Abstract
The physical processes we study in this text are modeled using models including stochastic terms. Direct numerical simulations based on such stochastic models give results that are hard to interpret and it is therefore common to run many simulations and compute the average, and we have also seen that we can derive models governing the probability density functions. These are powerful tools that provide insight in the processes. In this chapter we will see that it is useful to have specific numbers that characterize stochastic variables and associated probability density functions. We encountered the equilibrium probability of being in the open or closed state (see, e.g., pageĀ 57) and we introduced probability density functions (see, e.g., pageĀ 30). Here we shall derive some specific (and common) characteristics of the probability density functions and discuss how these characteristics can be used to gain an understanding of calcium release. We will also show how the characteristics relate to the concepts already introduced and we will discuss how the characteristics vary as functions of the mutation severity index. Finally, we will show how the statistical characterizations can be used to evaluate the properties of theoretical drugs.
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The physical processes we study in this text are modeled using models including stochastic terms. Direct numerical simulations based on such stochastic models give results that are hard to interpret and it is therefore common to run many simulations and compute the average, and we have also seen that we can derive models governing the probability density functions. These are powerful tools that provide insight in the processes. In this chapter we will see that it is useful to have specific numbers that characterize stochastic variables and associated probability density functions. We encountered the equilibrium probability of being in the open or closed state (see, e.g., pageĀ 57) and we introduced probability density functions (see, e.g., pageĀ 30). Here we shall derive some specific (and common) characteristics of the probability density functions and discuss how these characteristics can be used to gain an understanding of calcium release. We will also show how the characteristics relate to the concepts already introduced and we will discuss how the characteristics vary as functions of the mutation severity index. Finally, we will show how the statistical characterizations can be used to evaluate the properties of theoretical drugs.
4.1 Probability Density Functions
Let us briefly recall the models under consideration. We consider the model
of the calcium concentration of the dyad (see Fig.Ā 2.1). Recall that v r denotes the speed of release from the sarcoplasmic reticulum (SR) to the dyad, v d denotes the speed of diffusion from the dyad to the cytosol, c 0 is the concentration of calcium ions in the cytosol, and c 1 is the calcium concentration in the SR; both c 0 and c 1 are assumed to be constant. The stochastic function \(\bar{\gamma }=\bar{\gamma } (t)\) can be either zero (closed state) or one (open state) and the state is governed by the Markov model
where k oc and k co are the rates associated with the Markov model. As discussed above, the probability density functions of the states of the Markov model are governed by the following system of partial differential equations:
where, as above, Ļ o and Ļ c are the probability density functions of the open and closed states, respectively. Furthermore, we recall that
The system of partial differential equations given byĀ (4.3) andĀ (4.4) is solved on the computational domain given by \(\Omega = [c_{0},c_{+}]\), where
and the boundary conditions are set up to ensure that there is no leak of probability across the boundaries (see pageĀ 37).
4.2 Statistical Characteristics
For the probability density functions given by the systemĀ (4.3) andĀ (4.4), we can introduce the common statistical concepts of probability, expectation, and standard deviation. The probabilities of being in the open and closed states are given by
respectively. It is worth noting that these values are time dependent but independent of space (concentration). Furthermore, the sum of these probabilities adds up to one,
for all time. The expected values of the concentration are given by
under the condition that the channels are open and closed, respectively. Finally, the standard deviations \(\sigma _{o}\) and \(\sigma _{c}\) are given by
We will show below how changes in the Markov model affect these characteristics and how the characteristics are influenced by the theoretical drugs. Generally, we have to solve the systemĀ (4.3) andĀ (4.4) and then compute the statistical properties. However, we will see that in the special case in which the rate functions defining the Markov model, k oc and k co , are constant; we can compute some of the characteristics analytically. We will therefore start by considering such a case.
4.3 Constant Rate Functions
We consider the systemĀ (4.3) andĀ (4.4) in the special case that both k oc and k co are constants (independent of the concentration x). If we integrateĀ (4.3) andĀ (4.4) over the interval \(\Omega\), we obtain the system
where we use the boundary conditions that state that there is no flux of probability across the boundaries.
