CN107423554A - One kind realizes markovian design method with reversible monomolecular reaction - Google Patents
One kind realizes markovian design method with reversible monomolecular reaction Download PDFInfo
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Abstract
Markovian design method is realized with reversible monomolecular reaction the invention discloses one kind, characterizing the markovian probability of stability using the final concentration of reactant is distributed, the Markov Chain of Markov Chain and continuous time suitable for discrete time.The inventive method, which solves to be reacted with DNA in the prior art, estimates that the method for Markov Chain steady-state distribution can not realize the species for continuous time markovian calculating, simultaneously effective reducing required reactant and reaction number.
Description
Technical field
The invention belongs to the network that chemically reacts (CRNs) calculating field, more particularly to a kind of reversible monomolecular reaction realization
Markovian design method.
Background technology
Under normal circumstances, Markov Chain is used to the random process in analytical chemistry reaction network.But
S.A.Salehi, M.D.Riedel, and K.K.Parhi are being published in IEEE International Conference on
Digital Signal Processing (DSP), 2015, pp.689-693 paper " Markov chain
Computations using molecular reactions ", and M.Cardona, M.Colomer, J.Conde,
J.Miret, J.Miro, and A.Zaragoza, it is being published in Biosystems, vol.81, no.3, pp.261-266,2005
Paper " Markov chains:In Computing limit existence and approximations with DNA "
Opposite work is done：Markovian steady-state distribution is estimated with DNA reactions.S.A.Salehi et al. devises shape first
DNA is used to represent markovian end points and side by formula chemical reaction network, M.Cardona et al., but above-mentioned
Method is all not carried out continuous time markovian calculating.So this invention address that design chemical reaction network (CRNS)
To calculate including continuous markovian calculating, and the complexity of forefathers is reduced to a certain extent.
Deterministic models based on ordinary differential system (ODEs) can express the dynamics of chemical reaction network well
Characteristic, so the present invention is emulated using ODE models.According to the law of mass action, the speed for reacting generation is proportional to instead
Answer the concentration and speed constant of thing.It is such as follows for reaction A+B → C+D, ODE models：
2010, D.Soloveichik et al. was proposed, any bimolecular or monomolecular reaction may map to DNA
Displacement reaction, it is possible to arbitrarily design imaginary chemical reaction network.Noteworthy point is that the speed under the system quantified
The measurement of rate constant and concentration keeps constant.So the emulation concentration and time in the present invention are all no units.Last
Point, the speed constants of all reactions is all 1 in the present invention, is hereinafter repeated no more.
The content of the invention
Goal of the invention：It can not be realized for the method for reacting estimation Markov Chain steady-state distribution with DNA in the prior art
To continuous time markovian calculating, the present invention proposes a kind of chemical reaction network formed with unimolecule reversible reaction
Markovian design method is realized, this method is simultaneously suitable for the Markov Chain and the Ma Er of continuous time of discrete time
Section's husband's chain.
Technical scheme：To achieve these goals, markovian set is realized with unimolecule reversible reaction in the present invention
Meter method, comprises the following steps：
(1) design chemical reaction network, the different conditions in target Markov Chain are represented with different reactant species,
The initial concentration of respective reaction thing is set according to the initial probability distribution of each state of target Markov Chain；
(2) transition probability is represented with the reaction rate constant between reactant, sets reaction rate normal according to transition probability
Several values；
(3) State Transferring is represented with reversible monomolecular reaction；
(4) markovian steady-state distribution is calculated with designed reaction network, the ultimate density of all reactants is
For steady-state distribution.
Wherein, the initial concentration of reactant sets and is necessarily equal to or is proportional to phase in target Markov Chain in step (1)
Answer the initial probability distribution of state.
Wherein, the reaction rate constant value in step (2) between all reactants is necessarily equal to or is proportional to target Ma Er
Transition probability in section's husband's chain between corresponding state.
Beneficial effect：Realize that markovian design method is single by structureization with unimolecule reversible reaction in the present invention
Relation between the chemical reaction network and Markov Chain of molecule reversible reaction composition, utilize the reaction speed of monomolecular reaction thing
Rate constant characterizes the transition probability between state two-by-two, efficiently utilizes the reversible essence that chemically reacts, effectively reduces
The species and reaction number of required reactant.
