, η(ω), with n0 = 146 cells/mL; and symbol (4), n0 = 115 cells/mL. Solid line (—) represents to the solution of the deterministic model in dimensionless form, γ(ω).

ð31Þ

where u is a realization of the uniform random variable, U, on interval [0, 1). No event takes place during the time interval (t, t + τ), thereby rendering the state of the system unchanged, and thus, n r n. At the end of this time interval, a single death event occurs, and thus, the state of the system decreases exactly by one, and thus, n r (n 1).

’ RESULTS AND DISCUSSION To validate the model formulated, the results of analytical solution are compared with those yielded by Monte Carlo simulation, often regarded as numerical experiments. They are also compared with the very limited number of the available experimental data in the termination period of the disinfection of E. coli4 at two distinct experimental conditions. Because the number concentration of bacteria in the termination period is indeed minute, the ﬂuctuations of the process about its mean are signiﬁcantly magniﬁed. The two values of the initial number concentration of bacteria, n0, at the outset of the termination period have been identiﬁed as 115 and 146 cells per milliliter. Moreover, the regression of eq 14 for m(t) on the available experimental data4 via the adaptive random search procedure20 has resulted in the values of k as 2.31 103 min3 with n0 = 115 cells per mL and 2.65 103 min3 with n0 = 146 cells per mL. The mean, m(t), standard deviation, σ(t), and coeﬃcient of variation, CV(t), have been computed from eqs 14, 22, and 25, respectively, with these values of n0 and k. Moreover, the sample mean, m(t), sample standard-deviation, s(t), and sample coeﬃcient of variation, CV(t), for N(t) have been estimated via Monte Carlo simulation by resorting to the event-driven approach with the same values of n0 and k. Mean and Standard Deviation. Figure 1 compares the dimensionless mean, π(ω), computed from eq 15 of the analytical solution and the dimensionless sample mean, m(ω)/n0, estimated by Monte Carlo simulation. In this figure, these quantities are also compared with the available experimental data4 and the dimensionless deterministic values, γ(ω), computed from eq 7.

Figure 2. Temporal evolution of the coeﬃcient of variation, CV(ω), and the sample coeﬃcient of variation, CV(ω), of random variable N(ω) in the termination period of photoelectrochemical disinfection of E. coli4 with n0 = 146 cells per mL. Symbol (b) represents the normalized experimental data, ν(ω).

The dimensionless standard deviation, ζ(ω), and dimensionless sample standard deviation, s(ω)/n0, have been superimposed in Figure 1 in terms of the dimensionless standard deviation envelopes, [π(ω) ( ζ(ω)] and [m(ω) ( s(ω)]/n0, respectively. These quantities signify the extent of the inherent ﬂuctuations of the process about its mean as predicted by the stochastic model. In Figure 1, the experimental data have been rendered dimensionless by dividing them by n0; the resulting values are denoted by η(ω). The estimated values, m(ω) and s(ω), have been obtained by averaging 100 Monte Carlo simulations via the event-driven approach. Note that the analytical and simulated means are in excellent accord; moreover, they generally follow the trend of the available experimental data.4 Coefficient of Variation. Figure 2 compares the coefficient of variation, CV(ω), computed from eq 26 and the sample coefficient of variation, CV(ω), estimated via Monte Carlo simulation. CV(ω) is defined as the ratio between the sample standard 1700

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Comparison between Earlier and Present Models. Figure 3 compares the values of the dimensionless sample mean, m(t)/n0, estimated by Monte Carlo simulation based on the present model with those obtained based on our earlier nonlinear stochastic model for bacterial disinfection.1 In this figure, the Monte Carlo estimates correspond to the experimental data4 with n0 = 146 cells/mL (also see Figure G2 in Appendix G with n0 = 115 cells/mL). These data have been rendered dimensionless by dividing them by n0; the resulting values are denoted by η(ω). The sample standard deviation envelopes, [m(t) ( s(t)]/ n0, are not depicted in the figure for clarity. In our earlier model,1 the intensity of transition, μn(t), and the corresponding deterministic expression are given, respectively, by Figure 3. Comparison of the Monte Carlo estimates for the dimensionless sample mean, m(t)/n0, based on our present and earlier1 models in the termination period of photoelectrochemical disinfection of E. coli4 with n0 = 146 cells per mL. Symbol (b) represents the dimensionless experimental data, η(ω).

