reliable kinetic monte carlo simulation based on random...

1 Copyright © 2011 by ASME

Proceedings of ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference

IDETC/CIE 2011 August 29-31, 2011, Washington, DC, USA

DETC2011-48575

RELIABLE KINETIC MONTE CARLO SIMULATION BASED ON RANDOM SET SAMPLING

Yan Wang

Woodruff School of Mechanical Engineering Georgia Institute of Technology

Atlanta, GA 30332

ABSTRACT To simulate the dynamic behaviors of large molecular

systems, approaches that solve ordinary differential equations such as molecular dynamics (MD) simulation may become inefficient. The kinetic Monte Carlo (KMC) method as the alternative has been widely used in simulating rare events such as chemical reactions or phase transitions. Yet lack of complete knowledge of transitions and the associated rates is one major challenge for accurate KMC predictions. In this paper, a reliable KMC (R-KMC) mechanism is proposed to improve the robustness of KMC results, where propensities are interval estimates instead of precise numbers and sampling is based on random sets instead of random numbers. A multi-event algorithm is developed and implemented. The weak convergence of the multi-event algorithm towards traditional KMC is demonstrated with a proposed generalized Chapman-Kolmogorov Equation.

1. INTRODUCTION To simulate the dynamic behaviors of large molecular

systems in multiscale materials and product design, approaches that solve ordinary differential equations such as molecular dynamics (MD) simulation become inefficient. In contrast, atomic scale kinetic Monte Carlo (KMC) [1] is more efficient in simulating the infrequent transition processes with longer times than thermal vibrations. To simulate the infrequent or rare events of transitions or reactions, several improvements of MD have been proposed to bridge the gap of time scale and accelerate the simulation speed of rare events, such as by running multiple trajectories [2], introducing bias potentials [3], or increasing temperatures [4]. However, the inherent inefficiency of MD is that computational time is spent on trajectory prediction, which is not important for rare event simulations.

KMC does not simulate a system based on its continuous evolvement along time as in MD. Instead, it defines a discrete

set of states of the system (i.e. all possible configurations). It simulates state transitions between states which are triggered by events (also called processes in chemistry-oriented literature) that cause state changes. For instance, in the vapor deposition process of crystal growth shown in Fig. 1-(a), each “snapshot” of the system at a time where atoms are located in space is a state of the KMC model. In Fig. 1-(b), the major events that change the system’s state include adsorption (particles in vapor are attracted to solid surface) with transition rate a1 (transition or reaction rates indicate the frequencies of events), desorption (particles previously absorbed on the solid surface escape and are vaporized) with rate a7, surface diffusion (particles on the surface move to a different location) with rates a2 to a5, and vertical diffusion (particles on the surface move into the solid) with rate a6. In chemical reactions, the numbers of different species naturally form the states of KMC models. Reactions themselves are the events.

Fig. 1: Example states and events in vapor deposition process

simulation During the KMC simulation, events are randomly selected

to occur sequentially based on their respective probabilities of occurrence, which are exponential distributions in general. When an event occurs or is fired, the system state is updated according to the nature of the event. At the same time, the system clock that keeps track of the current time of the system needs to be advanced by a certain period, which is also randomly generated based on probabilistic principles. The system clock provides information of how long the processes

(a) discrete states (b) possible events

a1

a2 a3

a4a5

a7

a6

solid surface

particles


last. Therefore, KMC essentially simulates state transitions or reactions. Clock advancement is not based on a fixed time period but directly determined by when the next event occurs. Trajectories of atoms are not simulated. Time does not evolve continuously. This discrete-event approach allows us to simulate much larger atomic systems for much longer times than those MD can provide.

Although the importance of KMC in rare-event simulation has been widely recognized, there are three major challenges in KMC simulation. The first challenge of accuracy for KMC is that ideally all states and events have to be known a priori and listed in an event catalog so that the dynamics of physical process can be simulated accurately. That is, we should know all possible events and the associated probabilities of occurrence under all configurations when building KMC models. However, this is not realistic because the lack of full knowledge of physical systems does not allow us to know all possible states and transition pathways.

The second challenge of robustness is that during simulation the occurring probabilities of events are assumed to be accurate and fixed. In the real world, this is not true. Uncertainties are always involved in estimating the transition rates thus the probabilities of events. When the rates are estimated by physical experiments, measurement errors are unavoidable. When rates are estimated from first-principles calculation, numerical setup and approximations for computability bring unintentional uncertainties, such as the exchange-correlation treatment in density functional theory and sampling limitation in quantum Monte Carlo simulation. Model uncertainty is also inevitable when mathematical models are used to describe physical phenomena. In addition, the transition rates are dynamically changing over time. For instance, in diffusions, when the material structure is under dynamic mechanical load, the stress will change the diffusion rate. Furthermore, bulk diffusion or inter-layer mass transport is very sensitive to temperature change compared to surface diffusion or intra-layer mass transport. Temperature variation will affect the accuracy of KMC prediction. In biochemical processes, the crowding effect, where macromolecules block reaction paths, changes kinetic constants of reactions.

The third challenge of efficiency for KMC is that the time scales of the events vary significantly. Thus it is possible that the frequencies or probabilities of those events are in very different scales. For instance, in simulating of diamond growth in chemical vapor deposition, surface diffusion could be several orders faster than adsorption and desorption. Computational time is not optimized to focus more on slower but critical events.

In this paper, a reliable KMC (R-KMC) simulation framework is proposed to address the second challenge of KMC, which is to improve the robustness of KMC prediction under input uncertainty. The need to quantify variability and incertitude separately has been well recognized. Variability or aleatory uncertainty is the inherent randomness in the system because of fluctuation and perturbation, whereas incertitude or

epistemic uncertainty is due to lack of perfect knowledge or enough information about the system. The sources of epistemic uncertainty in modeling and simulation include lack of data or missing data, conflicting information, conflicting beliefs, lack of introspection, measurement and numerical errors, and lack of information about dependency.