4.3.1 Equilibrium Probabilities
When this system reaches equilibrium, the probabilities satisfy
and since \(\pi _{o} +\pi _{c} = 1,\) we find that
which we recognize as the probabilities o and c, respectively, derived directly from the equilibrium of the Markov model on pageĀ 57. This relation explains the connection between these two ways of considering the probability of being in a given state of the Markov model, but it is important to note that this relation only holds when the rate functions are constant.
4.3.2 Dynamics of the Probabilities
In the special case with only two states of the Markov model and constant rate functions, we can analytically compute how the probabilities evolve in time. If we use the fact that \(\pi _{o}\left (t\right ) +\pi _{c}\left (t\right ) = 1\) for all time, we find that the systemĀ (4.11) andĀ (4.12) can be reduced to one equation written in the form
Suppose we know that the channel is closed at tā=ā0;āthen Ļ o (0)ā=ā0 and we find the solution
We note that if the channel is closed at tā=ā0,āthe open probability reaches the equilibrium given by
at an exponential rate in time and the exponent is given by \(k_{co} + k_{oc}\) so that equilibrium is reached faster for higher rates.
4.3.3 Expected Concentrations
We still consider constant rate functions. In that case, we will show that the expected concentration in the case of open or closed channels can be obtained by solving a 2 Ć 2 linear system of ordinary differential equations. We start by considering the system defining the probability density functions,
Since
we find, usingĀ (4.18), that
Here the integral can be handled using integration by parts. The domain \(\Omega\) is defined by the interval \(\left [x_{-},x_{+}\right ] = [c_{0},c_{+}]\) and we recall that \(a_{o}\rho _{o} = a_{c}\rho _{c} = 0\) at \(x = x_{-}\) and at \(x = x_{+}.\) Therefore, by using the definition of a o given inĀ (4.5), we obtain
Consequently, we obtain
Similarly, we have
and, by the definitionĀ (4.6) of a c , we find that
We therefore obtain
Since we have already found explicit formulas for Ļ o and Ļ c ,āwe can define
and solve the system
When \(\pi _{o},\pi _{c}\), and \(e_{o},e_{c}\) are computed, of course computing the expectations E o and E c is straightforward.
4.3.4 Numerical Experiments
In Figs.Ā 4.1 andĀ 4.2, we illustrate the properties derived above by presenting the results of numerical computations. The parameters used in the computations are given in TableĀ 4.1. In Fig.Ā 4.1, we show how the probability defined byĀ (4.7) evolves as a function of time. The solid line is the exact solution given by the formulaĀ (4.17) and the crosses are based on the numerical solution of the systemĀ (4.3) andĀ (4.4), where the probability defined byĀ (4.7) is replaced by a Riemann sum based on the numerical solution. In Fig.Ā 4.2, we show the evolution of the expected concentration for the open (solid) or closed (dashed) state, based on solving the system of ordinary differential equations given byĀ (4.34) andĀ (4.35) and then computing the expectations fromĀ (4.33) and the solution ofĀ (4.16). The crosses are based on the numerical solution of the systemĀ (4.3) andĀ (4.4) and the expected values of the concentration defined byĀ (4.8) are again replaced by a Riemann sum based on the numerical solution.
4.3.5 Expected Concentrations in Equilibrium
In the case of constant rates, we derived the following system describing the evolution of the expected concentrations for open or closed channels, respectively,
where we recall that
The stationary solution of this system is given as the solution of the following linear 2 Ć 2 system of equations:
where Ļ o and Ļ c are equilibrium probabilities given byĀ (4.14) andĀ (4.15). The solution of this system in terms of a formula becomes messy, but if we consider the specific parameters used in the computations (see TableĀ 4.1), we find that the equilibrium expectations are given by
which compares well with our observations in Fig.Ā 4.2.
4.4 Markov Model of a Mutation
Mutations may change the release mechanism and thus seriously alter the function of the calcium-induced calcium release. Mutations in the RyR2 gene can lead to changes in the receptor function, increasing the open probability.
As mentioned above, one way to model the increased open probability is to define
where Ī¼ is referred to as the mutation severity index. This is a CO-mutation (see pageĀ 16) and it does not affect the mean open time. The parameter Ī¼ā=ā1 denotes the wild type case and larger values of Ī¼ indicate more severe mutations. Basically, since \(k_{co,\mu }> k_{co}\) for Ī¼ā>ā1, the mutation will lead to an increased probability of being in the open state.