deviation, s(ω), and the sample mean, m(ω). In this figure, the experimental data4 for which n0 = 146 cells per mL are superimposed for comparison. These experimental data have been normalized by dividing them by their corresponding values of m(ω) obtained from the analytical solution; the normalized experimental data are denoted by ν(ω) (also see Figure G1 in Appendix G for CV(ω) with n0 = 115 cells per mL). Moreover, the normalized mean is obtained as m(ω)/m(ω), or 1. In general, CV(ω) provides a relative measure of the variability or dispersion of the values of a random variable about their mean, which can be more meaningful than the standard deviation, σ(ω). Note that the smaller the values of random variable N(t) signifying the population size, the greater the extent of the expected fluctuations about their mean. This behavior manifests itself for the bacterial disinfection of concern: the temporal decrease in the population of bacteria is accompanied by the vivid magnification of such fluctuations. It is entirely logical for the internal noises, i.e., fluctuations, of any process portrayed by a stochastic model to be magnified when the process’ number concentration, or population density, is minute, e.g., in the period near the outset of a growth process or near the tail end of a death process for which the bacterial disinfection of the present work is a typical instance. Obviously, any given fluctuating event will impact more severely a smaller population than a larger population.9 What is described above is vividly revealed in terms of the coeﬃcient of variation, CV(ω), plotted in Figure G1 in Appendix G and Figure 2 for the termination period of disinfection. Expectedly, it is revealed more distinctly in the former with a smaller number concentration (n0 = 115 cells/mL) than in the latter with a larger number density (n0 = 146 cells/mL). It is also worth noting that the deviations or ﬂuctuations of the available experimental data4 are substantially more pronounced than those predicted by the stochastic model: the overall deviations of the experimental data include those attributable not only to the internal, or inherent, noises of the process, as characterized by its stochastic model, but also to the external noises arising from inevitable manual and/or instrumental measurement errors, which can never be totally circumvented. It is logical to expect that the complexity involved in experimental measurement would be increasingly magniﬁed as the number of bacteria to be detected becomes exceedingly minute at the tail end of the termination period.

μn ðtÞ ¼

dn ¼ knt dt

and t2 xðtÞ ¼ n0 exp k 2

ð32Þ

! ð33Þ

The proﬁles corresponding to the deterministic expressions for both models, i.e., x(t) and y(t), are also superimposed in the ﬁgure for comparison. Clearly, both models follow the trend of the experimental data;4 nevertheless, our earlier model1 tends to ﬁt these data better than the present model: the residual sum of squares (RSS)9 for our earlier model1 is 462 (cells/mL),2 and that for the present model is 1399 (cells/mL).2 For the experimental data with n0 = 115 cells/mL as depicted in Figure G2, the RSSs for our earlier and present models are 178 (cells/mL)2 and 659 (cells/mL),2 respectively. This might imply that the eﬀect of aging in terms of contact time of the bacteria with the disinfecting agent on the death rate is not pronounced for this particular set of experimental data obtained under well-controlled or deﬁned laboratory conditions:4 it is not accelerated as predicted by the present model. Nevertheless, it is highly likely that the present model would be viable in representing sets of experimental data exhibiting accelerated rates of bacteria disinfection, which could be the case for disinfection or sterilization processes eﬀected by novel techniques and/or materials, e.g., nanoparticles of metal oxides.2 Furthermore, the death rate of bacteria in disinfection might also be accelerated in practical implementation particularly in relatively open settings, e.g., animal shelters and cattle feedlots. The inﬂuences of a myriad of incessantly varying environmental conditions, such as wind, solar radiation and temperature, and compositions of media in which bacteria reside, cannot be circumvented. It is a tenet of stochastic modeling to embody all such inﬂuences collectively in terms of a limited number of parameters, namely, intensity functions, distributed according to probabilistic laws, thereby giving rise to inherent noises of the resultant model.

’ CONCLUDING REMARKS A stochastic model for bacterial disinfection has been derived as a pure-death process based on a nonlinear rate law to account for the acceleration in the rate of disinfection. The model has given rise to the process’ master equation, from which the analytical expressions for the process’ mean, variance (or standard deviation), and coeﬃcient of variation have been obtained. Moreover, the master equation has been simulated by means of 1701

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Industrial & Engineering Chemistry Research the Monte Carlo method via the event-driven approach, thereby yielding the numerical estimates of these quantities. For illustration, the mean, standard deviation, and coeﬃcient of variation of the number concentration of E. coli in the termination period of disinfection at two experimental conditions have been computed from their analytical expressions and estimated via Monte Carlo simulation, generally regarded as numerical experimentation. The estimates’ values are in excellent accord with those computed from the analytical expressions. In addition, the analytical and estimated means follow the temporal trend of the available experimental data reasonably well. Moreover, the simulated mean values have been compared with those obtained from our earlier nonlinear stochastic model. As expected, the data’s ﬂuctuations around the means are more noticeable than those predicted by the model: in addition to the process’ internal noises, the deviations of the experimental data also account for the external noises arising from inevitable manual and instrumental errors, which can never be totally eliminated. Such external noises are expected to be pronounced in the termination period of disinfection in which the number concentration of bacteria tends to be exceedingly low.

