J. Chem. Sci. Vol. 127, No. 6, June 2015, pp. 1025–1034. c Indian Academy of Sciences. DOI 10.1007/s12039-015-0871-5
Mathematical description of the nonlinear chemical reactions with oscillatory inflow to the reaction field
ALDONA KRUPSKA Institute of Molecular Physics PAS, M. Smoluchowskiego 17, 60-179 Poznan,´ Poland e-mail: [email protected]
MS received 5 January 2015; revised 3 February 2015; accepted 3 February 2015
Abstract. In this paper the arduous attempt to find a mathematical solution for the nonlinear autocatalytic chemical processes with a time-varying and oscillating inflow of reactant to the reaction medium has been taken. Approximate analytical solution is proposed. Numerical solutions and analytical attempts to solve the non-linear differential equation indicates a phase shift between the oscillatory influx of intermediate reaction reagent X to the medium of chemical reaction and the change of its concentration in this medium. Analytical solutions indicate that this shift may be associated with the reaction rate constants k1 and k2 and the relaxation time τ. The relationship between the phase shift and the oscillatory flow of reactant X seems to be similar to that obtained in the case of linear chemical reactions, as described previously, however, the former is much more complex and different. In this paper, we would like to consider whether the effect of forced phase shift in the case of nonlinear and non-oscillatory chemical processes occurring particularly in the living systems have a practical application in laboratory.
Keywords. Autocatalytic reactions; oscillatory inflow; forced phase delay effect; approximate solution.
PACS Nos *43.25.-x, 43.25. y; 05.45.-a; 02.30.Ks +
1. Introduction hydroxylamine-nitric acid,22 Fe(I1)-HNO23 reactions. Autocatalytic processing of β-pro’-tryptase to β-pro’- Nonlinear chemical reactions, especially autocatalytic tryptase is described.24 These autocatalytic processes type, have been studied experimentally by many possess bistable steady states observed in a stirred researchers.1,2 Reactions of this type can be reversible tank reactor (CSTR) experiments.25 Minimal self- or irreversible, with or without the inflow of the inter- replicating systems based on autocatalytic processes mediate product X. Many processes in nature are auto- were analyzed.26 Some autocatalytic reactions, for catalytic, such as the self-replication exhibited by ligase example, may exhibit propagating reaction fronts.27–33 ribozyme3 and peptides,4 self-perpetuating conforma- A theoretical work is concerned with a model tional changes such as those known to be exhibited of the enzymatic system of two coupled autocat- by prions5 or reproduction events in biological popu- alytic reactions based on three variables.34–36 A three- lations.6 The effect of a reversible autocatalytic process variable model of deterministic chaos in the Belousov– of spontaneous decay of the catalyst population was Zhabotinsky reaction, has been described too.37 The considered.7 The authors consider the system for which autocatalytic stage appears also in the hipercycle model the spontaneous production of the autocatalytic reagent described by Eigen.38 A simple three-variable model is exceedingly rare over the time scale of the experi- for complex oscillations with an autocatalytic stage in ment. Micro-fluid experiments and blood coagulation an isothermal chemical system was considered.39 The- have proved that the rate of mixing controls the rate and oretical approaches used in the Belousov–Zhabotinsky can regulate the rate and outcome of both biological and (BZ) reactions40 and arsenite-iodate waves,41 iodate non-biological autocatalytic reaction systems.8 oxidation,42 aerobic oxidation of NADH by horseradish The autocatalytic stage is involved in many chem- peroxidase enzyme model43 and many others. Time- ical and biochemical cycles. Well known and often periodic reaction-diffusion systems with application to considered by many researches are the following: the the autocatalysis in chemistry were considered too.44 Belousov-Zhabotynski reactions9–17 oxidation of bro- The simulation of a complex kinetic scheme and the mates by nitric acid18 or Landolt reactions,19 ferrous- results verified considering the thermal decomposition bromate,20 iodide – areniate,21 chlorite-iodide, and of cumene hydroperoxide can be easily generalized
