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in nature , there exist various systems that involve chemical reactions . some systems are on a geographical scale while others are on a nanoscale , in particular , the biochemical reactions in a cell . to study the dynamics of reaction systems , we often adopt rate equations in order to observe the change in the chemical concentrations . in rate equations , we consider the concentrations to be continuous variables and the rate of each reaction as a function of the concentrations . in fact , in macroscopic systems , there are a large number of molecules ; therefore , continuous representations are usually applicable .    when the concentration of a certain chemical is low , fluctuations in the reactions or flows can not be negligible . they are usually treated by employing stochastic differential equations , in which the noise is used as a continuum description of the fluctuations @xcite . the employment of stochastic differential equations has led to some important discoveries such as noise - induced transitions @xcite , noise - induced order @xcite , and stochastic resonance @xcite .    in stochastic differential equations , the quantities of chemicals are still regarded as continuous variables . at a microscopic level , however , we need to seriously consider the fact that the number of molecules is an integer ( @xmath0 , @xmath1 , @xmath2 , @xmath3 ) that changes in a discrete manner . fluctuations originate from the discrete stochastic changes ; thus , continuum descriptions of fluctuations are not always appropriate . biological cells appear to provide a good example for such discreteness in molecule numbers . the size of the cells is of the order of microns , in which nanoscale `` quantum '' effects can be ignored . however , in these cells , some chemicals act at extremely low concentrations of the order of pm or nm . assuming that the typical volume of a cell ranges from @xmath1 to @xmath4 @xmath5 , the concentration of one molecule in the cell volume corresponds to @xmath6 pm to @xmath6 nm . it is possible that the molecule numbers of some chemicals in a cell are of the order of @xmath1 or sometimes even @xmath0 . if such chemicals play only a minor role in a cell , we can safely ignore these chemicals to study intracellular chemical processes . however , this is not always the case . in biological systems , chemical species with a small number of molecules may critically affect the behavior of the entire system . as an extreme example , there exist only one or a few copies of genetic molecules such as dna , which are important to characterize the behavior of each cell . further , some experiments show that doses of particular chemicals at concentrations of the order of pm or fm may alter cell behavior ( e.g. , @xcite ) . biological systems also include positive - feedback mechanisms such as autocatalytic reactions , which may amplify single molecular changes to a macroscopic level . it is only recently that the stochastic effect due to small molecule numbers in cells has been noticed both theoretically @xcite and experimentally @xcite .    in this paper , we focus on the possible effects of molecular discreteness . through stochastic simulations , we showed that the discreteness can induce transitions to novel states in autocatalytic systems @xcite , which may affect macroscopic chemical concentrations @xcite . in the first part of this paper , we briefly review these studies and explain other aspects of such effects . see also @xcite for recent advances in the present topic by analytic methods and numerical simulations and @xcite for simulation methods concerned .    in some cases , the discreteness in the molecule numbers may cause switches between two or more states with distinct concentrations and dynamical behaviors . further , even though the concentration of chemicals is sufficiently high for one state , the concentration could be low in another state , in which a chemical with a very low concentration could work as a stochastic switch . in the second part of this paper , we discuss how molecular discreteness leads to switch among states with distinct dynamical behaviors in an autocatalytic chemical reaction network system . this spontaneous switching is characterized as an alteration ( i.e. , disconnection and reconnection ) of the acting reaction paths . for @xmath7 , @xmath8 , and @xmath9 . in this case , @xmath10 can reach @xmath0 , and the switching states appear . in the 1 - 3 rich state , the system successively switches between the @xmath11 and @xmath12 states . the interval of switching is considerably longer than the period of continuous vibration ( @xmath13 ) . at around @xmath14 , a transition occurs from the 1 - 3 rich state to the 2 - 4 rich