The system governing the open and closed probability densities now takes the form
where, as above, we have
Note that in this model the opening rate depends on the concentration x. Model parameters are given in TableĀ 4.2.
In Fig.Ā 4.3, we show the results of Monte Carlo simulations (histograms) and solutions of the probability density systemĀ (4.43) andĀ (4.44) (red solid line) for the wild type case (Ī¼ā=ā1) and mutant case (Ī¼ā=ā3). As above, we see that these two computational approaches give more or less the same answer. It is more interesting to observe the effect of the mutation. We see that the mutation tends to shift the open probability density function toward the upper boundary, where the function becomes very large. This shows that, in the case of mutation, it is very likely to have a high concentration and an open channelāmuch more likely than in the wild type case.
The statistical characteristics introduced above are given in TableĀ 4.3. We note that the total open probability Ļ o increases from 0.811 for the wild type to 0.962 for the mutant. Also, we note that the expected concentration, E o , for open channels is given by 81.91 \(\upmu\) M for the wild type and 87.95 \(\upmu\) M for the mutant. The standard deviation, on the other hand, is significantly reduced (by a factor of three) in the mutant case compared to the wild type. The probability of being in the closed state decreases by a factor of five in the mutant case compared to the wild type, whereas the expected concentration is doubled and the standard deviation is reduced by a factor of seven.
4.4.1 How Does the Mutation Severity Index Influence the Probability Density Function of the Open State?
We have seen a few examples indicating how changes in the reaction rates k co and k oc change the probability density functions. Since we are able to solve the stationary case analytically, this issue can be studied in great detail. Let us start by recalling that we model the effect of the mutation by introducing a severity index Ī¼. The stationary model is then
where we recall that Ī¼ā=ā1 is the wild type case. We discussed above how to solve the steady state model analytically (see Sect.Ā 2.6, pageĀ 41) and we can use the analytical solution to investigate how the mutation affects the probability density functions. Since the steady state open probability density function is given by the solution of
with
where
we have solutions of the form
where K Ī¼ is a constant given by the somewhat complicated expressionā
Here \(_{1}\!F_{1}\) is Kummerās regularized hypergeometric function and
It is useful to consider the ratio of the mutant solution to the wild type solution and we find that
In Fig.Ā 4.4, we graph this relation as a function of the severity index Ī¼ and the concentration x.āWe observe that, close to the maximum concentration, the open probability density function of the mutant is much larger than for the wild type.
4.4.2 Boundary Layers
As seen in both the numerical and analytical solutions above, the probability density functions may have singularities at the endpoints. It is easily seen fromĀ (4.48) that Ļ o,āĪ¼ has a singularity at the endpoint \(x = c_{+}\) whenever
Similarly, we find that the closed probability density function is given by
which has a singularity at xā=āc 0 whenever
4.5 Statistical Properties as Functions of the Mutation Severity Index
We have seen, using numerical computations and analytical considerations, how the mutation severity index changes the probability density functions. In this section, we shall look with more detail into how the index changes the statistical properties of the probability density functions. Again, we consider a case where the rates k oc and k co are constants.
4.5.1 Probabilities
We recall that the open probability, defined as
evolves as
for wild type parameters in the case of Ļ o (0)ā=ā0. If we introduce the mutation severity index in the Markov model (seeĀ (4.42)), we find that the open probability evolves as
and thus the mutant case shows faster convergence toward a higher probability than the wild type case. In Fig.Ā 4.5, we show the graphs of Ļ o and Ļ o,āĪ¼ in the case of Ī¼ā=ā3 and Ī¼ā=ā10;āthe other parameters are given in TableĀ 4.4.