1025 1026 Aldona Krupska to different kinetic models and is devoted to identify 2. Theory autocatalytic processes by means of dynamical DSC experiment.45 Autocatalytic set concept introduced in Nonlinear reactions with oscillatory inflow of reactant 1860 by Darwin was recently used to describe the to the reaction medium are considered. chemical reactions in combustion process in circulating The main subject is the simplest nonlinear autocat- fluidized bed boiler.46 alytic reaction with inflow IX of intermediate reactant The model of a simple autocatalytic type reac- to the reaction field: tion with oscillatory, cosine-like inflow of intermediate k1k2 A X 2X (1) product X to the reaction medium (reaction field) is pre- + ←−−−→ sented in this paper. Numerical solutions and analyti- Where, A stands for the initial product and X is the cal attempts to solve the non-linear differential equation intermediate product. indicates a phase shift between the oscillatory influx The kinetic equation of reaction (1) takes the form: of intermediate reagent X to the reaction medium and dX change of its concentration in this medium. Attempts 2 k1AX k2X IX (2) for analytical solutions indicate that this shift may be dt = − + associated with the reaction rate constants k1 and k2 We assume that inflow IX of the intermediate product and the relaxation time τ. The relationship between X is not constant in time, but it changes periodically the phase shift and the oscillatory flow of reactant X (cosine-like), according to the following function: seems to be similar to that obtained in the case of linear 47 2 chemical reactions, as described previously, however, IX a cos (ωt) (3) = is much more complex (table 1). Where, a is the oscillation amplitude and ω is the rate This raises the question: is it possible to use a forced (frequency) of X inflow to the reaction medium. phase shift effect in the simplest nonlinear chemical The equation (2) takes the form of: processes occurring especially in the biological sys- tems, for example DNA replication or formation of dX 2 2 thiodiglycol from thioglycolates or that which occurs k1AX k2X a cos (ωt) (4) in the initial transcript of rRNA and many other auto- dt = − + processing ones? In this paper, we would like to give an The equation (4) is nonlinear and non-homogenous. answer to this question. This work is a supplement to The equation (4) has two variables: concentration of our earlier work.47 intermediate product X and time variable t. For this
Table 1. Comparison between the linear and nonlinear processes with oscillatory, cosine-like inflow of X reactant to the reaction medium. Linear reaction type: Nonlinear reaction type: A X PAnX (n 1)X with oscillatory+ ↔ inflow of X47 with oscillatory+ inflow↔ of+ X [this work] analytical solution strictly possible strictly impossible, only default delta time independent time dependent phase shift tan φ 2ωτ,tanφ 2ωτ = = where τ 1/k1A where τ 1/√ = = possibility of determination possible impossible a rate constant and a relaxation time dependence of the phase increases with increasing decreases with increasing shift from supply-frequency X flow-frequency X flow-frequency X dependence on initial condition independent dependent number of possible behaviours one more than one stabilized by the forced delay effect. stabilized not stabilized bifurcation point not occurs occurs dependence of the amplitude decreases with increases of X decreases with increases of X on frequency of X inflow inflow frequency inflow frequency or does not change influence of increasing frequency tends to π/2 but cannot tends to zero of X inflow on the phase shift exceed this value Forced phase delay in nonlinear processes 1027 reason the analytical solution of this type equation is The relaxation time τ follows from the eq. (6) and takes impossible. In the case of equation (4) it can only find a the form: default solution of change of X product. It is true, that 1 1 the reaction (4) has two changes over time in oscillating τ (9) = √ = (k A)2 4k a cos2(ωt) manner stationary states: 1 ± 2 As follows from eqs. (9) and (8) the determination of