The proposed R-KMC mechanism is to perform sampling based on imprecise probability. Transition rates used in R-KMC are intervals with lower and upper bounds in stead of precise numbers. As a result, the probability of firing an event becomes imprecise and is also an interval. The interval width captures the epistemic portion of uncertainty. Therefore, in random variate generation, a pair of cumulative distribution functions as the lower and upper bounds are used, and the random variates in R-KMC are random sets. A random set is a compact set of random variables. Given the sample space Ω, a random set can be regarded as a statistical sample of the power set 2Ω. Thus instead of sampling one random value for each sampling step, multiple random values are possible. Based on the multi-event selection and clock advancement algorithm proposed in this paper, either one or two random values are sampled, which means that either one or two events occur at a time. The variation of the number of random values captures epistemic uncertainty, whereas the random values themselves correspond to aleatory uncertainty. As a result, the two types of uncertainties are represented explicitly so as to increase the confidence of simulation results when inputs are uncertain.

Notice that discrete events are not unique in KMC. The generic concept has also been widely used in simulating discrete-event dynamic systems such as queueing networks in operations research, also called discrete-event simulation (DES) [5,6]. The event generation and clock advancement mechanisms in commonly used DES are different from those in KMC. Nevertheless, the proposed reliable simulation concept is also applicable to DES [7,8,9].

In the remainder of the paper, Section 2 gives a brief review of the classical KMC simulation algorithm, some approaches to address the challenge of uncertain rate, and imprecise probability that the R-KMC is based upon. In Section 3, the proposed R-KMC mechanism is introduced. In Section 4, the multi-event algorithm as the core part of R-KMC is described. In Section 5, the R-KMC is demonstrated by examples. And finally in Section 6, the convergence of R-KMC towards traditional KMC is studied.

2. BACKGROUND The major concept of the proposed R-KMC mechanism is

that since lack of complete knowledge on the underlying physics of reaction or transition processes poses challenges on the robustness of KMC prediction, reliable KMC simulation should be conducted such that more information about imprecise inputs can be provided without significantly increasing computational load in order to improve robustness. In this section, after the introduction of the most used stochastic simulation algorithm in KMC, the existing


approaches to address the uncertain rates will be reviewed, followed by the introduction of imprecise probability. 2.1 Kinetic Monte Carlo

In KMC, how often event j occurs or the probability that even j is fired is determined by its associated rate aj (also called propensity in chemistry-oriented literature) with respect to other events. Among all M possible transitions leaving current state, the probability pj of the transition from current state to state j is the probability of firing event j, calculated as

1/

M

j j iip a a

== ∑ . Because the time between transition events

Xj is exponentially distributed as exp( )j jX a∼ , the probability

that the inter-arrival time of event j is the minimum among M independent events is

1 1Pr[ min( , , )] / ( )

j M j MX X X a a a= = + +… . Thus, for

each KMC step, an event is randomly selected based on the

proportions 1 1 1

( / ), ,( / )M M

i M ii ia a a a

= =∑ ∑… . In addition, once

an event is fired, the clock of the system is advanced by a

second random value ( )1exp

M

iit a

=∑∼ , which is the earliest

occurring time for any event out of M independent ones that are exponentially distributed. In other words, when the time between events

jT are exponentially distributed, i.e.

( ) ( )1 expj j

P T a≤ = − −τ τ for 1, ,j M= … , the earliest time

( ) ( )1

1min , , MT T T= … of any event occurs is also

exponentially distributed as

( )( ) ( )( ) ( )( )

1 1

1

1

1 1

1 exp

M

jj

M

jj

P T P T P T

a

=

=

≤ = − > = − >

⎛ ⎞= − −⎜ ⎟⎝ ⎠

∏∑

τ τ τ

τ

with the rate of 1

M

jja

=∑ . Therefore, the clock is advanced by

the random variant 1

ln /M

jjT a

== − ∑ρ , where ( )0,1Uρ ∼

is a random number sampled from the standard uniform distribution.

The above is the most used so-called rejection-free method [10] or stochastic simulation algorithm (SSA) [11], among others [1]. In the classical rejection-free method, selecting an event to fire has the linear time complexity with respect to the number of possible events during simulation. To improve the computational efficiency, searching methods with special data structures such as two-level binning [12], binary tree [13], and distinct rate list [14] have been developed. 2.2 Existing Approaches to Account for Uncertain

Rates The uncertainty associated with probabilities in KMC

simulation was recognized recently. It was shown that the kinetic rate in biochemical reactions varies over time instead of a constant because of the macromolecular crowding effect

[15,16]. It causes mutual impenetrability of solute molecules and phase separation, thus increased folding and refolding rates or reduced diffusion rates. To model the so-called spatial challenge and time dependency of kinetic rates, fractal and Zipf-Mandelbrot relationships were used. A combination of spatial lattice diffusion and chemical reaction simulations was proposed to model the actual crowding effect.

In addition, delays associated with transcription and translation are essential in modeling cellular pathways and regulatory networks, where biochemical processes such as diffusion, translocation, protein synthesis and folding do not occur instantaneously and often affected by spatial homogeneities. Delay stochastic simulation algorithms [17,18,19] were proposed to capture delays in temporal models, where the quantities of reactants and products are updated separately with a delayed period of time.

The above approaches incorporated input uncertainty for application-specific KMC models to improve the robustness of KMC. In contrast, the proposed R-KMC mechanism is generic and simple enough to be applied to various domains, which is based on the theory of imprecise probability. 2.3 Imprecise Probability

The imprecise probability is a generalization of the traditional precise probability where probability values are no longer precise numbers. That is, instead of a precise value of the probability ( )P E p= associated with an event E , a pair of lower and upper probabilities ( ) [ , ]P E p p= are used to include a set of probabilities and quantify the uncertainty. Imprecise probability differentiates epistemic and aleatory uncertainties qualitatively. The range of the interval [ , ]p p captures the

epistemic component, whereas the probability captures the aleatory component. When p p= , the degenerated interval

probability becomes a precise one. When the traditional precise probability is used to analyze systems, sensitivity analysis is usually applied to assess the robustness. Interval-valued probability that considers a collection of scenarios simultaneously thus is an efficient alternative.

