Random reaction diffusion two-dimensional modeling method for cardiac muscle cell calcium holes

Abstract

The invention discloses a two-dimensional modeling method for random reaction diffusion of cardiac muscle cell calcium holes, which is used for researching a dynamic mechanism of calcium ions in a myoplasmal network at a subcellular level from the calcium holes. According to the characteristics of a sarcoplasmic reticulum final pool and the Reynodine receptors, free Reynodine receptors randomly distributed on the surface of a sarcoplasmic reticulum final pool membrane are integrated into a model, and based on a random opening and closing mode of the Reynodine receptors, the diffusion process of calcium ions and a calcium fluorescent agent and the chemical reaction process of the calcium ions, a calcium buffering agent and the calcium fluorescent agent are considered; and establishing a novel two-dimensional model of random reaction diffusion of the cardiac muscle cell calcium cavity by using a boundary effect of releasing calcium ions to the outside from the myoplasmic reticulum final pool and transporting the calcium ions into the myoplasmic reticulum final pool from the free myoplasmic reticulum. A discrete method is used for carrying out numerical analysis on the reaction diffusion model, three modes of the calcium holes are represented by numerical values, and the time progress and the release strength of mini calcium holes and typical calcium holes are well matched with experimental results.

Description

Technical Field

[0001] The invention relates to the field of myocardial cell calcium cavities, and in particular to a random reaction-diffusion two-dimensional modeling method of myocardial cell calcium cavities. Background Art

[0002] A single or a small number of free and randomly distributed ranolin clusters are called free ranolin receptors; neatly arranged in a two-dimensional lattice array are called clustered ranolin receptors. Free ranolin receptors in myocardial cells may increase sarcoplasmic reticulum calcium leakage in the form of calcium quarks at rest and accelerate the propagation of calcium waves, playing an important role in the disorder of the calcium balance system during heart failure. Therefore, free ranolin receptors play an important role in calcium signal transduction in myocardial cells.

[0003] At present, the model study of the effect of free ranolin receptor on myocardial calcium signal is only at the cellular level, that is, calcium spark events or calcium wave events in the cytoplasm, and there is no relevant report on the subcellular level, that is, calcium cavitation events in the sarcoplasmic reticulum. As a sister event of calcium sparks, exploring the effect of free ranolin receptor on myocardial calcium cavitation at the subcellular level can provide a new theoretical basis for further understanding the dynamic mechanism of microscopic conduction of calcium ions in the sarcoplasmic reticulum. And through the dynamic modeling analysis of calcium ion concentration in the terminal pool of the sarcoplasmic reticulum of myocardial cells at the subcellular level, it is possible to numerically reproduce calcium cavitation events and quantitatively characterize the spatial distribution of free calcium ion concentration. Summary of the invention

[0004] The purpose of the present invention is: the present invention takes into account the existence of free ranolin receptors and proposes a random reaction-diffusion two-dimensional modeling method for calcium cavities in myocardial cells. The constructed two-dimensional model starts with calcium cavities and studies the dynamics of calcium ions in the terminal pool of the sarcoplasmic reticulum at the subcellular level, thereby providing a new theoretical basis for more detailed exploration of the termination mechanism of calcium sparks in the process of myocardial cell excitation-contraction coupling.

[0005] The technical solution of the present invention is:

[0006] The random reaction-diffusion two-dimensional modeling method of calcium vacancies in myocardial cells includes the following steps:

[0007] S1: Establish a reaction-diffusion model of calcium ions in the terminal pool of the sarcoplasmic reticulum of myocardial cells: the reaction-diffusion model includes a calcium fluorescence system, a calcium buffer system, and a calcium release system;

[0008] The reaction-diffusion model is:

[0009]

[0010] J RyR =J Rogue +J Clustered (1.2)

[0011]

[0012]

[0013] Among them, [Ca 2+ ] SR , [CaF], [CaB] represent the concentrations of free calcium ions, calcium fluorescent agent bound to calcium ions, and calcium buffer bound to calcium ions in the terminal cisterna of the sarcoplasmic reticulum, respectively; D Ca and D F represent the diffusion coefficients of free calcium ions and calcium fluorescent agent in the terminal cisterna of the sarcoplasmic reticulum, respectively; J dye , J buffer , J RyR J represents the calcium ion flux caused by the reaction of calcium ions with calcium fluorescent agent, the reaction of calcium ions with calcium buffer, and the release of calcium ions. RyR It consists of two parts: Rogue and J Clustered represent the calcium ion release flux caused by the opening of free ranolidine receptors and clustered ranolidine receptors, respectively, and t represents time;

[0014] S2: Numerical methods for reaction-diffusion modeling;

[0015] S3: Three modes of calcium vacancies are reproduced based on the model numerical values: mini calcium vacancies, medium calcium vacancies, and typical calcium vacancies.

[0016] Preferably, in the calcium fluorescence system described in S1, a low-affinity calcium fluorescent agent Fluo-5N is used to label calcium ions in the sarcoplasmic reticulum:

[0017] Assume that F is the calcium fluorescent agent Fluo-5N, then the binding-dissociation reaction expression of the calcium fluorescent agent and calcium ions is:

[0018]

[0019] Wherein, CaF represents a calcium fluorescer bound to calcium ions;

[0020] The calcium ion flux J caused by the reaction between the calcium fluorescent agent Fluo-5N and calcium ions is dye It is expressed as:

[0021]

[0022] Among them, [F] T represents the total concentration of calcium fluorescer, They represent the forward and reverse reaction rates of the calcium fluorescent agent respectively. The concentration of the calcium fluorescent agent that is not bound to calcium ions can be expressed as [F] T -[CaF].