state.,width=302 ]     in the autocatalytic loop system for @xmath15 and @xmath16 with nonequivalent reaction constants . for small @xmath17 , the flow of molecules dominates the system . thus , @xmath18 , which simply reflects @xmath19 ; this does not depend on how the continuum limit is imbalanced by the reactions . ( a ) @xmath20 and @xmath21 . ( b ) @xmath22 and @xmath23.,title="fig:",width=241 ]    in the autocatalytic loop system for @xmath15 and @xmath16 with nonequivalent reaction constants . for small @xmath17 , the flow of molecules dominates the system . thus , @xmath18 , which simply reflects @xmath19 ; this does not depend on how the continuum limit is imbalanced by the reactions . ( a ) @xmath20 and @xmath21 . ( b ) @xmath22 and @xmath23.,title="fig:",width=241 ]    we have previously reported that the discrete nature of molecules may induce transitions to novel states , which are not reproduced by the continuous descriptions of the dynamics ( stochastic differential equations ) @xcite . here , we briefly review that result by including some novel results . we consider a simple autocatalytic network ( loop ) with four chemicals @xmath24 ( @xmath25 ) . we assume the reactions @xmath26 ( with @xmath27 ) between these chemicals . all the reactions are irreversible . we assume that the reactor is a well - stirred container with a volume @xmath17 . the set of @xmath10 , the number of @xmath24 molecules , determines the state of the system . the container is in contact with a chemical reservoir in which the concentration of @xmath24 is fixed at @xmath28 . the flow rate of @xmath24 between the container and the reservoir is @xmath29 , which corresponds to the probability of the flow - out of a molecule per unit time . is the diffusion rate across the surface of the container . here , we choose a flow proportional to @xmath17 in order to obtain a well - defined continuum limit . ] we can consider the continuum limit as @xmath30 . in this limit , the change in @xmath31 , the chemical concentration of @xmath24 in the container , obeys the following rate equation : @xmath32 where @xmath33 is the rate constant of the reaction @xmath26 , and @xmath34 .    in @xcite , we considered a case with four equivalent chemical species , given as @xmath35 , @xmath36 , and @xmath37 for all @xmath38 ( @xmath39 , @xmath40 , @xmath41 ) , @xmath42 . in the continuum limit , the dynamics is represented by the rate equation , which has only one attractor : a stable fixed point @xmath43 for all @xmath38 . around the fixed point , @xmath31 vibrates with the frequency @xmath44 . if the number of molecules is finite but fairly large , we can estimate the dynamical behavior of the system using the langevin equation , which is obtained by adding a noise term to the rate equation . each concentration @xmath31 fluctuates and vibrates around the fixed point . an increase in the noise ( corresponding to a decrease in the number of molecules ) merely amplifies the fluctuation . however , as we have shown in @xcite , when the number of molecules is small , novel states that do not exist in the continuum limit are observed . two chemicals are dominant and the other two are mostly extinct ( @xmath45 ) . figure [ fig : a-032 ] shows the time series of @xmath10 in such a case . at @xmath46 , @xmath47 and @xmath48 dominate the system and @xmath49 for the most part ( the 1 - 3 rich state ) . once the system reaches @xmath49 , all the reactions stop . the system remains at @xmath49 for a long time as compared with the ordinary time scale of the reactions ( @xmath50 ) . conversely , at @xmath51 , @xmath52 and @xmath53 are large and usually @xmath54 ( the 2 - 4 rich state ) . in the 1 - 3 or 2 - 4 rich states , the system alternately switches between either @xmath11 and @xmath12 or @xmath55 and @xmath56 . we name these states the `` switching states . '' the appearance of discreteness - induced novel states is described as a phase transition with a decrease in the system size ( or flow rate ) , where the histogram of @xmath57 exhibits a change from single - peaked to double - peaked distribution .    