Several theories and representations of imprecise probability have been proposed. Yet the computation based on these structures is inefficient. Recently a generalized interval probability [20] based on generalized interval [21,22] was proposed. In generalized interval, an interval [ , ]x x=x is called proper if x x≤ , and called improper if x x≥ . The introduction of improper intervals greatly simplifies the calculus structure of interval probability and makes it very similar to the precise probability in the traditional probability theory. The calculation of generalized intervals is based on the Kaucher interval arithmetic. Particularly, for two intervals [ , ]x x and [ , ]y y , addition and subtraction are defined as [ , ] [ , ] [ , ]x x y y x y x y+ = + + and [ , ] [ , ] [ , ]x x y y x y x y− = − − respectively. The width of interval [ , ]x x=x is defined as


wid : | |x x= −x (1) The relationship between proper and improper intervals is

established by a dual operator defined as dual[ , ] : [ , ]x x x x= (2)

The most important property of the generalized interval probability is the logic coherence constraint (LCC). It is stated that for a mutually disjoint event partition

1

n

iiE

== Ω∪ , the

sum of interval probabilities is always one, i.e.

( )11

n

iiE

==∑ p . The LCC ensures that the imprecise

probability is logically coherent with precise probability. For instance, for a system’s working status with ( ) [0.2, 0.3]down =p , ( ) [0.3, 0.5]idle =p , ( ) [0.5, 0.2]busy =p ,

it is interpreted as

( ) ( ) ( )( )

1 2 3

1 2 3

[0.2, 0.3] [0.3, 0.5] [0.2, 0.5]

1

p p p

p p p

∀ ∈ ∀ ∈ ∃ ∈

+ + =

based on the Theorems of Interpretability [21]. See references [20,23] for more information of the generalized interval probability.

3. PROPOSED RELIABLE KINETIC MONTE CARLO (R-KMC) SIMULATION

Without loss of generality, the vapor deposition process in Fig. 1 is used to illustrate the R-KMC. The vapor deposition process is characterized by several events such as adsorption (with rate a1), intra-layer surface diffusion (with rates a2, …, a5), inter-layer diffusion (with rate a6), and desorption (with rate a7). Each of these rates is an interval value [ , ]

j j ja a=a for

j=1,2,…. If a null-event KMC algorithm [1] without differentiating different locations is applied, the imprecise probability that event j is chosen to be the next event is

0[ , ] /

j j ja a a=p where a0 is the upper bound of the sum of all

propensities. When choosing which event to fire, multiple events can be selected because of the imprecise probability. This is called multi-event algorithm.

A simplified example is used to illustrate the multi-event algorithm. Suppose that there are three events A, B, and C that may occur, which have the respective rates of a1=[1,3], a2=[1,3] and a3=[4,5]. First, the rates are sorted based on the uncertainty levels, i.e. the widths of the intervals, in an ascending order. Next, the lower and upper bounds of intervals are “flipped” in an alternating pattern. Now the sequence of rates is a'3=[4,5], a'1=[3,1], a'2=[1,3]. Then, a null event N is introduced with rate a'N=[1,0]. The general principle of introducing null events is to make sure the interval sum a0=a'1+a'2+a'3+a'N degenerates to a precise number (a0=[9,9] in this example). This is based on the logic coherence constraint mentioned in Section 2.3. The probability mass that event j occurs is pj= a'j/a0. As a result, an empirical cumulative distribution function (c.d.f.) can be built, as shown in Fig. 2. Based on the empirical interval c.d.f., the events are chosen to

fire by generating a random number u~U(0,1) uniformly distributed and selecting the random sets that u corresponds in the c.d.f. For instance, in Fig. 2, if u1 is generated, the next event is C. If u2 is generated, the next events are A, B. That is, a random set of events are chosen.

Fig. 2: An illustration of random set sampling based on the

empirical interval c.d.f. Once a random set of events are fired, the time should be

advanced. In the traditional KMC, the time increment is exponentially distributed with the rate

jja∑ and sampling of

time increment is based on the inverse function of exponential c.d.f. In R-KMC, the time to fire a random set of events has an interval value, which includes the earliest and latest possible times that multiple events occur.

The details of event selection and clock advancement in the multi-event algorithm are described in the following sections. 3.1 Event Selection

For M possible reaction or transition events, each event is characterized by an interval-valued rate or propensity function

ja ( 1, ,j M= … ). First,

1 2, , ,

Ma a a… are sorted based on the

widths of the intervals in the ascending order and become (1) (2) ( ), , , Ma a a… . That is,

( ) ( ) ( )(1) (2) ( )wid wid wid M≤ ≤ ≤a a a where wid( )⋅ is

defined in Eq.(1). Further, a *-sum, denoted by *∑ , is

defined recursively as

( ) ( )* * 1( ) ( ) ( )

1 1: dual 2, ,

J Jj j J

j jJ M

−

= == + =∑ ∑a a a … (3)

with * 1 ( ) (1)

1

J j

j

=

==∑ a a , where dual( )⋅ is defined in Eq.(2). The

semantics of interval propensities is maintained during event selection, shown as follows.

Based on operator “+” of the Kaucher interval arithmetic, if wid wid≤x y for any two intervals x and y that are not improper (i.e. proper intervals or point-wise real numbers), then dual +x y is not improper and

( )wid dual wid+ ≤x y y . Therefore,

1

0

Upper Bound

Lower Bound

A B C N

u2

u1


( ) ( )* 1 ( ) ( 1) ( )

1wid wid wid 2, ,

J j J J

jJ M

− −=

≤ ≤ =∑ a a a … (4)

Fig. 3 illustrates how the cumulative *-sum of the sorted interval propensities is constructed. Because of Eq.(4), the

lower and upper bounds of * ( )

1

J j

j =∑ a are always calculated

from the upper and lower bounds of * 1 ( )

1

J j

j

−

=∑ a respectively. At

the top of the accumulative *-sum, a null event is introduced

with the propensity of ( )* ( )

10,wid

M jnull j=

⎡ ⎤= ⎢ ⎥⎣ ⎦∑a a . Then the

real-valued upper limit of propensity sum is defined as

( ) ( )* *( ) ( )0 1 1

dual 0,widM Mj j

j ja

= =⎡ ⎤= + ⎢ ⎥⎣ ⎦∑ ∑a a (5)

Thus when a random fraction 0

ua is generated, either one or two events will be selected to fire, depending on the random number u . If the projected line of

0ua simultaneously

intersects with two propensities (J and J+1 shown in Fig. 3), the corresponding two events will be fired. Most importantly, the above selection process based on the cumulative *-sum indeed ensures that the probability that event J is selected is

( ) ( )0 0

/ , /J Ja a a a⎡ ⎤⎣ ⎦ , which is the intent of interval

propensities. Again, the introduction of null event is to satisfy LCC so that the sum is up to a real value. When event M and the null event are selected, only event M is fired whereas the null event does not change the state of the system.