[0023] Preferably, in the calcium buffer system described in S1, assuming that B is a calcium buffer CSQ, its binding-dissociation reaction equation is the same as that of the calcium fluorescent agent, as follows:

[0024]

[0025] Wherein, CaB represents the calcium buffer CSQ that binds to calcium ions;

[0026] The calcium ion flux J caused by the reaction between calcium buffer CSQ and calcium ions is buffer It can be expressed as:

[0027]

[0028] Among them, [B] T represents the total concentration of calcium buffer CSQ, They represent the forward and reverse reaction rates of the calcium buffer, respectively. The concentration of the calcium buffer that is not bound to calcium ions can be expressed as [B] T -[CaB].

[0029] Preferably, in the calcium release system described in S1, the calcium ion release flux generated during the occurrence and development of calcium vacancies is expressed as follows:

[0030]

[0031] Among them, J RyR =J Rogue +J Clustered It is the calcium ion flux released by calcium ion channels, including the calcium ion release flux caused by the opening of free ranodine receptors. Rogue and calcium ion release flux caused by opening of clustered ranoldin receptors J Clustered ;(x i ,y j ) is the spatial coordinate of the location of a single ranolin receptor; δ is the Dirac delta function, that is, all values ​​except zero are equal to zero; S(x i ,y j ,t;T open ) is a random function that controls the on / off state of the Ranodine receptor, taking values ​​of 1 or 0. When S takes a value of 0, the Ranodine receptor is in a closed or inactivated state. When S takes a value of 1, it is located at (x i ,y j ) single ranoldin receptor opens and lasts for T open time;

[0032] σ(x i ,y j ) means that the i ,y j) when a single ranodine receptor is opened, the calcium ion flux released into the adjacent cell of the cytoplasm (volume is V0) is expressed as follows;

[0033]

[0034] Where Far is the Faraday constant; i RyR is the initial current when a single ranoldin receptor opens; [Ca 2+ ] SR (x i ,y j ) and [Ca 2+ ] cyto (x i ,y j ) represent the coordinates (x i ,y j ) at the ranoldin receptor, the free calcium ion concentration inside and outside the terminal cisterna tum of the sarcoplasmic reticulum (outside is the cytoplasm), [Ca 2+ ] SR (x i ,y j ) and [Ca 2+ ] cyto (x i ,y j ) changes with time, and the values ​​at different times are determined by synchronously calculating the correlation models of calcium sparks and calcium holes;

[0035] The process of activating the ranodine receptor is a random process, and its open probability per unit time is expressed as:

[0036]

[0037] Among them, P max =0.3 / RyR / ms is the maximum probability of single ranolin receptor opening, K = 15 μM is the calcium ion sensitivity parameter, and m-RyR is the Hill coefficient representing the cooperativity of ranolin receptors.

[0038] Preferably, initial conditions and boundary conditions are also set in S1; the method for setting the initial conditions is:

[0039] The initial value of the free calcium ion concentration in the sarcoplasmic reticulum and cytoplasm is the concentration value in its resting state. and

[0040] The initial value of free calcium ion concentration can be expressed as:

[0041]

[0042]

[0043] The initial values ​​of [CaF] and [CaB] are calculated from equations 1.6 and 1.8:

[0044]

[0045] Therefore, the initial values ​​of [CaF] and [CaB] are t＝0 , [CaB] t＝0 They are respectively expressed as follows:

[0046]

[0047] Preferably, the boundary condition is set as follows:

[0048] The boundary of the sarcoplasmic reticulum terminal pool is divided into two parts: the boundary Γ1 connected to the free sarcoplasmic reticulum and the remaining boundary Γ2. Assuming that the calcium ion concentration in the free sarcoplasmic reticulum remains unchanged, the free calcium ion concentration at the boundary Γ1 of the sarcoplasmic reticulum terminal pool connected to the free sarcoplasmic reticulum is always the initial value of the resting state. Considering that the membrane of the terminal cistern of the sarcoplasmic reticulum is impermeable to ions, the permeability of calcium ions at the other boundary Γ2 of the terminal cistern of the sarcoplasmic reticulum is 0, which is expressed as follows:

[0049]

[0050] Where n represents the normal vector at the boundary.

[0051] Preferably, the numerical method for establishing the reaction-diffusion model in S2 includes:

[0052] First, integrate equation 1.1 over any triangular mesh area Ω to obtain:

[0053]

[0054] Where C represents the free calcium concentration in the terminal cisterna of the sarcoplasmic reticulum [Ca 2+ ] SR , S is the area of ​​the grid Ω, and J is the sum of all source and sink terms, that is, J = J dye +J buffer +J RyR ;

[0055] Secondly, the mesh Ω is a polygon with 3 edges, so Equation 2.1 can be written as:

[0056]

[0057] in, represents the average free calcium ion concentration within the grid Ω;

[0058] Finally, solve equation 2.2, which is to establish the relationship between the normal derivative and the average value on the boundary; let the left unit of a boundary be E1, and the right unit be E2, approximate the average value of the unit to the value at the unit centroid, and let the unit vector of the line connecting the centroids of units E1 and E2 be n', then we have

[0059]

[0060] Among them, r1 and r2 represent the centroid positions of elements E1 and E2 respectively; the normal derivative The projection of formula 2.3 in the direction of the boundary unit normal vector n can be used, that is,

[0061]

[0062] On each edge, we can introduce a quantity Where l is the boundary length;