k1A √ k1A √ τ is impossible in this case, namely, nonlinear chemical x1 + and x1 − (5) processes with the oscillatory, cosine-like inflow to the = 2k2 = 2k2 reaction medium. The problem here is the time variable where: delta. This factor indicates the fundamental difference between linear and nonlinear reaction with oscillatory in time inflow of chemical reagent X to the reaction 2 2 medium. √ (k1A) 4k2a cos (ωt) in case outflow of X, = + >0(5a)As in the linear reaction with oscillatory inflow of X to the reaction medium47 the sum of two trigonometric functions in brackets in eq. (6) can be replaced by a sin- gle trigonometric function after introducing an auxiliary 2 2 √ (k1A) 4k2a cos (ωt) in case inflow of X, = − angle φ defined as: >0or<0 (5b) 2ω tan φ 2ωτ (10) = √ = The formula (5) contains changing time t variable where, and it is not possible to solution. For this reason the 2ω eq. (4) is also impossible to analytical resolve. We φ arctan arctan 2ωτ (11) assume that is positive as in a constant inflow of X, = √ = because it has a definite physical meaning. Based on After substitution eq. (11) into eq. (6) and some mathe- the analytical solution of nonlinear equation with a con- matical transformations we obtain: stant inflow of product X to the reaction medium (see Appendix) and on the analytical solve of linear type k A √ (k A √) c exp(√t)√ equation with a periodically inflow of X to the reaction X 1 − −[ 1 + ] medium described in our earlier paper and confirmed = 2k2 (1 c exp( √t) 47 [ − − ] experimentally we assume that approximate solution 1 cos(2ωt φ) (12) of the eq. (4) can take the following form: ×cos φ − where, k A √ (k A √) c exp(√t) X 1 − −[ 1 + ] = 2k2 (1 c exp( √t) √ [ − − ] cos φ (12a) = (√)2 4ω2 (√ cos 2ωt 2ω sin 2ωt) (6) + × + Let us consider now a high-order autocatalysis level Where, pre-exponential factor c is, reaction type: k k A 2X 1 2 3X (13) + ←−−−→ X02k2 k1A √ c − + − (7) with a time constant inflow IX of X reagent to the = X 2k (k A √)√ − 0 2 + 1 + reaction medium is described by the kinetic equation: dX X – is the initial concentration of the product X. X k AX2 k X3 I (14) 0 0 dt = 1 − 2 + X is calculated from the following initial conditions: when The reaction has two stable stationary states. The ana- X X0 t 0 and it takes the form: = → = lytical solution for these states is given by eq. (5). When the inflow is oscillatory and time dependent, k1A √ (k1A √)c√ described by the eq. (3) approximate solution of the X0 − − + (8) = 2k2 c2k2 eq. (14) takes the form similar to eq. (6). The solution − 1028 Aldona Krupska of eq. (14) applies only for one of the stable station- (a) ω = 0.2 ary state. It is known that the system tends to only one 1,8 k A = 0.5 1 k = 0.3 stable stationary state. 1,6 2
For the higher degrees of the nonlinear and oscilla- 1,4 tory autocatalytic processes, seems to be that there is 1,2 a dependency similar to eq. (2). For the higher degrees 1,0 k A = 0.3 of autocatalysis in real space, i.e., three-dimensional 0,8 1 Amplitude k = 0.5 space, there are only two stable steady states and the 0,6 2 system seeks one of them. 0,4
It seems that the formula (6) also can apply to the 0,2 2 one of the stable stationary state to the high-order 0,0 acos (ωt) autocatalytic reactions with the general formula: 0 102030405060 Time [a.u.]
dX (n 1) n 2 k1AX − k2X a cos (ωt) (15) (b) ω = 0.6 dt = − + k A =0.5 1,8 1 k = 0.3 In general the formula (6) takes the form: 1,6 2 1,4
n n n 1,2 k1A √ (k1A √) c exp(√t) X − −[ + ] 1,0 n k A =0.3 = 2k2 (1 c exp( √t) 0,8 1 Amplitude k = 0.5 n [ − − ] 2 (√ cos 2ωt 2ω sin 2ωt) (16) 0,6 × + 0,4 where, n denotes the order of autocatalysis. 0,2 2 acos (ωt) X0 and c are as described for n 2 by eq. (7) and 0,0 n = eq. (8) but with √. 0102030405060 The equation (4) can be numerically solved using Time [a.u] four-order Runge-Kutta algorithm (figure 1). Figures (c) ω = 1.2 1a, 1b and 1c present the numerical analysis (based 1,8 k A = 0.5 1 on the Runge-Kutta algorithm) of a delay effect and 1,6 k2 = 0.3 an amplitude modification in the case of high-order 1,4 nonlinear autocatalytic processes. All simulations were 1,2 performed for the initial conditions: X 1, t 0and 1,0 = = k A = 0.3 an integration step h 0.01. The numerical calcula- 0,8 1 Amplitude k = 0.5 = 2 tions show a delay between the concentration of inflow 0,6 product X to the reaction medium (field) and its con- 0,4 centration in this medium. The amplitude of X in the 0,2 2 acos (ωt) reaction field is modified too. The phase shift and 0,0 amplitude modification depends on frequency ω of X 0 102030405060 inflow, as well as on values of k1A and k2.Foragiven Time [a.u.] level of autocatalysis the numerical solution shows the following dependence: when the influx of reagent X to Figure 1. The results of numerical calculation for the auto- the field (medium) reaction is more intense, the delay in catalytic reactions with oscillatory stimulus (described by eq. (4)) both for k A>k (upper lines) and k A