Fig. 3: The cumulative *sum to choose events randomly based

on interval propensities The pseudo-code of event selection in the multi-event

algorithm is listed in Table 1. In each iteration of R-KMC, the cumulative *sum is updated first. With a random number generated, the corresponding random set of events is selected. These events then are fired and the system’s state is updated.

Table 1: The pseudo-code of event selection in the multi-event algorithm during each R-KMC iteration

INPUT: The list of propensity_interval OUTPUT: The list of events to fire Sort propensity_interval in the ascending order based on the widths of intervals; // calculate the *sum of interval propensities to build the cumulative distributive function sum_interval[1]=propensity_interval[1];

FOR i=2:numEvents

sum_interval[i]=dual(sum_interval[i−1])+propensity_interval[i]; END sum_interval[numEvents+1]=

dual(sum_interval[numEvents])+propensityOfNullEvent; // sampling a random set of events Generate a random number u ~ Uniform(0,1);

Fraction=u×sum_interval[numEvents+1]

FOR i=1:numEvents

Find the smallest i as index_L such that

Fraction<=sum_interval[index_L].UBound; END FOR i=numEvents:1

Find the largest i as index_U such that

Fraction>=sum_interval[index_U−1].LBound; END RETURN the random set of events corresponding to propensity_interval[index_L:index_U]

3.2 Clock Advancement

In the multi-event algorithm, a random set of multiple events are chosen and fired at a time. Uncertainties are associated with the clock advancement for firing the random set of events. That is, how much time is needed for these random events to occur varies and includes both components of epistemic and aleatory uncertainties. To estimate the epistemic uncertainty component, an interval approach is used here. The worst and best scenarios need to be given. The least time for the set of events to fire is when all of them occur at the same time. Therefore, the lower bound of clock advancement is

1

ln /M

L jjT a

== − ∑ρ (6)

The longest possible time for the set of events to fire is when the events occur consecutively one by one. Suppose that the respective rates of the firing sequence of n out of N events from a random set are (1) (1) ( ) ( )[ , ], ,[ , ]n na a a a… . In the worst-case scenario, the total elapsed time for the sequence of n events is ( ) (1) (2) ( )n nT X X X= + + + , where inter-arrival times (1) (2) ( ), , , nX X X… are exponentially distributed with the respective rates of

0 1 1, , ,

n−α α α… where

* 1 ( )

1

J j

j

−

=∑ a

* ( )

1

J j

j =∑ a Lower bound of (J)

Upper bound of (J)

J+1 J J−1

* ( )

1

M j

j =∑ a

M null

a0

0

u×a0


0 1

(1)1 1

1 ( )1 1 1

...

N

iiN

ii

N n jn ii j

a

a a

a a

=

=

−

− = =

⎧ =⎪⎪ = −⎪⎨⎪⎪ = −⎪⎩

∑∑

∑ ∑

α

α

α

(7)

The rates are changing because after each event is fired, the earliest time for any of the left events to occur depends on the number of events that are left. Then the c.d.f. of ( )nT is

( )( )

( )( ) ( )1

01 1 exp

n

nn

j jj

P T

P T A−

=

≤

= − > = − −∑τ

τ α τ (8)

where 1

0

ni

ji i ji j

A−

=≠

=−∏α

α α. Instead of sampling based on Eq.(8),

it is easier to sample each of ( )jX ’s as exponential distributions and the sum of the sample values will be a sample of ( )nT . Therefore, the upper bound of clock advancement is

1

0ln 1/

n

U jjT

−

=⎡ ⎤= − ⎢ ⎥⎣ ⎦∑ρ α (9)

where ρ is the same random number as in Eq.(6) and j

α ’s are

defined as in Eq.(7). With the interval time increment [ , ]

L UT T , the current

system time [ , ]L Ut t in the simulation clock as an interval is

updated to [ , ]L L U Ut T t T+ + . The purpose of keeping track of

an interval time is to estimate the best-case and worst-case scenarios that how fast a system evolves. The interval clock indirectly provides an estimation of lower and upper bounds of system states, which forms the basis of R-KMC to accommodate the uncertain simulation parameters, particularly

transition rates.

4. IMPLEMENTATION AND DEMONSTRATION The proposed R-KMC based on random set sampling is

implemented in C++ and integrated with SPPARKS [24], which is an open-source KMC toolbox developed at the Sandia National Laboratories. R-KMC is demonstrated with two examples, an E. coli reaction network and the microbial fuel cell (MFC) simulation. 4.1 E. coli Reaction Network

The R-KMC is first demonstrated by the classical model of the E. coli reaction network from Kierzek’s paper [25]. As shown in Fig. 4, nodes represent species. Arrows denote reaction directions and connect reactants to products, associated with real-valued reaction rates. In R-KMC, the rates are intervals because of uncertainties involved. With the interval rates, interval propensities and probabilities are calculated. Within each iteration, a random set of reaction events are selected based on the algorithm in Section 3.1. The interval time advancement is sampled based on the method in Section 3.2.

At 0t = , the initial counts of species are 1 for PLac, 35 for RNAP, and 350 for Ribosome. All others are zeros. Based on the nominal values of rates in Fig. 4, interval rates are used. Interval rates for ±1% and ±10% of the nominal values in the respective experiments are chosen and the results are compared with that from the traditional KMC. For instance, the rates of the reaction PLacRNAP→TrLacZ1 at the right-bottom corner of Fig. 4 become [0.99,1.01] and [0.9,1.1] respectively. For each of the three scenarios (original real-valued rates, ±1% interval rates, and ±10% interval rates), the model runs 20 times.