[0063] Substituting equation 2.4 into equation 2.2, we have

[0064]

[0065] In summary, within each grid, given the average value of the free calcium ion concentration as the unknown quantity, Equation 2.5 is a system of ordinary differential equations, which is numerically solved using the fourth-order Runge-Kutta method;

[0066] Then, integrating equation 1.3 over any triangular mesh area Ω gives:

[0067]

[0068] Among them, C F represents the concentration of calcium fluorescer bound to calcium ions [CaF], S is the area of ​​the grid Ω;

[0069] Secondly, the mesh Ω is a polygon with 3 sides, so Equation 2.6 can be written as:

[0070]

[0071] in, represents the average value of the concentration of calcium fluorescer bound to calcium ions within the grid Ω;

[0072] Finally, solve equation 2.7, which is to establish the relationship between the normal derivative and the average value on the boundary; let the left unit of a boundary be E1, and the right unit be E2, approximate the average value of the unit to the value at the unit centroid, and let the unit vector of the line connecting the centroids of units E1 and E2 be n', then we have

[0073]

[0074] Among them, r1 and r2 represent the centroid positions of elements E1 and E2 respectively; the normal derivative The projection of formula 2.8 in the direction of the boundary unit normal vector n can be used, that is,

[0075]

[0076] On each edge, we can introduce a quantity Where l is the boundary length;

[0077] Substituting equation 2.9 into equation 2.7, we have

[0078]

[0079] In summary, within each grid, the average value of the concentration of the fluorescent agent bound to calcium ions is given as the unknown quantity, then Equation 2.10 is a system of ordinary differential equations, which is numerically solved using the fourth-order Runge-Kutta method.

[0080] The advantages of the present invention are:

[0081] The present invention takes into account for the first time the existence of free and randomly distributed ranolidine receptors on the membrane surface of the sarcoplasmic reticulum terminal cistern. Based on the random opening and closing patterns of ranolidine receptors, the release of calcium ions from the sarcoplasmic reticulum terminal cistern to the cytoplasm through ranolidine receptors, and the boundary effect of the transport of calcium ions from the free sarcoplasmic reticulum to the sarcoplasmic reticulum terminal cistern, a new two-dimensional model of random reaction-diffusion of calcium vacancies in the sarcoplasmic reticulum terminal cistern was established. Three modes of calcium vacancies were numerically simulated: mini calcium vacancies, medium calcium vacancies, and typical calcium vacancies. The time course and release intensity of mini calcium vacancies and typical calcium vacancies were consistent with the experimental results. The spatiotemporal dynamic characteristics of the three modes were numerically analyzed. BRIEF DESCRIPTION OF THE DRAWINGS

[0082] The present invention will be further described below in conjunction with the accompanying drawings and embodiments:

[0083] Figure 1 This is a schematic diagram of the open and closed states of the ranodine receptor when a mini calcium hole occurs;

[0084] Figure 2 To numerically reproduce the time course of mini calcium cavitation;

[0085] Figure 3 The experimentally observed calcium quark QCR in the sarcoplasmic reticulum and the corresponding small-amplitude calcium signal event QCD in the terminal pool of the sarcoplasmic reticulum;

[0086] Figure 4 The spatial distribution of free calcium ion concentration in the terminal cisterna of the sarcoplasmic reticulum at different times during the occurrence and development of mini calcium cavities numerically reproduced;

[0087] Figure 5This is a schematic diagram of the open and closed states of the ranodine receptor when a medium calcium vacancy occurs;

[0088] Figure 6 The time course of the medium calcium vacancy is numerically reproduced;

[0089] Figure 7 This is a schematic diagram of the open and closed states of the ranodine receptor when a typical calcium vacancy occurs;

[0090] Figure 8 The time course of the medium calcium vacancy is numerically reproduced;

[0091] Fig. 9 is the time course of typical calcium hole calcium fluorescence ΔF / F0;

[0092] Fig.10 This is the spatial distribution of free calcium ion concentration in the terminal cisterna of the sarcoplasmic reticulum at different times during the numerical reproduction process. DETAILED DESCRIPTION

[0093] The random reaction-diffusion two-dimensional modeling method of myocardial cell calcium vacancies of the present invention comprises the following steps:

[0094] S1: Establish a reaction-diffusion model of calcium ions in the terminal pool of the sarcoplasmic reticulum of myocardial cells: The model includes a calcium fluorescence system (calcium ions react with calcium fluorescent agents), a calcium buffer system (calcium ions react with calcium buffer agents), and a calcium release system (the terminal pool of the sarcoplasmic reticulum releases calcium ions into the cytoplasm through the ranolidine receptor). The calcium fluorescence system uses a low-affinity calcium fluorescent agent Fluo-5N to label and display the calcium ions in the experiment; the calcium buffer system is when the ranolidine receptor is turned on, the sarcoplasmic reticulum terminal pool first releases calcium ions into the cytoplasm, binds to the calcium buffer, and the calcium buffer binds and dissociates with calcium ions very quickly, which greatly improves the efficiency of calcium ion release; the calcium release system is the release of calcium ions from the sarcoplasmic reticulum of myocardial cells into the cytoplasm, and affects the spatiotemporal characteristics of calcium ions in the terminal pool of the sarcoplasmic reticulum (or the free sarcoplasmic reticulum connected to it) according to the state of the ranolidine receptor (open, closed, inactivated). Boundary conditions, the calcium ion concentration at the boundary connected to the free sarcoplasmic reticulum is maintained at the initial value, and the free sarcoplasmic reticulum continuously provides sufficient calcium ions to the terminal pool of the sarcoplasmic reticulum. At other boundaries, the sarcoplasmic reticulum is impermeable to ions and the flux is zero.