in this example , although the state at each instant exhibits a clear transition , the average concentrations of the chemicals are not altered since the system resides equally in the 1 - 3 rich and 2 - 4 rich states over a long time span . on the other hand , we found examples in @xcite , where the long - term average concentration of the molecule species is altered by the discreteness as well . a simple example is provided by considering the same reaction model as that depicted by eq . ( [ eqn : tr1-rate ] ) , but by considering the case where the parameters @xmath29 , @xmath28 , or @xmath33 are species dependent . note that the rate equation ( [ eqn : tr1-rate ] ) obtained in the continuum limit does not contain the volume @xmath17 ; hence , the average concentrations should be independent of @xmath17 . here , we seek the possibility of the change in the average concentrations depending on the decrease in the system size @xmath17 by taking advantage of the switching states . recall that for the transitions to the switching states to occur in @xcite , it was necessary for the interval of the inflow to be greater than the time scale of the reactions . in the present case , the inflow interval of @xmath24 is @xmath58 , and the time scale of the reaction @xmath26 in order to consume @xmath24 is @xmath59 . if the conditions of all the chemicals are equivalent , the discreteness of all the chemicals takes effect equally and the 1 - 3 and 2 - 4 rich states coordinately appear at @xmath60 . now , since the parameters are species dependent , the effect of the discreteness may be different for each species . for example , by assuming that @xmath61 , the inflow interval of @xmath62 is greater than that of @xmath63 . thus , the discreteness in the inflow of @xmath62 can be significant for larger @xmath17 .    in the previous paper @xcite , we studied the case in which only the external concentrations ( chemical inflows ) @xmath28 were dependent on the species . based on the degree of @xmath64 , the discreteness - induced transition occurs successively with the decrease in @xmath17 , and the average concentrations of the chemicals take distinct values from those of the continuum limit case . similarly , we can study the effect of the discreteness when each of the chemical reaction rates @xmath33 is species dependant . in fact , the dependence of @xmath65 on @xmath17 in this case is different from the previous study in which only @xmath28 was species dependent . for example , we consider the case that @xmath66 and @xmath67 . figure [ fig : tr2-average - r ] shows the dependence of @xmath65 on @xmath17 . recall that the concentrations should not depend on @xmath17 as long as the continuum representations hold ( eq . ( [ eqn : tr1-rate ] ) does not contain @xmath17 ) . here , in the continuum limit or in the case of large @xmath17 , @xmath68 , as shown in fig . [ fig : tr2-average - r ] ( a ) . in contrast , when @xmath17 is small , @xmath18 . if @xmath17 is very small , so that the total number of molecules is mostly @xmath0 or @xmath1 , the reactions rarely occur and the flow of chemicals dominates the system . thus , @xmath69 . if both the reactions and the flows are species dependent , we can simply expect the behavior to be a combination of the above mentioned cases . even this simple system can exhibit a multi - step change in the concentrations along with the change in @xmath17 . furthermore , the present behavior is not limited to the simple autocatalytic reaction loop . in fact , we observe this type of change in randomly connected reaction networks . for a large reaction network with multiple time scales of reactions and flows , the discreteness effect may bring about behaviors that are more complicated , although our discussion is generally applicable to such cases if the time scales are appropriately defined .    as seen above , the discreteness of molecules can alter the average concentrations . when the rates of inflow and/or the reaction are species dependent , the transitions between the discreteness - induced states are imbalanced . this may drastically alter the average concentrations from those of the continuum limit case . note that although the concentrations are altered in both cases , their dependence on @xmath17 is different . if the system is extremely small ( @xmath70 ) , the frequency of the reaction event , in comparison to the diffusion , is low . the reaction is limited by the inflows , and therefore , the system is dominated by diffusion . the average concentrations @xmath65 depend on @xmath17 , but the dependence is quite different from the case with uniform reaction rates ( @xmath33 ) and imbalanced inflows ( @xmath28 ) , which were previously reported by us . ( substrate ) . ( i ) if all the chemical species exist , all the reactions may occur . the system exhibits damped oscillations . ( ii , iii ) if the system lacks one or more chemicals , some of the reactions can not proceed . the portion of the reaction path beyond the stalled reaction is disconnected ; consequently , the actual topology of the network may change . , width=483 ]    , the number of molecules . ( a ) @xmath71 , ( b ) @xmath72 , and ( c ) @xmath73.,title="fig:",width=241 ] , the number of molecules . ( a ) @xmath71 , ( b ) @xmath72 , and ( c ) @xmath73.,title="fig:",width=241 ] + , the number of molecules . ( a ) @xmath71 , ( b ) @xmath72 , and ( c ) @xmath73.,title="fig:",width=340 ]     and @xmath74 with different @xmath17 . transition is observed between the @xmath75 and @xmath76 cases.,title="fig:",width=241 ]   and @xmath74 with different @xmath17 . transition is observed between the @xmath75 and @xmath76 cases.,title="fig:",width=241 ]    molecular discreteness may not only affect the chemical concentrations but also the network of reactions . as seen above , if the number of molecules required for a certain reaction is @xmath0 , the reaction can not take place at all . if such a situation continues for a long time , when compared to the time scales of other reactions , the system behaves as if the reaction never existed , i.e. , the reaction is virtually eliminated from the network . furthermore , the existence of even one or a few molecules could cause the resumption of the reaction and the recovery of the network . in contrast to the continuum limit , where decay or recovery of the chemical is gradual , such changes in the network structure are discrete and therefore quick .    here , we show an example in which the discreteness alters the actual network and switches the dynamical behavior . we adopt a simple model with four chemicals and five reactions among them ( see fig . [ fig : net2 ] ) such that @xmath77 again , we assume a well - mixed reactor of volume @xmath17 in contact with a chemical reservoir , where the concentration of @xmath24 is maintained at @xmath28 ( @xmath29 is the flow rate of @xmath24 ) . in the continuum limit , the system is governed by the following rate equations : @xmath78 where @xmath31 is the concentration of the chemical @xmath24 . this reaction network mainly comprises constant flows of chemicals ( @xmath79 and @xmath80 ) and an autocatalytic loop ( @xmath81 , @xmath82 , and @xmath83 ) . here , we set @xmath84 ( for all @xmath38 ) , @xmath85 , @xmath86 , and @xmath87 . with these settings , generally , @xmath62 molecules flow into the container and serve as substrates . they are then converted into other chemicals , following which they flow out ; this maintains the nonequilibrium condition . in the continuum limit , the concentrations @xmath31 vibrate and converge to the fixed point .    to elucidate the behavior at a condition distant from the continuum limit , we have investigated the dynamical behavior in such a condition by stochastic simulation . [ fig : ts - v ] shows the time series of @xmath10 , the number of @xmath24 molecules . when @xmath17 is large , generally , @xmath10 remains large . this behavior is similar to the rate equation with the addition of noise . however , when @xmath17 is small , @xmath10 may reach @xmath0 . in our model , if the system lacks a substrate or a catalyst for a certain reaction , the reaction ceases completely . consequently , the dynamics of the system with such a small @xmath17 are qualitatively different . we define the state of the system based on the combination of the reactions that cease . a system has the following three distinct states ( see fig . [ fig : net2 ] ) :    state i. : :    @xmath88 for all @xmath38 , and all the reactions    occur . +    this state is determined by the fixed point concentrations obtained by    the continuum limit , and the system converges to the fixed point ,    while the vibration around it is sustained when the number of    molecules is finite . : :    for the majority of the time , @xmath89 , and reactions    @xmath80 , @xmath81 , and @xmath83    cease . +    the reaction loop can not proceed , while reaction @xmath79    continuously converts @xmath62 into @xmath90 . state iii . : :    for the majority of the time , @xmath91 , and reaction    @xmath79 ceases . +    in the absence of any reactions , the @xmath62 molecules    accumulate . an @xmath63 molecule flowing in may trigger    reactions @xmath80 and @xmath81 and convert    @xmath62 into @xmath92 .    in the continuum limit , the concentrations can not reach @xmath0 due to the constant inflows , and the system remains at state i when @xmath17 is sufficiently large , as shown in fig . [ fig : ts - v ] ( a ) , even though the concentrations fluctuate and vibrate around the fixed point .    