Fig. 4: The reaction channels of LacZ and LacY proteins in E. coli [25]

PLac

RNAP

PLacRNAP

TrLacZ1

RbsLacZ

TrLacZ2 TrLacY1 TrLacY2

RbsLacY

Ribosome

RbsRibosomeLacZ

RbsRibosomeLacY

TrRbsLacZ

TrRbsLacY

LacY

LacZ

dgrLacZ

dgrRbsLacZ

dgrLacY

dgrRbsLacY

0.17

10

1

10.0151

0.36

0.17 0.45

0.17

0.45

0.4

0.4

0.036

0.015

6.42E-5

6.42E-5

0.3

0.3

lactose

LacZlactose

14

product

9.52E-5

431


Fig. 5: Comparisons of the numbers of species over time in the

E. coli reaction network between the KMC and R-KMC simulations, where interval reaction rates are ±1% and ±10%.

Fig. 5 compares the average values of some species over time, including product, Ribosome, dgrLacY, and LacZ. In the charts, red solid curves represent the average values in the real case. Blue dotted and dash curves are the lower and upper bounds of those for the ±1% interval case. For the lower-bound curves, the upper-bound time increment

UtΔ is used in the plot,

whereas LtΔ is used for the upper-bound curves. Similarly,

green dotted and dash curves are the lower and upper bounds of those for the ±10% interval case. It is observed that the ±10% interval case enclose the real case better than the ±1% interval case. That is, at a particular time the likelihood that the predicted lower and upper bounds for the number of a species in R-KMC enclose the real-valued prediction in KMC is higher in the ±10% interval case. The reliability or robustness of R-KMC is quantified by the probability that the results from the real-valued KMC are enclosed by those from R-KMC, which will be more rigorously defined in Eq.(14). It is seen that the wider the input interval rates are, the more robust the prediction would be. 4.2 Microbial Fuel Cell (MFC)

The MFC is a type of fuel cell that converts the chemical energy contained in organic matter to electricity using microorganisms (bacteria) as a biocatalyst from organic waste and renewable biomass. A MFC typically consists of two chambers, an anaerobic anode chamber and an aerobic cathode chamber separated by a proton exchange membrane (PEM). For scaling up the MFC technology to the commercial scale, fine-grained multi-physics simulation of electrochemical processes for the entire fuel cell system is necessary for better understanding of the system. Here the R-KMC mechanism is demonstrated with a simplified example of MFC reaction networks adopted from Refs. [26,27] as in Table 2, where the estimated rates used in the R-KMC simulation are also listed. The first nine reactions (R1-R9) occur in the anode chamber whereas the last four (R10-R13) in the cathode chamber. Two example outputs are shown in Fig. 6. Similar to the previous example, it is highly likely that interval-valued predictions enclose real-valued ones when systems are stabilized. The engineering design procedure involves the optimization of geometries or chemical compositions of electrode, PEM, reduction-oxidization mediator, etc. for desirable rates.

5. WEAK CONVERGENCE The random set based R-KMC can be viewed as a

generalization of the traditional KMC. In this section, it will be shown that the multi-event algorithm in R-KMC converges to the traditional SSA in KMC smoothly with respect to both distributions and expected values as the widths of interval transition rates reduce towards zeros, when epistemic uncertainty vanishes.

(b) Ribosome

(c) dgrLacY

(d) LacZ

(a) product


Table 2: The modeled reactions of a two-chamber MFC

Event type Species and reactions Rate constant

R1: water dissociation H2O ↔ OH− + H+ 101 R2: carbonic acid dissociation CO2 + H2O ↔ HCO3

− + H+ 101 R3: acetic acid dissociation AcH ↔ Ac− + H+ 101 R4: reduced thionine first dissociation

MH3+ ↔ MH2 + H+ 101

R5: reduced thionine second dissociation

MH42+ ↔ MH3

+ + H+ 101

R6: acetate with oxidized mediator

Ac− + MH+ + NH4+ + H2O →

XAc + MH3+ + HCO3

− + H+ 101

R7: oxidation double protonated mediator

MH42+ → MH+ + 3H+ + 2e− 101

R8: oxidation single protonated mediator

MH3+ → MH+ + 2H+ + 2e− 101

R9: oxidation neutral mediator

MH2 → MH+ + H+ + 2e− 101

R10: proton diffusion through PEM

H+ → H_+ 10−2

R11: electron transport from anode to cathode

e− → e_− 10−2

R12: reduction of oxygen with current generated

2H_+ + 1/2O2_ + 2e_− → H2O_ 105

R13: reduction of oxygen with current generated

O2_ + 4e_− + 2H2O_ → 4OH_− 103

Fig. 6: Comparisons of the numbers of species over time in the

MFC reaction network between the traditional KMC and R-KMC simulations, where interval reaction rate is ±10%.

Suppose that a system consisting of N species

1 2, , ,

NS S S… has the state variable

( )1( ), , ( ) N

NX t X t += ∈ Z…X at time t where 0+ = ∪Z N is

the set of nonnegative integers. There are a total of M reaction channels

jR ’s ( 1, ,j M= … ), each of which is characterized

by an interval-valued propensity function ( )ja x given the

current state ( )t =X x , where 2: Nj + +→Z Ra , and its state

change vector ( )1, ,

j j jNv v= …v . From the sum of propensity

function 0( )a x defined in Eq.(5), the probability of firing

jR is

( )( )1 0 0min , , / , /

j M j jP T T T a a a a⎡ ⎤= = ⎣ ⎦… .

As a special case of the generalized Chapman-Kolmogorov equation (Eq.(1.8) in Appendix), the interval master equation that describes the evolution of the distribution of the system states is

( ) ( ) ( )

( ) ( )

0 0 0 01

0 01

, | , , | ,

dual , | ,

M

j j jj

M

jj

dt t t t

dt

t t

=

=

= − −

⎡ ⎤− ⎢ ⎥

⎢ ⎥⎣ ⎦

∑

∑

p x x W x v p x v x

W x p x x(10)

where ( )0 0, | ,t tp x x is the probability of ( )t =X x given the

initial distribution 0 0

( )t =X x , and 0

( ) ( ) / ( )j j

a=W x a x x is

the transition rate. Because ja ’s are interval propensities, the

solution of Eq.(10) is a set of probability evolution paths. In the multi-event algorithm in Section 3, the probability

density function of time between jR is fired at the current state

x and next jR is

( ) ( )0 0 0

( , )

( ) ( )exp ( ) ( )exp ( )j

j j

t

a a t a t= ⋅ − = −

f x

W x x x a x x (11)

which converges to the one in the traditional SSA as the widths of the interval propensities

ja ’s reduce to zeros.