[0095] The reaction-diffusion model is:

[0096]

[0097] J RyR =J Rogue +J Clustered (1.2)

[0098]

[0099] Among them, [Ca 2+ ] SR , [CaF], [CaB] represent the concentrations of free calcium ions, calcium fluorescent agent bound to calcium ions, and calcium buffer bound to calcium ions in the terminal cisterna of the sarcoplasmic reticulum, respectively; D Ca and D F represent the diffusion coefficients of free calcium ions and calcium fluorescent agent in the terminal cisterna of the sarcoplasmic reticulum, respectively; J dye , J buffer , J RyR J represents the calcium ion flux caused by the reaction of calcium ions with calcium fluorescent agent, the reaction of calcium ions with calcium buffer, and the release of calcium ions. RyR It consists of two parts: Rogue and J Clustered represents the calcium ion release flux caused by the opening of free ranolidine receptors and clustered ranolidine receptors, respectively, and t represents time.

[0100] In the calcium fluorescence system described in S1, the low-affinity calcium fluorescent agent Fluo-5N is used to label calcium ions in the sarcoplasmic reticulum:

[0101] Assume that F is the calcium fluorescent agent Fluo-5N, then the binding-dissociation reaction expression of the calcium fluorescent agent and calcium ions is:

[0102]

[0103] Wherein, CaF represents a calcium fluorescer bound to calcium ions;

[0104] The calcium ion flux J caused by the reaction between the calcium fluorescent agent Fluo-5N and calcium ions is dye It is expressed as:

[0105]

[0106] Among them, [F] T represents the total concentration of calcium fluorescer, They represent the forward and reverse reaction rates of the calcium fluorescent agent respectively. The concentration of the calcium fluorescent agent that is not bound to calcium ions can be expressed as [F] T -[CaF].

[0107] In the calcium buffer system described in S1, assuming that B is the calcium buffer CSQ, its binding-dissociation reaction equation is the same as that of the calcium fluorescent agent, as follows:

[0108]

[0109] Wherein, CaB represents the calcium buffer CSQ that binds to calcium ions;

[0110] The calcium ion flux J caused by the reaction between calcium buffer CSQ and calcium ions is bufferIt can be expressed as:

[0111]

[0112] Among them, [B] T represents the total concentration of calcium buffer CSQ, They represent the forward and reverse reaction rates of the calcium buffer, respectively. The concentration of the calcium buffer that is not bound to calcium ions can be expressed as [B] T -[CaB].

[0113] In the calcium release system described in S1, the calcium ion release flux generated during the occurrence and development of calcium holes is expressed as follows:

[0114]

[0115] Among them, J RyR =J Rogue +J Clustered It is the calcium ion flux released by calcium ion channels, including the calcium ion release flux caused by the opening of free ranodine receptors. Rogue and calcium ion release flux caused by opening of clustered ranoldin receptors J Clustered ;(x i ,y j ) is the spatial coordinate of the location of a single ranolin receptor; δ is the Dirac delta function, that is, all values ​​except zero are equal to zero; S(x i ,y j ,t;T open ) is a random function that controls the on / off state of the Ranodine receptor, taking values ​​of 1 or 0. When S takes a value of 0, the Ranodine receptor is in a closed or inactivated state. When S takes a value of 1, it is located at (x i ,y j ) single ranoldin receptor opens and lasts for T open Time, set to 1ms in calculation.

[0116] σ(x i ,y j ) means that the i ,y j ) when a single ranodine receptor is opened, the calcium ion flux released into the adjacent cell of the cytoplasm (volume is V0) is expressed as follows;

[0117]

[0118] Where Far is the Faraday constant; i RyR is the initial current when a single ranoldin receptor opens; [Ca 2+ ] SR (x i ,y j ) and [Ca2+ ] cyto (x i ,y j ) represent the coordinates (x i ,y j ) at the ranoldin receptor, the free calcium ion concentration inside and outside the terminal cisterna tum of the sarcoplasmic reticulum (outside is the cytoplasm), [Ca 2+ ] SR (x i ,y j ) and [Ca 2+ ] cyto (x i ,y j ) changes with time, and the values ​​at different times are determined by synchronously calculating the correlation models of calcium sparks and calcium holes;

[0119] The process of activating the ranodine receptor is a random process, and its open probability per unit time is expressed as:

[0120]

[0121] Among them, P max =0.3 / RyR / ms is the maximum probability of single ranolin receptor opening, K = 15 μM is the calcium ion sensitivity parameter, and m-RyR is the Hill coefficient representing the cooperativity of ranolin receptors.

[0122] S1 also sets initial conditions and boundary conditions; the method for setting the initial conditions is:

[0123] The initial values ​​of the concentration of free calcium ions in the sarcoplasmic reticulum (terminal cisternae and free sarcoplasmic reticulum) and the cytoplasm are the concentration values ​​in their resting state. and

[0124] The initial value of free calcium ion concentration can be expressed as:

[0125]

[0126]

[0127] The initial values ​​of [CaF] and [CaB] are calculated from equations 1.6 and 1.8:

[0128]

[0129] Therefore, the initial values ​​of [CaF] and [CaB] are t＝0 , [CaB] t＝0 They are respectively expressed as follows:

[0130]

[0131] The boundary conditions are set as follows:

[0132] The boundary of the sarcoplasmic reticulum terminal pool is divided into two parts: the boundary Γ1 connected to the free sarcoplasmic reticulum and the remaining boundary Γ2. Assuming that the calcium ion concentration in the free sarcoplasmic reticulum remains unchanged, the free calcium ion concentration at the boundary Γ1 of the sarcoplasmic reticulum terminal pool connected to the free sarcoplasmic reticulum is always the initial value of the resting state. Considering that the membrane of the terminal cistern of the sarcoplasmic reticulum is impermeable to ions, the permeability of calcium ions at the other boundary Γ2 of the terminal cistern of the sarcoplasmic reticulum is 0, which is expressed as follows:

[0133]

[0134] Where n represents the normal vector at the boundary.