with a small @xmath17 , however , the other states appear . for example , the time series of @xmath10 with @xmath72 is shown in fig . [ fig : ts - v ] ( b ) . at around @xmath93 , the system is in state i , and it switches to state ii at around @xmath94 . it then alternates between states ii and iii . the system spontaneously switches among these states . if @xmath17 is considerably smaller , state i is rarely observed , as shown in fig . [ fig : ts - v ] ( c ) . the distribution of @xmath31 is shown in fig . [ fig : xdist ] . a transition is observed with a decrease in @xmath17 . for a large @xmath17 , the distribution shows a peak at around @xmath95 and @xmath96 , corresponding to the fixed point of the rate equation . for a small @xmath17 , the distribution of @xmath74 shows peaks at around @xmath97 and @xmath98 , corresponding to state ii ( @xmath99 , @xmath100 ) and state iii ( mostly @xmath98 ) , respectively . as mentioned above , these states are classified based on the reactions that cease ; in other words , the states are classified based on the part of the network that actually functions . in state i , all the reactions in the network function ; in state ii , the autocatalytic loop does not function ; and in state iii , the conversion of @xmath62 into @xmath90 ceases . the transitions to states ii or iii can be viewed as the disconnection of some parts in the reaction network . such transitions are possible only if @xmath10 reaches @xmath0 , and therefore , molecular discreteness is essential . the extinction of the @xmath63 and @xmath90 molecules makes the system switch to states ii and iii , respectively . the question that arises here is as follows : in general , which chemicals can switch states in a network ? in our model , molecule @xmath62 can not serve as a switch , even though @xmath47 sometimes reaches @xmath0 in the case of @xmath72 . first , for a molecule species to function as a key for switching , @xmath10 should be maintained at @xmath0 for a longer time than that for other reactions . for @xmath62 , there is considerable inflow , and the inflow rate is not affected if @xmath47 reaches @xmath0 . thus , @xmath47 can not remain at @xmath0 for a long time , and @xmath62 can not switch the dynamics . second , a key chemical for a switch should be located within the reaction paths and the extinction of the molecule disconnects some reaction paths . stochasticity in gene expression is widely studied with regard to the problem of a small number of molecules in a biological system . it is often assumed that two states  on and off  are switched by a single regulatory site . the controlling chemicals and controlled chemicals can be clearly separated . in contrast , our result shows that chemical species , which are usually abundant , may sometimes work as a stochastic switch . in this sense , molecules that are common or ions such as ca@xmath101 ( see @xcite ) may cause stochastic effects . the role of a chemical may change with time . in this study , we have demonstrated that molecular discreteness may induce transitions to novel states in autocatalytic reaction systems , which may result in an alteration of macroscopic properties such as the average chemical concentrations .    in biochemical pathways , it is not uncommon to find that the number of molecules of a chemical is of the order of @xmath102 or less in a cell . there are thousands of protein species , and the total number of protein molecules in a cell is not very large . for example , in signal transduction pathways , some chemicals work at concentrations of less than @xmath103 molecules per cell . there exist only one or a few copies of genetic molecules such as dna ; furthermore , mrnas are not present in large numbers . thus , regulation mechanisms involving genes are quite stochastic . naturally , molecular discreteness involves such rare chemicals .    in the second part of this paper , we have shown that the molecular discreteness may change the dynamical behavior of reaction networks . the reaction network is virtually disconnected by the extinction of certain chemicals , which is not possible in the continuum limit . although the network studied here is a small model , similar phenomena can exist in a complex reaction network with a large number of chemicals and reactions . we have also investigated random networks of catalytic reactions @xmath104 ( @xmath105 ) . in such systems , the dynamics of chemical concentrations also depend on the system size @xmath17 . in a small system , many of the chemical species become extinct ( @xmath45 ) , and the actual reaction network is disconnected into fragments , which may be occasionally reconnected by inflow or generation of a molecule . the onset of change in the concentrations due to disconnection or reconnection is stochastic and sudden . in the continuum limit , in contrast , the concentrations gradually converge to a fixed point in most cases ( or to a limit cycle or other attractors ) . the simple model in this paper can be viewed as a switching element of such a network ; however , the exact conditions that determine whether a chemical works as a stochastic switch or not should be addressed in future . we observed the transitions in the distribution of the concentrations @xmath31 with respect to the change in the system size @xmath17 . multiple transitions can also occur , especially if there are many chemical species for which the number of molecules is sometimes ( but not necessarily always ) small .    