ω = 0.2 (a) delay maximum for k A = 0.5 and k = 0.3 1 2 ω = 0.4 1,6 ω = 0.6 1,4 ω = 0.9 ω = 1.2 1,2
1,0 φ
0,8
0,6 phase delay
0,4
0,2
0,0 23456 degree of autocatalysis
(b) delay minimum for k A = 0.5 and k = 0.3 ω = 0.2 1 2 ω = 0.4 1,6 ω = 0.6 1,5 ω = 0.9 1,4 ω = 1.2 1,3 1,2 1,1
φ 1,0 0,9 0,8 0,7 0,6 phase delay 0,5 0,4 0,3 0,2 0,1 0,0 23456 degree of autocatalysis
Figure 2. Illustration of a) the phase delay φ maximum and b) the phase delay minimum, dependence on frequency ω; 0.2, 0.4, 0.6, 0.9, 1.2 of the X inflow and on autocatalysis degree for k1A>k2 k1A 0.5, k2 0.3. : = =
In the presented model √ probably depends on minimum of X inflow (figures 2a–b). As shown in the the order of autocatalysis in case when √ is positive figures 2a and 2b, when the k1A is greater than k2 the this is, for k1A>k2. For higher levels of autocataly- phase shift depends strictly on the frequency ω of sup- sis a sign of the delta (positive or negative) probably ply of X reagent to the reaction medium – decreases determines whether a system oscillations disappear or not with increasing of ω inflow of reactant X at the given (at a fixed frequency inflow of X reagent, see figures degree of autocatalysis. As shown in the figures 2a and 1a–c,). When the k1A is greater than k2 (delta is 2b, with the increase in the degree of autocatalysis, positive) the damping of the amplitude of the oscilla- phase shift between the concentration of the reactant X tions and decrease in the phase shift are observed in and the inflow is reduced. At six-order of autocatalysis the system with an increase in the degree of autocatal- the phase shift does not seem to depend on frequency ysis (figures 1a–c (upper lines) and figures 2a–b and ω of flow of X. A similar correlation is observed in the table 2) (the shift between the oscillatory inflow of X amplitude values obtained from numerical calculation reagent to the reaction medium and its concentration as show in table 2. When the k1A is smaller than the in this medium). This is visible both in maximum and k2 ( is negative) the oscillation of X are sustained (with 1030 Aldona Krupska
ω = 0.2 (a) delay maximum for k A = 0.3 and k = 0.5 1 2 ω = 0.4 ω = 0.6 1,8 ω = 0.9 1,6 ω = 1.2
1,4
1,2 φ
1,0
0,8
phase delay delay phase 0,6
0,4
0,2
0,0 23456 degree of autocatalysis
(b) delay minimum for k A = 0.3 and k = 0.5 1 2 ω = 0.2 ω = 0.4 2,0 ω = 0.6 ω = 0.9 ω = 1.2
1,5 φ
1,0 phase delay phase
0,5
0,0 23456 degree of autocatalysis
Figure 3. Illustration of a) the phase delay φ maximum and b) the phase delay minimum, dependence on frequency ω; 0.2, 0.4, 0.6, 0.9, 1.2 of the X inflow and on autocatalysis degree for k1A Table 2. Juxtaposition of the values of amplitude obtained from the numerical cal- culations for the case: k1A>k2, k1A 0.5, k2 0.3. ω is a frequency of X supply, 2X, 3X, 4X, 5X and 6X denote the degree= of autocatalysis.= Degree of autocatalysis ω 0.2 ω 0.4 ω 0.6 ω 0.9 ω 1.2 = = = = = 2X 0.14 0.1 0.08 0.05 0.04 3X 0.1 0.08 0.07 0.04 0.03 4X 0.06 0.06 0.05 0.04 0.03 5X 0.03 0.03 0.03 0.02 0.02 6X 0.02 0.02 0.02 0.02 0.01 increase of degree of autocatalysis ((figure 1a), figure does not change with increasing of degree of autocatal- 1b), figure 1c) – lower graphs)) and the phase delay ysis at the given frequency ω of X inflow (figures 3a Forced phase delay in nonlinear processes 1031 and 3b). But, in this case, the phase shift depends on common characteristic for the nonlinear systems, both frequency ω of X inflow at the given degree of auto- oscillatory and non oscillatory, forced by oscillatory catalysis (figures 3a and 3b). In both cases (when k1A is stimulus. In this work we present the non-oscillating greater than k2 or k1A is smaller than the k2) the phase and nonlinear system forced by oscillatory stimulus. shift depends on frequency ω of inflow of X at the given Parameters of the forced oscillation