[0135] S2: Numerical methods for reaction-diffusion models, including:

[0136] Equation 1.1 and Equation 1.3 use the equation discretization method to discretize the partial differential equation;

[0137] First, integrate equation 1.1 over any triangular mesh area Ω to obtain:

[0138]

[0139] Where C represents the free calcium concentration in the terminal cisterna of the sarcoplasmic reticulum [Ca 2+ ] SR , S is the area of ​​the grid Ω, and J is the sum of all source and sink terms, that is, J = J dye +J buffer +J RyR ;

[0140] Secondly, the mesh Ω is a polygon with 3 edges, so Equation 2.1 can be written as:

[0141]

[0142] in, represents the average free calcium ion concentration within the grid Ω;

[0143] Finally, solve equation 2.2, which is to establish the relationship between the normal derivative and the average value on the boundary; let the left unit of a boundary be E1, and the right unit be E2, approximate the average value of the unit to the value at the unit centroid, and let the unit vector of the line connecting the centroids of units E1 and E2 be n', then we have

[0144]

[0145] Among them, r1 and r2 represent the centroid positions of elements E1 and E2 respectively; the normal derivative The projection of formula 2.3 in the direction of the boundary unit normal vector n can be used, that is,

[0146]

[0147] On each edge, we can introduce a quantity Where l is the boundary length;

[0148] Substituting equation 2.4 into equation 2.2, we have

[0149]

[0150] In summary, within each grid, given the average value of the free calcium ion concentration as the unknown quantity, Equation 2.5 is a system of ordinary differential equations, which is numerically solved using the fourth-order Runge-Kutta method;

[0151] Then, integrating equation 1.3 over any triangular mesh area Ω gives:

[0152]

[0153] Among them, C F represents the concentration of calcium fluorescer bound to calcium ions [CaF], S is the area of ​​the grid Ω;

[0154] Secondly, the mesh Ω is a polygon with 3 sides, so Equation 2.6 can be written as:

[0155]

[0156] in, represents the average value of the concentration of calcium fluorescer bound to calcium ions within the grid Ω;

[0157] Finally, solve equation 2.7, which is to establish the relationship between the normal derivative and the average value on the boundary; let the left unit of a boundary be E1, and the right unit be E2, approximate the average value of the unit to the value at the unit centroid, and let the unit vector of the line connecting the centroids of units E1 and E2 be n', then we have

[0158]

[0159] Among them, r1 and r2 represent the centroid positions of elements E1 and E2 respectively; the normal derivative The projection of formula 2.8 in the direction of the boundary unit normal vector n can be used, that is,

[0160]

[0161] On each edge, we can introduce a quantity Where l is the boundary length;

[0162] Substituting equation 2.9 into equation 2.7, we have

[0163]

[0164] In summary, within each grid, the average value of the concentration of the fluorescent agent bound to calcium ions is given as the unknown quantity, then Equation 2.10 is a system of ordinary differential equations, which is numerically solved using the fourth-order Runge-Kutta method.

[0165] S3: Three modes of calcium vacancies are reproduced based on the model numerical values: mini calcium vacancies, medium calcium vacancies, and typical calcium vacancies.

[0166] Among them, the time course and release intensity of mini calcium vacancies and typical calcium vacancies are consistent with the experimental results. The time course and spatial distribution of free calcium ion concentration in the terminal pool of the sarcoplasmic reticulum during the occurrence and development of calcium vacancies are given, and the spatiotemporal dynamic characteristics of the three modes of calcium vacancies are analyzed.

[0167] S3-1, mini calcium hole

[0168] When a small amount of activated free Ranodin receptors fails to successfully trigger the opening of clustered Ranodin receptors within the Ranodin receptor cluster, that is, only free Ranodin receptors are open, a mini calcium cavitation event with an amplitude 2-3 times smaller than that of the calcium cavitation corresponding to the calcium spark is formed in the terminal cisterna of the sarcoplasmic reticulum. Figure 1 This is a schematic diagram of the open and closed states of the Ranolidine receptor when mini calcium holes occur, where the red dots represent free Ranolidine receptors with calcium release channels in the open state, and the black dots represent Ranolidine receptors with calcium release channels in the closed (or inactivated) state.