in this paper , we consider reactions in a well - stirred medium , where only the number of molecules is taken into account for determining the system behavior . however , if the system is not mixed well , we need to take into account the diffusion of molecules in space . from a biological viewpoint , the diffusion in space is also important because the diffusion in cells is not always fast when compared with the time scales of the reactions . if the reactions are faster than the mixing , we should consider the system as a reaction - diffusion system , with discrete molecules diffusing in space . the relation between these time scales will be important , as indicated by mikhailov and hess @xcite . with regards to these time scales , we recently found that the spatial discreteness of molecules within the so - called kuramoto length @xcite , over which a molecule diffuses in its lifetime ( lapses before it undergoes reaction ) , can yield novel steady states that are not observed in the reaction - diffusion equations @xcite . spatial domain structures due to molecular discreteness are also observed @xcite . see also @xcite for relevance of the discreteness in a replicating molecule system . the discreteness - induced effect present here does not depend on the characteristics of the reactions . furthermore , it may be applicable to systems beyond reactions , such as ecosystems or economic systems . the inflow of chemicals in a reaction system can be seen as a model of intrusion or evolution in an ecosystem : for both systems , discrete agents ( molecules or individuals ) may become extinct . in this regard , our result is relevant to the studies of ecosystems , e.g. , extinction dynamics with a replicator model by tokita and yasutomi @xcite ; strong dependence of the survival probability of new species in evolving networks on the population size was reported by ebeling _ the discreteness of agents or operations might also be relevant to some economic models , e.g. , artificial markets . most mathematical methods that are applied to reaction systems can not appropriately describe the discreteness effect . although the utility of simulations with the progress in computer technology has become convenient , it would also be important to develop a theoretical formulation applicable to discrete reaction systems . on the other hand , in recent years , major advances have been made in the detection of a small number of molecules and the fabrication of small reactors , which raises the possibility to experimentally demonstrate the discreteness effect predicted here . we believe that molecular discreteness has latent but actual importance with respect to biological mechanisms such as pattern formation , regulation of biochemical pathways , or evolution , which will be pursued in the future . this research is supported by a grant - in - aid for scientific research from the ministry of education , culture , sports , science and technology of japan ( 11ce2006 , 15 - 11161 ) and research fellowships from the japan society for the promotion of science ( 15 - 11161 , abroad h17 ) . x. wang , g. z. feuerstein , j. gu , p. g. lysko , and t. yue , `` interleukin-1@xmath106 induces expression of adhesion molecules in human vascular smooth muscle cells and enhances adhesion of leukocytes to smooth muscle cells '' , atherosclerosis * 115 * , 89 ( 1995 ) . w. ebeling , r. feistel , i. hartmann - sonntag , l. schimansky - geier , and a. scharnhorst , `` new species in evolving networks stochastic theory of sensitive networks and applications on the metaphorical level '' , biosystems * 85 * , 65 ( 2006 ) .
to study the fluctuations and dynamics in chemical reaction processes , stochastic differential equations based on the rate equation involving chemical concentrations are often adopted . 
 when the number of molecules is very small , however , the discreteness in the number of molecules can not be neglected since the number of molecules must be an integer . 
 this discreteness can be important in biochemical reactions , where the total number of molecules is not significantly larger than the number of chemical species . to elucidate the effects of such discreteness , we study autocatalytic reaction systems comprising several chemical species through stochastic particle simulations . the generation of novel states is observed ; it is caused by the extinction of some molecular species due to the discreteness in their number . 
 we demonstrate that the reaction dynamics are switched by a single molecule , which leads to the reconstruction of the acting network structure . 
 we also show the strong dependence of the chemical concentrations on the system size , which is caused by transitions to discreteness - induced novel states .