strictly depend on degree of autocatalysis. When the frequency ω of X the initial conditions. Nonlinear systems are sensitive inflow to the reaction medium is greater, the phase shift to the initial conditions. The oscillations have the fre- (phase delay) of the concentration of X in this medium quency of driving force and are characterized by ampli- is reduced at a given order of autocatalysis as shown in tude and phase shift (as in the linear systems). There is the figure 2 and figure 3. It is opposite to the effect in no one set rule which describes the behaviour of the linear reactions.47 Apart from that, bifurcation point is nonlinear systems. This is due to the fact that the non- visible in figures 1a–c. linear systems are far from thermodynamic equilibrium. The forced phase shift in the nonlinear systems 3. Results and Discussion is even less frequent than in the linear ones. There are known examples of nonlinear physical systems in 3.1 Physical interpretation - connection between the which there is a forced phase shift effect. It is diffi- exponential relaxation and the forced delay effect in cult to find examples of nonlinear chemical systems in the linear and nonlinear systems and some examples which the forced phase shift effect is observed. A spe- cial class of reaction-diffusion equations to a sys- In nature the most of relaxation processes have an tem incorporating time delay effects were theoretically exponential character. In the biological and chemical considered.48 The equations applied e.g., to a nonlinear systems, relaxation takes place exponentially in almost reaction-diffusion Lotka-Volterra system. 70–80%. In chemistry the first order and autocatalytic non oscillating type of reactions have an exponential relaxation. Especially the nonlinear autocatalytic pro- 3.2 Discussion cesses play an important role in biological systems. If (in the non-oscillating and exponentially relaxing The nonlinear, non-oscillating and an auto-processing- systems) intermediate product X flows in the oscilla- autocatalytic chemical reactions are considered in tory manner to the reaction medium (as a variable time this paper. The simplest nonlinear autocatalytic type cosine type function) it will be seen the phase shift in reaction with the oscillatory inflow of X reagent to the the concentration of X in response to the supply of this reaction medium (reaction field) is much more compli- component to the medium. This process is called as a cated in comparison to the linear first-order reac- forced (by the oscillating stimulus), the effect of phase tion with the X reagent inflow to the reaction medium. shift. Phase shift (phase delay) is strictly connected with In our case, the delay in the concentration of X chem- the relaxation process. ical reagent in the reaction medium is connected with There is a one common formula for all oscillating and its frequency of inflow to the reaction medium. As in non-oscillating linear systems relaxing exponential and the linear first-order process we also observed the phase forced by a periodical stimulus: tan ωt, where ω – shift between the X inflows to the reaction medium and is a frequency of stimulus, τ – is a relaxation= time. This change its concentration in this medium. The amplitude formula is mainly used for determination of the relax- modification is observed too. However in the nonlin- ation time τ, especially in physical systems. In the lin- ear autocatalytic processes the is connected with an ear first-order chemical reactions, the forced phase delay oscillatory time dependent inflow of X and is variable in effect turned out to as a good method for determination time (Eqs. 5a and 5b). Connected with the the relax- 47 of the rate constant k1 and the relaxation time τ. ation time τ is also variable in time. Apart from and There are common factors relating to all the linear τ are also dependent on k1A and k2. For this reason the relaxing exponentially systems both oscillatory and non relaxation time τ as well as k1A and k2 are not possible oscillatory forced by oscillatory stimulus. Parameters of to determination. Nonlinear autocatalytic processes of the forced oscillation do not depend on the initial con- high-order have also a one stable stationary state, which ditions. These oscillations have the frequency of driv- rule this same