[0169] The reaction-diffusion model was established because the presence of free ranolidine receptors was added to the model, which reproduced the mini calcium cavitation corresponding to the experiment. The calcium cavitation model with only free ranolidine receptors open was numerically calculated, and the time course of the mini calcium cavitation was given, such as Figure 2 As shown, Figure 2 (a) represents the average free calcium concentration in the terminal cisterna of the sarcoplasmic reticulum Figure 2 (b) represents calcium fluorescence ΔF / F0. The low-affinity fluorescent agent Fluo-5N was used to load the sarcoplasmic reticulum of rabbit cardiomyocytes. Combined with laser confocal microscopy technology, small-amplitude calcium signal events, called mini calcium holes, were synchronously observed in the terminal pool of the sarcoplasmic reticulum. Figure 2 (b) and Figure 3 It was found that the numerical simulation of mini calcium cavities was consistent with the time course and release intensity. Figure 3 As shown, Figure 3(a) shows the experimentally measured calcium fluorescence ΔF / F0 time course, the time course of the two QCR events (blue curve) and the corresponding two QCD events (red curve) observed at 1 and 2 indicated by the elliptical dashed lines, and the curve graph in the box is the enlarged image, where QCR represents the calcium quark event in the cytoplasm, and QCD represents the calcium signal event in the sarcoplasmic reticulum terminal pool corresponding to the calcium quark QCR; Figure 3 (b) represents the amplitude of calcium fluorescence ΔF / F0; Figure 3 (c) represents the time t required for calcium fluorescence ΔF / F0 to reach the peak value peaknadir And the time constant t 67 , t 67 represents the experimentally measured recovery time constant, where the blue column represents QCR and the red column represents QCD. Figure 2 and Figure 3 It can be seen that mini calcium vacancies have the following characteristics during their occurrence and development as shown in Table 1.

[0170] Table 1: Numerical results and experimental data of mini calcium vacancies

[0171]

[0172] Figure 2 The numerically reproduced time course of the occurrence and development of mini calcium holes is given, which clearly shows the change trend of the average free calcium ion concentration in the terminal pool of the sarcoplasmic reticulum over time. In order to more intuitively analyze the spatial dynamics of mini calcium holes in myocardial cells, we numerically calculated the spatial distribution of free calcium ion concentration in the terminal pool of the sarcoplasmic reticulum at different times during the occurrence and development of mini calcium holes, such as Figure 4 As shown. It can be seen that mini calcium vacancies have the following characteristics during their occurrence and development:

[0173] First, at t = 10 ms, the local free calcium ion concentration in the terminal cisterna of the sarcoplasmic reticulum reached its lowest value.

[0174] Second, at t = 70 ms, the mini calcium vacancies were basically restored, that is, the free calcium ion concentration in the terminal pool of the sarcoplasmic reticulum basically recovered to 95% of the resting concentration.

[0175] Third, when mini calcium vacancies occur, the spatial distribution of free calcium ion concentration is more symmetrical.

[0176] S3-2, medium calcium vacancy

[0177] When a small amount of activated free ranolin receptors triggers the opening of clustered ranolin receptors within a nearby ranolin receptor cluster, a medium-sized calcium cavitation event with a larger amplitude than the mini calcium cavitation can be induced, which is called medium calcium cavitation. Figure 5This is a schematic diagram of the open and closed states of the Ranolidine receptor when a medium calcium hole occurs, where the red dots represent the Ranolidine receptor with the calcium release channel in the open state, and the black dots represent the Ranolidine receptor with the calcium release channel in the closed (or inactivated) state.

[0178] The calcium cavitation model of free ranolin receptor and clustered ranolin receptor opening in a ranolin receptor cluster was numerically calculated, and the time course of the intermediate calcium cavitation was given, as Figure 6 As shown, Figure 6 (a) represents the average free calcium concentration in the terminal cisterna of the sarcoplasmic reticulum Figure 6 (b) represents calcium fluorescence ΔF / F0. Figure 6 It can be seen that medium calcium vacancies have the following characteristics during their occurrence and development as shown in Table 2.

[0179] Table 2: Numerical results for medium calcium vacancies

[0180]

[0181]

[0182] S3-3. Typical calcium vacancies

[0183] A small amount of activated free ranolin receptors may successfully trigger the opening of clustered ranolin receptors within the two ranolin receptor clusters, inducing typical calcium cavitation events. Figure 7 This is a schematic diagram of the open and closed states of the Ranolidine receptor when a typical calcium vacancy occurs, where the red dots represent the Ranolidine receptor with the calcium release channel in the open state, and the black dots represent the Ranolidine receptor with the calcium release channel in the closed (or inactivated) state.

[0184] The calcium cavitation model of free ranolin receptor and clustered ranolin receptor opening in two ranolin receptor clusters was numerically calculated, and the time course of typical calcium cavitation was given, such as Figure 8 As shown, Figure 8 (a) represents the average free calcium concentration in the terminal cisterna of the sarcoplasmic reticulum Figure 8 (b) represents calcium buffer [CaB] / [CaB]0. Using low-affinity fluorescent agent Fluo-5N, combined with laser confocal microscopy, typical calcium cavitation events were found in rabbit cardiomyocytes. The time course is shown in Figure 2. Fig. 9 (b) Fig. 9 (a) Typical calcium hole calcium fluorescence ΔF / F0 time course of numerical simulation, Fig. 9 It can be found that the typical calcium vacancies simulated by numerical simulation are consistent with the time course and release intensity. Figure 8 , Fig. 9It can be seen that typical calcium vacancies have the following characteristics during their occurrence and development as shown in Table 3.

[0185] Table 3: Numerical results and experimental data for typical calcium vacancies

[0186]

[0187] The spatial distribution of free calcium ion concentration in the terminal pool of the sarcoplasmic reticulum at different times during the occurrence and development of a typical calcium vacancy was numerically calculated. Fig.10 As shown. It can be seen that typical calcium vacancies have the following characteristics during their occurrence and development:

[0188] First, at t = 10 ms, the local free calcium ion concentration in the terminal cisterna of the sarcoplasmic reticulum reached its lowest value.