4.5.2 Expected Calcium Concentrations
We defined the expected calcium concentrations in the case of open and closed channels as
Recall that Ļ o and Ļ c ,āare given by explicit formulas and that we introduced
For constant rates k oc and k co , the expectations can be found by solving the system of ordinary differential equations
and then computing
In Fig.Ā 4.6, we show the expected values of the calcium concentration for wild type data when the channel is open (solid, red) and closed (solid, blue), as well as mutant-type data (Ī¼ā=ā3) when the channel is open (dotted, red) and closed (dotted, blue). In the computation using mutant data, we simply replace k co with Ī¼ k co . However, keep in mind that this affects the formulas defining the probabilities Ļ o and Ļ c as well.
4.5.3 Expected Calcium Concentrations in Equilibrium
As explained above, the equilibrium version of the expected concentrations E o and E c can be found by solving the following 2 Ć 2 linear system of equations:
and then computing
where Ļ o and Ļ c are equilibrium probabilities given byĀ (4.14) andĀ (4.15),
In Fig.Ā 4.7, we plot the expectations as a function of the mutation severity index. The red line represents the expected value of the calcium concentration when the channel is open and the blue line represents the expected value of the calcium concentration when the channel is closed. Here, we use the parameters given in TableĀ 4.4. The graphs start at Ī¼ā=ā1, which represents the wild type case.
4.5.4 What Happens as \(\mu \longrightarrow \infty\)?
When the mutation severity index goes to infinity, we force the channel to be open more or less all the time. If we consider the stochastic model
as \(\mu \longrightarrow \infty,\) we know that the channel is generally open, so we have \(\bar{\gamma }(t) \approx 1.\) Therefore, we obtain the model
As we have seen earlier, the equilibrium version of this equation is given by
and this is what we see from the graphs of Fig.Ā 4.7.
We can also see this from systemĀ (4.56). For the parameters given in TableĀ 4.4, we have
and therefore the systemĀ (4.56) takes the form
which, in terms of E o and E c , reads
If we let \(\mu \longrightarrow \infty,\) we obtain the system
and the solution
4.6 Statistical Properties of Open and Closed State Blockers
We have seen above that open and closed state theoretical blockers can significantly reduce the effect of the mutation. Computations have shown that closed state blockers repair the effect of the mutation as the parameter k bc goes to infinity. This effect is also shown by a direct mathematical argument. For the open state blocker, we have seen that fairly good results can be obtained when the parameters of the drug are optimized, but perfect results can probably not be obtained for a CO-mutation because of the change of the mean open time described above. In this section, we present the statistical properties of the two types of drugs. The properties are presented in TableĀ 4.5. In the table we observe that the total open probability (see Sect.Ā 4.2, pageĀ 72) of the open state in the wild type case is 0.811. This increases to 0.962 for the mutant case (Ī¼ā=ā3). When the closed state blocker is applied and the factor k bc is increased, we see that the open probability is repaired by the drug. The same effect holds for the expected concentration E o of the open state; it is completely repaired by the closed state blocker for large values of k bc . This also holds for the standard deviation. For the open state blocker, we do not obtain a sufficient effect by increasing k bo , but when both parameters of the drug are optimized, the open probability and the expected concentration of the open state are almost completely repaired. The open state blocker is, however, unable to repair the standard deviation.
4.7 Stochastic Simulations Using Optimal Drugs
We derived closed state and open state blockers with the parameters summarized in TableĀ 3.2 In Fig.Ā 4.8, we show the solutions of the stochastic model
computed using the scheme
where the dynamics of the stochastic function Ī³ are given by the Markov model. The wild type solution is given in the upper-left part of the solution and we observe significantly larger variations than for the solution in the mutant case (upper right). The effect of the mutation is well repaired by both drugs. Note that since a random number is used in every time step, the solutions will never coincide, no matter how good the drug is. This illustrates the difficulty of comparing stochastic solutions and shows that comparison using probability density functions and derived statistics is much easier.
References
G.S.B. Williams, M.A. Huertas, E.A. Sobie, M. Saleet Jafri, G.D. Smith, Moment closure for local control models of calcium-induced calcium release in cardiac myocytes. Biophys. J. 95, 1689ā1703 (2008)
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Tveito, A., Lines, G.T. (2016). Properties of Probability Density Functions. In: Computing Characterizations of Drugs for Ion Channels and Receptors Using Markov Models. Lecture Notes in Computational Science and Engineering, vol 111. Springer, Cham. https://doi.org/10.1007/978-3-319-30030-6_4
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