Brief description of the drawings
Fig. 1 is the flow chart for realizing markovian design method in the present invention with unimolecule reversible reaction；
Fig. 2 is a discrete Markov Chain and the chemical reaction network for realizing the chain, and Fig. 2 (a) is a discrete Ma Er
The state transition graph of section's husband's chain, Fig. 2 (b) are to realize markovian chemical reaction network in Fig. 2 (a)；
Fig. 3 is markovian ODE simulation results in Fig. 2；
Fig. 4 is a continuous Markov Chain and the chemical reaction network for realizing the chain, and Fig. 4 (a) is a continuous Ma Er
Section's husband's chain, Fig. 4 (b) are to realize markovian chemical reaction network in Fig. 4 (a)；
Markovian simulation result in Fig. 4 when Fig. 5 is λ=0.5；
Markovian simulation result in Fig. 4 when Fig. 6 is λ=1.
Embodiment
As shown in figure 1, realize markovian design method, including following step with unimolecule reversible reaction in the present invention
Suddenly：
(1) with different reactant species νi(i=1,2 ..., k) νi, i=1,2 ... k represent different states, according to
The initial probability distribution of target Markov Chain (Markov Chain of steady-state distribution to be solved) sets the initial concentration of reactant；
The initial concentration of i.e. all reactants is necessarily equal to or is proportional to the markovian initial state distribution of target；
(2) transition probability k is represented with reaction rate constantij(i=1,2 ..., k j=1,2 ..., k), according to transition probability
The value of reaction rate constant is set；
(3) with reversible monomolecular reactionRepresent State Transferring；
(4) it is as steady with the markovian steady-state distribution of designed Response calculation, the ultimate density of all reactants
State is distributed.
In above-mentioned steps (4), for continuous Markov Chain, reactant concentration versus time curve is the chain
Transient solution, the ultimate density of all reactants is steady-state distribution.
The inventive method illustrates probability of stability distribution with the final concentration of reactant, in continuous Markov Chain, appoints
The concentration of meaning moment t reactant illustrates the stateful probability distribution of t.That is, this method can use chemistry anti-
Answer the markovian steady-state distribution of network calculations and transient solution.
For discrete time Markov Chain (DTMC), the inventive method is specifically described as follows：
Discrete time Markov Chain (DTMC) is to all n ∈ Ν0,si∈ S meet
P(Xn+1=sn+1|Xn=sn,Xn-1=sn-1,...,X0=s0)=P (Xn+1=sn+1|Xn=sn) random process
{X0,X1,...,Xn+1,...}.In the case of homogeneous, the transition probability from state i to state j is defined as pij=P (Xn+1=j
|Xn=i).Vectorial ν (n)=(ν0(n),ν1(n),ν2(n) n moment stateful probability distribution ...) is represented.
In order to realize corresponding function, the design method is with different each state of chemically reactive species analoglike.
Each monomolecular reaction represents a kind of State Transferring.All rate constant values are equal to or are proportional to transition probability.It is all
Reactant initial concentration be equal to or be proportional to the markovian initial probability distribution of target.DTMC and CRN mapping is closed
System is summarised in table 1.Realize that the reaction network calculated is as follows：
The DTMC of form 1 and CRNs simple mapping relations
For continuous time Markov Chain (DTMC), the inventive method is specifically described as follows：
Continuous time Markov Chain (CTMC) is to any
ti∈R0 +, 0=t0＜ t1＜ ... ＜ tn＜ tn+1,si∈ S meet
Random process { Xt:t∈T}.Vectorial π (u)=(π0(u),π1(u),π2(u) u moment stateful probability point ...) is represented
Cloth.Real-time transfer rate is defined as
It is similar when continuous markovian design method is with discrete case in order to realize objective function.CTMC's and CRN
Mapping relations are summarised in table 2.Different each state of chemically reactive species analoglike.Each monomolecular reaction represents a kind of
State Transferring.All rate constant values are equal to or are proportional to real-time transfer rate.All reactant initial concentrations are all etc.
In or be proportional to the markovian initial probability distribution of target.Realize that the reaction network calculated is as follows：
The CTMC of form 2 and CRNs simple mapping relations
Fig. 2 (a) is markovian state transition graph, it can be seen that this chain one shares 100 states,
It is state 1,2,3 respectively ... 100.If it is currently state 1, then next moment state is constant.If being currently state 2,
The probability that 0.6 is carved with when so next jumps to state 1, and the possibility for having 0.4 jumps to state 3, by that analogy.Profit now
Steady-state distribution with above-mentioned mapping method with the chemical reaction network calculations chain, with 100 kinds of reactant ν1,ν2..., ν100Represent
100 kinds of states of the chain.It is assumed that original state is 99, so ν99Initial concentration be 1, the initial concentration of remaining material is 0.