[0189] Second, at t = 110 ms, the typical calcium vacancies were basically restored, that is, the free calcium ion concentration in the terminal pool of the sarcoplasmic reticulum basically recovered to 95% of the resting concentration.

[0190] Third, when typical calcium vacancies occur, the spatial distribution of free calcium ion concentration is relatively symmetrical.

[0191] By comparing the time course and spatial distribution of the three modes, it can be found that the average free calcium ion concentration in the terminal pool of the sarcoplasmic reticulum when the calcium vacancy reaches the lowest point is typical calcium vacancy < medium calcium vacancy < mini calcium vacancy. The calcium ions required to fill the calcium vacancy during the calcium recovery phase are typical calcium vacancy > medium calcium vacancy > mini calcium vacancy. The calcium ion filling rate during the calcium recovery phase is the same, so the time required for calcium vacancy recovery is typical calcium vacancy > medium calcium vacancy > mini calcium vacancy. The occurrence of mini calcium vacancy is based on the opening of a single or a small amount of free ranolin receptors. The occurrence of medium calcium vacancy requires the opening of multiple clustered ranolin receptors in a ranolin receptor cluster, while the occurrence of typical calcium vacancy requires the opening of multiple clustered ranolin receptors in two ranolin receptor clusters. From a theoretical perspective, the occurrence frequency of the three modes is mini calcium vacancy > medium calcium vacancy > typical calcium vacancy. From a numerical perspective, the occurrence frequency of the three modes is mini calcium vacancy > medium calcium vacancy > typical calcium vacancy.

[0192] The above embodiments are only for illustrating the technical concept and features of the present invention, and their purpose is to enable people familiar with the technology to understand the content of the present invention and implement it accordingly, and they cannot be used to limit the protection scope of the present invention. Any modifications made according to the spirit of the main technical solution of the present invention should be included in the protection scope of the present invention.

Claims

1. A two-dimensional random reaction-diffusion modeling method for calcium vacancies in myocardial cells, characterized in that: The steps include: S1: Establish a reaction-diffusion model of calcium ions in the terminal pool of the sarcoplasmic reticulum of myocardial cells: the reaction-diffusion model includes the calcium fluorescence system, the calcium buffer system, and the calcium release system; The reaction-diffusion model is: I RyR =J Rogue +J Clustered (1.2) Among them, [Ca 2+ ] SR , [CaF], [CaB] represent the concentrations of free calcium ions, calcium fluorescent agent bound to calcium ions, and calcium buffer bound to calcium ions in the terminal cisterna of the sarcoplasmic reticulum, respectively; D Ca and D F represent the diffusion coefficients of free calcium ions and calcium fluorescent agent in the terminal cisterna of the sarcoplasmic reticulum, respectively; J dye , J buffer , J RyR J represents the calcium ion flux caused by the reaction of calcium ions with calcium fluorescent agent, the reaction of calcium ions with calcium buffer, and the release of calcium ions. RyR It consists of two parts: Rogue and J Clustered represent the calcium ion release flux caused by the opening of free ranolidine receptors and clustered ranolidine receptors, respectively, and t represents time; S2: Numerical methods for reaction-diffusion modeling; S3: Three modes of calcium vacancies are reproduced based on the model numerical values: mini calcium vacancies, medium calcium vacancies, and typical calcium vacancies.

2. The random reaction-diffusion two-dimensional modeling method of myocardial cell calcium vacancies according to claim 1, characterized in that: In the calcium fluorescence system described in S1, the low-affinity calcium fluorescent agent Fluo-5N is used to label calcium ions in the sarcoplasmic reticulum: Assume that F is the calcium fluorescent agent Fluo-5N, then the binding-dissociation reaction expression of the calcium fluorescent agent and calcium ions is: Wherein, CaF represents a calcium fluorescer bound to calcium ions; The calcium ion flux J caused by the reaction between the calcium fluorescent agent Fluo-5N and calcium ions is dye It is expressed as: Among them, [F] T represents the total concentration of calcium fluorescer, They represent the forward and reverse reaction rates of the calcium fluorescent agent respectively. The concentration of the calcium fluorescent agent that is not bound to calcium ions can be expressed as [F] T -[CaF].

3. The random reaction-diffusion two-dimensional modeling method of myocardial cell calcium vacancies according to claim 1, characterized in that: In the calcium buffer system described in S1, assuming that B is the calcium buffer CSQ, its binding-dissociation reaction equation is the same as that of the calcium fluorescent agent, as follows: Wherein, CaB represents the calcium buffer CSQ that binds to calcium ions; The calcium ion flux J caused by the reaction between calcium buffer CSQ and calcium ions is buffer It can be expressed as: Among them, [B] T represents the total concentration of calcium buffer CSQ, They represent the forward and reverse reaction rates of the calcium buffer, respectively. The concentration of the calcium buffer that is not bound to calcium ions can be expressed as [B] T -[CaB].