For the state variable, the path-wise representation of the Poisson process with the integral form is

( ) ( )0 01

( )M t

j j jj

t d=

⎛ ⎞= + ⎜ ⎟⎝ ⎠∑ ∫P τ τX x a X v (12)

where ( ) ( )0

( )t

j j jt d⎛ ⎞= ⎜ ⎟

⎝ ⎠∫P P τ τa X is a generalized Poisson or

counting process of the number of events to fire jR and

( )0

( )t

jd∫ τ τa X is the mean firing rate of time t . Again, ( )tX

is a random interval with discrete integer values. The expectation is

(a) H2O in anode chamber

(b) H+ in cathode chamber


( ) 0 0

1

0 00 01 1

( ) ( )

( ) , ( )

M t

j jj

M Mt t

j j j jj j

t d

a d a d

=

= =

= +

⎡ ⎤= + +⎢ ⎥⎢ ⎥⎣ ⎦

∑ ∫

∑ ∑∫ ∫

E τ τ

τ τ τ τ

X x v a

x v x v (13)

Therefore, when ( ) ( )j ja a=x x for all x , ( )j

tP ’s are

degenerated to regular Poisson processes in the traditional KMC, and the expected state values in R-KMC are the same as the ones in KMC.

The R-KMC is to improve the robustness of the simulation given input uncertainties. During simulation, the imprecise rates in input are converted to imprecise times to achieve certain states in output. The robustness is measured by the time of enclosure as

( ) ( ) ( )11 1P X X X

−− −⎛ ⎞≤ ≤⎜ ⎟⎝ ⎠

x x x (14)

for a particular state x , where random variables ( )1X

−⋅ ,

( )1X − ⋅ , and ( )1

X−

⋅ are the times that x is reached for the

lower-bound, real-valued, and upper-bound scenarios respectively. All three time variables follow the Erlang distribution with c.d.f.’s as

( )( ) ( )011

001 / !i

kn a ti k

P X t e a t k− −−=

⋅ ≤ = −∑ (15)

where 1iX − is

1X

−, 1X − , or

1

X−

, correspondingly in is n ,

n , or n , which are the minimal, real-case, and maximal numbers of events that occur respectively.

6. CONCLUDING REMARKS In this paper, an R-KMC mechanism based on random-set

sampling is proposed to improve the KMC simulation robustness where uncertainty associated with transition rates and probabilities in simulation is considered. The new R-KMC mechanism considers a range of possible state updates for each simulation step based on a proposed multi-event algorithm. The multi-event algorithm uses generalized intervals and strictly follows the semantics of interval probability so that the sampling process is a generalization of the traditional SSA in KMC. The convergence of the multi-event algorithm towards the traditional SSA is demonstrated.

For simple chemical reactions, the uncertainty effect is captured by an interval system time, which indicates the earliest and latest possible times required for the system reach a particular state, as demonstrated by two examples. For more complex systems where complete chemical and physical processes at different locations need to be kept track of, the uncertainty effect can be similarly captured as the earliest and latest times that a number of certain species at a particularly location is observed.

In this paper, only the worst- and best-case of time advancement is considered in the multi-event algorithm. Future

work will include the extension to incorporate generalized intervals so that interval uncertainty of time can be reduced to zeros by calculating with improper intervals. For instance, when measurements are taken at certain time, they can serve as a reference point and precise numbers of species are given without interval uncertainty. This may generalize the procedure of R-KMC simulation and validation.

APPENDIX – DERIVATION OF THE GENERALIZED CHAPMAN-KOLMOGOROV EQUATION

With , nx y ∈ R , the time evolution of the probability ( , | , ')x t y tp is characterized as

0

( , | , ')

1lim ( , | , ') dual ( , | , ')t

x t y tt

x t t y t x t y ttΔ →

∂∂

+ Δ −Δ

p

p p (1.1)

Because of the logic coherent constraint, we have ( , | , ) 1dz z t t x t+ Δ =∫ p (1.2)

where nz ∈ R . In addition, ( , | , ') ( , | , ) ( , | , ')x t t y t dz x t t z t z t y t+ Δ = + Δ∫p p p (1.3)

Replace the first term in Eq.(1.1) by Eq.(1.3) and multiply Eq.(1.2) with the second term, Eq.(1.1) now becomes

0

( , | , ')

( , | , ) ( , | , ')1lim

dual ( , | , ) ( , | , ')t

x t y tt

x t t z t z t y tdz

z t t x t x t y ttΔ →

∂∂

⎧ ⎫⎡ ⎤+ Δ⎪ ⎪= ⎢ ⎥⎨ ⎬− + ΔΔ ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭∫

p

p pp p

(1.4)

Consider two subdomains x z− ≤ ε and x z− > ε

separately where ε is small enough. Eq.(1.4) can be regarded as

( , | , ')x z x z

x t y t I It − ≤ − >

∂= +

∂p

ε ε

where x zI

− ≤ε is the right-side integral in Eq.(1.4) within the

small neighborhood that captures continuous diffusion processes, whereas

x zI

− >ε is the one outside the neighborhood

that represents jump processes. Within the small neighborhood x z− ≤ ε , let x z h− =

and define ( , ) ( , | , ) ( , | , ')x h x h t t x t x t y t+ + Δf p p . Then ( , | , ) ( , | , ') ( , )z t t x t x t y t x h+ Δ = −p p f and ( , | , ) ( , | , ')

( , | , ) ( , | , ') ( , )

x t t z t z t y t

x t t x h t x h t y t x h h

+ Δ= + Δ − − = −p pp p f

.