Fig. 2 (b) is to realize the reaction network calculated, and the value of reaction rate constant is set according to transition probability, is set in the present embodiment anti-
Speed constant is answered to be equal to corresponding transition probability.
Emulated in the present invention with deterministic chemical reaction network model, to confirm the accuracy of design method.Tool
It is exactly to be emulated with ordinary differential system (ODEs) for body.After reaction network designs, according to material action law, row
Go out corresponding ODE equations, solve all substances concentration and change with time.Reaction network design after, using fromhttp://users.ece.utexas.edu/~soloveichik/Simulation software, CRN simulator, pass through
Mathematica is emulated.Input all reactions and reactant initial concentration, it is possible to obtain emulating image, such as Fig. 3.
From the point of view of the result of emulation, ν1And ν100Ultimate density tend to 0.33 and 0.67 respectively, the ultimate density of remaining material is 0, therefore
Omit in the picture.Illustrating the steady-state distribution of this chain includes ν1And ν100, probability is respectively 0.33 and 0.67, and remaining state is
0.This is consistent with the answer mathematically solved.
Markov Chain in Fig. 4 (a) has numerous state, wherein each state is jumped to down with identical speed λ
One state.Because utilizable chemical reactant is limited in practice, so with six reactant πi, i=0,1,2 ... 5 is near
Like representing this chain, from the point of view of the definition of continuous Markov Chain and its transient solution, now πi, five states of i=0,1,2,3,4
Transient solution with there is numerous state when preceding 5 states solution it is just the same, so emulation when only show this five states
Image.It is assumed that original state is π0, so π0Initial concentration be 1, the initial concentration of remaining material is 0.Design in Fig. 4 (b)
Reaction network, its reaction rate is identical with transfer rate, is λ.Simulation process is as above-mentioned discrete situation.Fig. 5 is shown
The markovian simulation result in Fig. 4 as λ=0.5, Fig. 6 illustrate the markovian emulation in Fig. 4 as λ=1
As a result, the concentration of wherein t all substances is the instantaneous state distribution of the chain.From on image, simulation result is with leading to
The transient solution for crossing mathematical method releasing is the same.
It the above is only the preferred embodiment of the present invention, it should be pointed out that above implementation column does not form restriction, phase to the present invention
Close staff in the range of without departing from the technology of the present invention thought, carried out it is various change and modifications, all fall within the present invention
Protection domain in.
Claims (3)
1. one kind realizes markovian design method with reversible monomolecular reaction, it is characterised in that this method includes following
Step：
（1）Design chemical reaction network, the different conditions in target Markov Chain are represented with different reactant species, according to
The initial probability distribution of each state of target Markov Chain sets the initial concentration of respective reaction thing；
（2）Transition probability is represented with the reaction rate constant between reactant, reaction rate constant is set according to transition probability
Value；
（3）State Transferring is represented with reversible monomolecular reaction；
（4）Markovian steady-state distribution is calculated with designed reaction network, the ultimate density of all reactants is as steady
State is distributed.
2. according to claim 1 realize markovian design method with reversible monomolecular reaction, it is characterised in that
Setting steps（1）The initial of corresponding state in target Markov Chain is made it equal to or is proportional to during the initial concentration of middle reactant
Probability distribution.
3. according to claim 1 realize markovian design method with reversible monomolecular reaction, it is characterised in that
Step（2）In reaction rate constant value between all reactants be equal to or be proportional in target Markov Chain corresponding state it
Between transition probability.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN110135636A (en) * | 2019-05-10 | 2019-08-16 | 北京理工大学 | A kind of acquisition methods, the apparatus and system of workshop operation status prediction information |
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| EP3045542A1 (en) * | 2008-03-28 | 2016-07-20 | Pacific Biosciences Of California, Inc. | Compositions and methods for nucleic acid sequencing |
| CN103620403A (en) * | 2011-01-11 | 2014-03-05 | 纽约市哥伦比亚大学理事会 | Systems and methods for single-molecule detection using nanotubes |
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN110135636A (en) * | 2019-05-10 | 2019-08-16 | 北京理工大学 | A kind of acquisition methods, the apparatus and system of workshop operation status prediction information |
| CN110135636B (en) * | 2019-05-10 | 2021-04-20 | 北京理工大学 | Method, device and system for acquiring workshop operation state prediction information |
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