4. The random reaction-diffusion two-dimensional modeling method of myocardial cell calcium vacancies according to claim 1, characterized in that: In the calcium release system described in S1, the calcium ion release flux generated during the occurrence and development of calcium holes is expressed as follows: Among them, J RyR =J Rogue +J Clustered It is the calcium ion flux released by calcium ion channels, including the calcium ion release flux caused by the opening of free ranodine receptors. Rogue and calcium ion release flux caused by opening of clustered ranoldin receptors J Clustered ;(x i ,y j ) is the spatial coordinate of the location of a single ranolin receptor; δ is the Dirac delta function, that is, all values ​​except zero are equal to zero; S(x i ,y j ,t;T open ) is a random function that controls the on / off state of the Ranodine receptor, taking values ​​of 1 or 0. When S takes a value of 0, the Ranodine receptor is in a closed or inactivated state. When S takes a value of 1, it is located at (x i ,y j ) single ranoldin receptor opens and lasts for T open time; σ(x i ,y j ) means located at (x i ,y j ) when a single ranodine receptor is opened, the calcium ion flux released into the adjacent cell of the cytoplasm (volume is V0) is expressed as follows; Where Far is the Faraday constant; i RyR is the initial current when a single ranoldin receptor opens; [Ca 2+ ] SR (x i ,y j ) and [Ca 2+ ] cy to(x i ,y j ) represent the coordinates (x i ,y j ) at the ranoldin receptor, the free calcium ion concentration inside and outside the terminal cisterna tum of the sarcoplasmic reticulum (outside is the cytoplasm), [Ca 2+ ] SR (x i ,y j ) and [Ca 2+ ] cyto (x i ,y j ) changes with time, and the values ​​at different times are determined by synchronously calculating the correlation models of calcium sparks and calcium holes; The process of activating the ranodine receptor is a random process, and its open probability per unit time is expressed as: Among them, P max =0.3 / RyR / ms is the maximum probability of single ranolin receptor opening, K = 15 μM is the calcium ion sensitivity parameter, and m-RyR is the Hill coefficient representing the cooperativity of ranolin receptors.

5. The random reaction-diffusion two-dimensional modeling method of myocardial cell calcium vacancies according to claim 1, characterized in that: S1 also sets initial conditions and boundary conditions; the method for setting the initial conditions is: The initial value of the free calcium ion concentration in the sarcoplasmic reticulum and cytoplasm is the concentration value in its resting state. and The initial value of free calcium ion concentration can be expressed as: The initial values ​​of [CaF] and [CaB] are calculated from equations 1.6 and 1.8: Therefore, the initial values ​​of [CaF] and [CaB] are t＝0 , [CaB] t＝0 They are respectively expressed as follows:

6. The random reaction-diffusion two-dimensional modeling method of myocardial cell calcium vacancies according to claim 5, characterized in that: The boundary conditions are set as follows: The boundary of the sarcoplasmic reticulum terminal pool is divided into two parts: the boundary Γ1 connected to the free sarcoplasmic reticulum and the remaining boundary Γ2. Assuming that the calcium ion concentration in the free sarcoplasmic reticulum remains unchanged, the free calcium ion concentration at the boundary Γ1 of the sarcoplasmic reticulum terminal pool connected to the free sarcoplasmic reticulum is always the initial value of the resting state. Considering that the membrane of the terminal cistern of the sarcoplasmic reticulum is impermeable to ions, the permeability of calcium ions at the other boundary Γ2 of the terminal cistern of the sarcoplasmic reticulum is 0, which is expressed as follows: Where n represents the normal vector at the boundary.

7. The random reaction-diffusion two-dimensional modeling method of myocardial cell calcium vacancies according to claim 1, characterized in that: The numerical methods for establishing the reaction-diffusion model in S2 include: First, integrate equation 1.1 over any triangular mesh area Ω to obtain: Where C represents the free calcium concentration in the terminal cisterna of the sarcoplasmic reticulum [Ca 2+ ] SR , S is the area of ​​the grid Ω, and J is the sum of all source and sink terms, that is, J = J dye +J buffer +J RyR ; Secondly, the mesh Ω is a polygon with 3 edges, so Equation 2.1 can be written as: in, represents the average free calcium ion concentration within the grid Ω; Finally, solve equation 2.2, which is to establish the relationship between the normal derivative and the average value on the boundary; let the left unit of a boundary be E1, and the right unit be E2, approximate the average value of the unit to the value at the unit centroid, and let the unit vector of the line connecting the centroids of units E1 and E2 be n', then we have Among them, r1 and r2 represent the centroid positions of elements E1 and E2 respectively; the normal derivative The projection of formula 2.3 in the direction of the boundary unit normal vector n can be used, that is, On each edge, we can introduce a quantity Where l is the boundary length; Substituting equation 2.4 into equation 2.2, we have In summary, within each grid, given the average value of the free calcium ion concentration as the unknown quantity, Equation 2.5 is a system of ordinary differential equations, which is numerically solved using the fourth-order Runge-Kutta method; Then, integrating equation 1.3 over any triangular mesh area Ω gives: Among them, C F represents the concentration of calcium fluorescer bound to calcium ions [CaF], S is the area of ​​the grid Ω; Secondly, the mesh Ω is a polygon with 3 sides, so Equation 2.6 can be written as: in, represents the average value of the concentration of calcium fluorescer bound to calcium ions within the grid Ω; Finally, solve equation 2.7, which is to establish the relationship between the normal derivative and the average value on the boundary; let the left unit of a boundary be E1, and the right unit be E2, approximate the average value of the unit to the value at the unit centroid, and let the unit vector of the line connecting the centroids of units E1 and E2 be n', then we have Among them, r1 and r2 represent the centroid positions of elements E1 and E2 respectively; the normal derivative The projection of formula 2.8 in the direction of the boundary unit normal vector n can be used, that is, On each edge, we can introduce a quantity Where l is the boundary length; Substituting equation 2.9 into equation 2.7, we have In summary, within each grid, the average value of the concentration of the fluorescent agent bound to calcium ions is given as the unknown quantity, then Equation 2.10 is a system of ordinary differential equations, which is numerically solved using the fourth-order Runge-Kutta method.

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