’ ASSOCIATED CONTENT

bS

Supporting Information. Appendices A through G. This material is available free of charge via the Internet at http:// pubs.acs.org.

’ AUTHOR INFORMATION

ARTICLE

(9) Casella, G.; Berger, R. L. Statistical Inference; Duxbury: Paciﬁc Grove, CA, 2002. (10) Gillespie, D. T. Markov Processes - An Introduction for Physical Scientists; Academic Press: San Diego, CA, 1992. (11) Chiang, C. L. An Introduction to Stochastic Processes and Their Applications; Robert E. Krieger Publishing Company: Huntington, NY, 1980. (12) Gillespie, D. T. A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. J. Comput. Phys. 1976, 22, 403–434. (13) Gillespie, D. T. Stochastic Simulation of Chemical Kinetics. Annu. Rev. Phys. Chem. 2007, 58, 35–55. (14) Shah, B. H.; Borwanker, J. D.; Ramkrishna, D. Monte Carlo Simulation of Microbial Population Growth. Math. Biosci. 1976, 31, 1–23. (15) Rajamani, K.; Pate, W. T.; Kinneberg, D. J. Time-Driven and Event-Driven Monte Carlo Simulations of Liquid-Liquid Dispersions: A Comparison. Ind. Eng. Chem. Fund. 1986, 25, 746–752. (16) Ullah, M.; Schmidt, H.; Cho, K. H.; Wolkenhauer, O. Deterministic Modelling and Stochastic Simulation of Biochemical Pathways using MATLAB. IEE Proc. Syst. Biol. 2006, 153, 53–60. (17) Rao, C. V.; Arkin, A. P. Stochastic Chemical Kinetics and the Quasi-Steady-State Assumption: Application to the Gillespie Algorithm. J. Chem. Phys. 2003, 118, 4999–5010. (18) Cao, Y.; Gillespie, D. T.; Petzold, L. R. Eﬃcient Step Size Selection for the Tau-Leaping Simulation Method. J. Chem. Phys. 2006, 124, 044109. (19) Ramaswamy, R.; Gonzalez-Segredo, N.; Sbalzarini, F. A New Class of Highly Eﬃcient Exact Stochastic Simulation Algorithms for Chemical Reaction Networks. J. Chem. Phys. 2009, 130, 244104. (20) Fan, L. T.; Chen, H. T.; Aldis, D. An adaptive random search procedure for large scale industrial and process systems synthesis. Proceedings of Symposium on Computers in Design and Erection of Chemical Plants, Karlovy Vary, 31 Aug.4 Sept., 1975; pp 279291.

Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT This is contribution 11-329-J from the Kansas Agricultural Experiment Station, Kansas State University, Manhattan, KS 66506. ’ REFERENCES (1) Argoti, A.; Fan, L. T.; Chou, S. T. Monte Carlo Simulation of Bacterial Disinfection: Nonlinear and Time-Explicit Intensity of Transition. Biotechnol. Prog. 2010, 26, 1486–1493. (2) Li, Q.; Li, Y. W.; Wu, P.; Xie, R.; Shang, J. K. Palladium Oxide Nanoparticles on Nitrogen-Doped Titanium Oxide: Accelerated Photocatalytic Disinfection and Post-Illumination Catalytic Memory. Adv. Mater. 2008, 20, 3717–3723. (3) Argoti, A.; Fan, L. T.; Cruz, J.; Chou, S. T. Introducing the Stochastic Simulation of Chemical Reactions using the Gillespie Algorithm and MATLAB: Revisited and Augmented. Chem. Eng. Educ. 2008, 42, 35. (4) Harper, J. C.; Christensen, P. A.; Egerton, T. A.; Curtis, T. P.; Gunlazuardi, J. Eﬀect of Catalyst Type on the Kinetics of the Photoelectrochemical Disinfection of Water Inoculated with E. Coli. J. Appl. Electrochem. 2001, 31, 623–628. (5) van Kampen, N. G. Stochastic Processes in Physics and Chemistry; North-Holland: Amsterdam, 1992. (6) Oppenheim, I.; Shuler, K. E.; Weiss, G. H. Stochastic Processes in Chemical Physics: The Master Equation; The MIT Press: Cambridge, MA, 1977. (7) Clarke, A. B.; Disney, R. L. Probability and Random Processes for Engineers and Scientists; John Wiley & Sons Inc.: New York, 1970. (8) Gardiner, C. W. Handbook of Stochastic Methods: For Physics, Chemistry, and the Natural Sciences; Springer: Berlin, 1985. 1702

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