or similar formula as the simplest auto- ing force and are characterized by amplitude and phase catalytic process one. Determination of the relaxation shift (with respect to the appropriate stimulus). time τ and rate constants k1 and k2 are impossible too. There are some differences characterizing the nonlin- We observe that with increasing frequency of X ear systems with respect to the linear ones. There are the inflow to the reaction medium the phase shift tends to 1032 Aldona Krupska zero. In the linear first-order reaction, the phase shift in frequency ω of X inflow to the reaction medium tends to π/2 with increasing frequency of X inflow. (figures 2a and 2b, figures 3a and 3b) at a given degree It may be connected with the fundamental difference of autocatalysis. For the higher order of autocatalysis between the systems linear and nonlinear. It is proba- (five and six), the phase delay seems to not depend on bly connected with oscillating in time. The fact that frequency ω of X inflow in some cases (when k1A>k2, the phase shift decreases with increasing ω inflow X see figures 2a and 2b). seems to be important especially in the biological sys- The following question arises: is it possible that any tems. High frequency ωX component supply needed for application of the forced phase delay affect the non- the occurrence of a chemical reaction reduces the delay linear autocatalytic processes? In order to answer this of its delivery. This is very important for the proper question let’s cite one researcher - Laister. Laister48 in conduct of the chemical process in a living organism. the abstract of his article writes: Most nonlinear biological processes, such as, enzymatic “For a significant class of nonlinear equations it reactions, DNA replication, biosynthesis of protein and is shown that an unstable equilibrium solution of the many others occurs with very great speed of 1012 s. It reaction-diffusion system cannot be stabilized by the follows that the reagent X must also be provided at introduction of delay”.48 In our case the nonlinear the same rate. The rate is 1000 times greater than the autocatalytic but without diffusion processes, cannot rate of transmission of any signal, which is 109 s. When be stabilized by forced delay effect. For this reason the the delivery rate of reagent X is very rapid, about 1012 forced delay effect in the nonlinear chemical processes s, a delay in phase is observed. Also, any oscillations cannot be applied for determination of a rate constant are extinguished. In the linear reactions it is inverse; the or a relaxation time. higher the frequency of reactant X supply to the reac- The phase delays’ phenomena seem also to play an tion medium, the greater is the phase delay, and tends important role in some biological processes. It has been to a maximum of 90◦ (but does not exceed this value). proved that the p19 protein induces G1/S phase delay, This explains the very important role of nonlinearity in thereby maintaining cells in a reversible quiescence biology. For a biological system to function effectively state and preventing entry into apoptosis49 N-methyl-D there can be no delay in the chemical process. If the Revised: aspartate (NMDA) induces a rapid phase shift reactant flow rate decreases, the delay may occur, which of mPer 1promotor gene in cultured SCN. It has been may have an adverse effect on the functioning of the shown that NMDA induces phase shifts at various time biological system. Presented model explains the behav- points.50 ior of the nonlinear system, depending on the frequency Autocatalytic processes are connected with the class of X reagent flow to the reaction field. In some cases, of reactions which are most closely associated with when k1A 13. Vanag V K, Zhabotinsky A M and Epstein I R 2000 J. 34. Decroly O and Goldbeter A 1985 J. Theoret. Biol. 113 Phys. Chem. A 104 11566 649 14. Asai T, Kanazawa Y, Hirose T and Amemiya Y 2005 35. Decroly O and Goldbeter A 1987 J. Theoret. Biol. 124 Int. J. Unconv. Comput. 1 123 219 15. Qiao Ch, Wang H and Ouyang Qi 2009 Phys. Rev. E 79 36. Decroly O and Goldbeter A 1984 Physics Letters 105A 016212 259 16. Flesselles J M, Belmonte A and Gaspar V 1998 J. Chem. 37. Györgi L and Field R J 1992 Nature 355 808 Soc. Faraday Trans. 94(7) 851 38. 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