Apply the Taylor alike expansion

( )2 3

1

( , )

1( , ) / /

2

n

i i i j i ji i j

x h h

x h x h x x h h O=

−

= − ∂ ∂ + ∂ ∂ ∂ +∑ ∑ ∑f

f f f ε,


where 0

( , ) / lim ( , ) dual ( , ) /i

i i ixx h x x x h x h x

Δ →∂ ∂ + Δ − Δf f f

and similarly 2

00

( , )

( , ) dual ( , )lim /

dual ( , ) dual ( , )i

j

i j

i j ii jx

jx

x h

x x

x x x h x x hx x

x x h x hΔ →Δ →

∂∂ ∂

⎧ ⎫+ Δ + Δ − + Δ⎪ ⎪ Δ Δ⎨ ⎬− + Δ +⎪ ⎪⎩ ⎭

f

f ff f

.

Then

0

0

1

2

1 10

( , | , ) ( , | , ')1lim

dual ( , | , ) ( , | , ')

1lim ( , ) dual ( , )

( , )

1 1lim

2

x z x zt

ht

n

iii

n n

i ji jti j

x t t z t z t y tI dz

z t t x t x t y tt

dh x h h x ht

x h hx

dh h ht x x

− ≤ − ≤Δ →

≤Δ →

=

= =Δ →

⎧ ⎫⎡ ⎤+ Δ⎪ ⎪= ⎢ ⎥⎨ ⎬− + ΔΔ ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎧ ⎫⎡ ⎤= − − −⎨ ⎬⎣ ⎦Δ ⎩ ⎭

∂−

∂∂

= +Δ ∂ ∂

∫

∫

∑

∑

p pp p

f f

ff

f

ε ε

ε

( )

( )

3

0

10

2

1 10

0

dual ( , )

1lim ( , ) dual ( , )

1lim

1 1lim

2

1lim

h

h ht

n

iihti

n n

i ji jhti j

t

x h O

dh x h dh x ht

dh ht x

dh h h Ot x x

dht x

≤

≤ ≤Δ →

=≤Δ →

= =≤Δ →

Δ →

⎧ ⎫⎡ ⎤⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎨ ⎬⎢ ⎥⎪ ⎪⎢ ⎥⎪ ⎪⎢ ⎥− − +⎪ ⎪⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎡ ⎤= − −⎢ ⎥Δ ⎣ ⎦

∂−

Δ ∂∂

+ +Δ ∂ ∂

∂= −

Δ ∂

∑∫

∫ ∫

∑∫

∑ ∑∫

f

f f

f

f

f

ε

ε ε

ε

ε

ε

ε

( )

1

2

1 1 0

1 1lim

2

n

ii hi

n n

i ji j hti j

h

dh h h Ot x x

= ≤

= = ≤Δ →

⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

⎡ ⎤∂+ +⎢ ⎥

Δ ∂ ∂⎢ ⎥⎣ ⎦

∑ ∫

∑ ∑ ∫f

ε

εε

since ( , ) ( , )h hdx x h dx x h

≤ ≤= −∫ ∫f f

ε ε. Furthermore,

1 1

1( , | , ) ( , | , ')

i ih hi i

ihi

dh h h dht x x t

h dh x h t t x t x t y tx t

≤ ≤

≤

⎡ ⎤∂ ∂= ⎢ ⎥Δ ∂ ∂ Δ⎣ ⎦

⎡ ⎤∂= + + Δ⎢ ⎥∂ Δ⎣ ⎦

∫ ∫

∫

ff

p p

ε ε

ε

,

and similarly

2

2

1

1( , | , ) ( , | , ')

i jhi j

i jhi j

dh h ht x x

h h dh x h t t x t x t y tx x t

≤

≤

∂Δ ∂ ∂

⎡ ⎤∂= + + Δ⎢ ⎥∂ ∂ Δ⎣ ⎦

∫

∫

f

p p

ε

ε

.

We thus define

0

1( , ) lim ( , | , )i ihtx t h dh x h t t x t

t ≤Δ →+ + Δ

Δ ∫A pε

(1.5)

0

1( , ) lim ( , | , )ij i jhtx t h h dh x h t t x t

t ≤Δ →+ + Δ

Δ ∫B pε

(1.6)

If ( )1( ) ( ), , ( )

nX t x t x t= … represents the state of the system at

time t , then ( , | , ) ( ) ( ) | ( )

i i ihh dhp x h t t x t x t t x t X t

≤⎡ ⎤+ + Δ = + Δ −⎣ ⎦∫ E

ε is

the expected state value change in the i th direction. Therefore, Eq.(1.5) is interpreted as the state value change rate along one direction, known as the drift vector. Similarly,

( ) ( )( , | , )

( ) ( ) ( ) ( ) | ( )

i jh

i i j j

h h dhp x h t t x t

x t t x t x t t x t X t

≤+ + Δ

⎡ ⎤= + Δ − + Δ −⎣ ⎦

∫Eε .

Eq.(1.6) is the combined area change rate in two directions, known as the diffusion matrix.

We now have

( )

1

1 1

( , )( , | , ')

( , )1( , | , ')

2

n ix z i

i

n n ij

i ji j

x tI x t y t

xx t

x t y t Ox x

− ≤ =

= =

∂= −

∂∂

+ +∂ ∂

∑

∑ ∑

Ap

Bp

ε

ε

where

0

( , ) / lim ( , ) dual ( , ) /i

i i i i i ixx t x x x t x t x

Δ →∂ ∂ + Δ − ΔA A A

and

2

00

( , )

( , ) dual ( , )1lim

dual ( , )

dual ( , )i

j

ij i j

ij ij i

xij ji j i jx

ij

x x x t

x t x x t

x x tx x x xx t

Δ →Δ →

⎧ ⎫+ Δ + Δ⎪ ⎪∂ − + Δ⎪ ⎪⎨ ⎬− + Δ∂ ∂ Δ Δ ⎪ ⎪⎪ ⎪+⎩ ⎭

BB B

BB

for the diffusion process. For the jump process

0

0

0

( , | , ) ( , | , ')1lim

dual ( , | , ) ( , | , ')

1lim ( , | , ) ( , | , ')

1dual lim ( , | , ) ( , | , ')

x z

x zt

x z t

x z t

I

x t t z t z t y tdz

z t t x t x t y tt

dz x t t z t z t y tt

dz z t t x t x t y tt

− >

− >Δ →

− > Δ →

− > Δ →

⎧ ⎫⎡ ⎤+ Δ⎪ ⎪= ⎢ ⎥⎨ ⎬− + ΔΔ ⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭⎡ ⎤

= + Δ⎢ ⎥Δ⎣ ⎦⎡ ⎤

− + Δ⎢ ⎥Δ⎣ ⎦

∫

∫

∫

p pp p

p p

p p

ε

ε

ε

ε

We define

0

1( | , ) lim ( , | , )

ty x t y t t x t

tΔ →+ Δ

ΔW p (1.7)

as the rate of transition from x to y . Then

( | , ) ( , | , ')

dual ( | , ) ( , | , ')

x z x z

x z

I dz x z t z t y t

dz z x t x t y t

− > − >

− >

⎡ ⎤= ⎣ ⎦⎡ ⎤− ⎣ ⎦

∫∫

W p

W p

ε ε

ε

As 0→ε , the subdomain x z− > ε becomes the complete

domain. As a result, the generalized differential Chapman-Kolmogorov equation is


2

1 1 1

( , )( , ) 1( , | , ') ( , | , ') ( , | , ')

2

( | , ) ( , | , ') dual ( | , ) ( , | , ')

n n n iji

i i ji i j

x tx tx t y t x t y t x t y t

t x x x

dz x z t z t y t dz z x t x t y t

= = =

∂∂∂= − +

∂ ∂ ∂ ∂

+ −

∑ ∑ ∑

∫ ∫

BAp p p

W p W p

(1.8)

REFERENCES [1] Chatterjee A. and Vlachos D.G. (2007) An overview of

spatial microscopic and accelerated kinetic Monte Carlo methods. Journal of Computer-Aided Materials Design, 14:253-308

[2] Voter, A.F. (1998) Parallel replica method for dynamics of infrequent events, Physical Review B, 57(22): R13985-R13988

[3] Voter, A.F. (1997) Hyperdynamics: Accelerated molecular dynamics of infrequent events, Physical Review Letters, 78(20): 3908-3911

[4] Sørensen, M.R. and Voter, A.F. (2000) Temperature-accelerated dynamics for simulation of infrequent events, Journal of Chemical Physics, 112: 9599

[5] Law A.M. and Kelton W.D. (2000) Simulation Modeling & Analysis. 3rd Ed., New York: McGraw-Hill.

[6] Banks J., Carson J.S., and Nelson B. (1996) Discrete-Event System Simulation. 2nd Ed. Upper Saddle River, NJ: Prentice-Hall.

[7] Batarseh O.G. (2010) An Interval-Based Approach to Model Input Uncertainty in Discrete-Event Simulation. Ph.D. dissertation, University of Central Florida, Orlando

[8] Batarseh O.G. and Wang Y. (2008) Reliable Simulation with Input Uncertainties Using an Interval-Based Approach. Proc. 2008 Winter Simulation Conference, Dec.7-10, 2008, Miami, FL, pp.344-352

[9] Batarseh O.G., Nazzal D., and Wang Y. (2010) An interval-based metamodeling approach to simulate material handling in semiconductor wafer fabs. IEEE Transactions on Semiconductor Manufacturing, 23(4): 527-537

[10] Bortz A.B., Kalos M.H., and Lebowitz J.L. (1975) A new algorithm for Monte Carlo simulation of Ising spin systems. Journal of Computational Physics, 17(1): 10-18

[11] Gillespie D.T. (1976) A general method for numerically simulating the stochastic evolution of coupled chemical reactions. Journal of Computational Physics, 22: 403-434

[12] Maksym PA (1988) Fast Monte Carlo simulation of MBE growth. Semiconductor Science & Technology, 3(6): 594-596

[13] Blue J.L., Beichl I., and Sullivan F.(1995) Faster Monte Carlo simulations. Physical Review E, 51(2): R867-868

[14] Schulze T.P. (2002) Kinetic Monte Carlo simulations with minimal searching. Physical Review E, 65(3): 036704(1-3)

[15] Berry H. (2002) Monte Carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation. Biophyiscal Journal, 83: 1891-1901

[16] Schnell S. and Turner T.E. (2002) Reaction kinetics in

intracellular environments with macromolecular crowding: simulations and rate laws. Progress in Biophysics & Molecular Biology, 85: 235-260

[17] Bratsun D., Volfson D., Tsimring L.S., and Hasty J. (2005) Delay-induced stochastic oscillations in gene regulation. Proceedings of the National Academy of Sciences of the U.S.A., 102(41): 14593-14598

[18] Barrio M., Burrage K., Leier A., and Tian T. (2006) Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation. PLoS Computational Biology, 2(9): 1017-1030

[19] Burrage K., Hancock J., Leier A., and Nicolau D.V. (2007) Modelign and simulation techniques for membrane biology. Briefing in Bioinformatics, 8(4): 234-244

[20] Wang Y. (2010) Imprecise probabilities based on generalized intervals for system reliability assessment. International Journal of Reliability & Safety, 4(4): 319-342

[21] Gardeñes E., Sainz M.A., Jorba L., Calm R., Estela R., Mielgo H., and Trepat A. (2001) Modal intervals. Reliable Computing, 7(2): 77-111

[22] Dimitrova N.S., Markov S.M., and Popova E.D. (1992) Extended interval arithmetics: new results and applications, Computer Arithmetic and Enclosure Methods, L. Atanassova and J. Herzberger (eds.), pp.225-232

[23] Wang Y. (2008) Imprecise probabilities with a generalized interval form. Proc. 3rd Int. Workshop on Reliability Engineering Computer (REC’08), Savannah, Georgia, pp.45-59

[24] SPPARKS Kinetic Monte Carlo Simulator, available at http://www.cs.sandia.gov/~sjplimp/spparks.html

[25] Kierzek A.M. (2002) STOCKS: STOChastic Kinetic Simulations of biochemical systems with Gillespie algorithm. Bioinformatics, 18(3): 470-481

[26] Picioreanu C., van Loosdrencht M.C.M., Curtis T.P., and Scott K. (2010) Model based evaluation of the effect of pH and electrode geometry on microbial fuel cell performance. Bioelectrochemistry, 78(1): 8-24

[27] Zeng Y., Choo Y.F., Kim B.-H., and Wu P. (2010) Modeling and simulation of two-chamber microbial fuel cell. Journal of Power Sources, 195: 79-89

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