A note on a method for determining advantageous properties of an anti-arrhythmic drug based on a mathematical model of cardiac cells

A computational method for identifying an optimal combination of existing drugs to repair the action potentials of SQT1 ventricular myocytes

Mutations are known to cause perturbations in essential functional features of integral membrane proteins, including ion channels. Even restricted or point mutations can result in substantially changed properties of ion currents. The additive effect of these alterations for a specific ion channel can result in significantly changed properties of the action potential (AP). Both AP shortening and AP prolongation can result from known mutations, and the consequences can be life-threatening. Here, we present a computational method for identifying new drugs utilizing combinations of existing drugs. Based on the knowledge of theoretical effects of existing drugs on individual ion currents, our aim is to compute optimal combinations that can ‘repair’ the mutant AP waveforms so that the baseline AP-properties are restored. More specifically, we compute optimal, combined, drug concentrations such that the waveforms of the transmembrane potential and the cytosolic calcium concentration of the mutant cardiomyocytes (CMs) becomes as similar as possible to their wild type counterparts after the drug has been applied. In order to demonstrate the utility of this method, we address the question of computing an optimal drug for the short QT syndrome type 1 (SQT1). For the SQT1 mutation N588K, there are available data sets that describe the effect of various drugs on the mutated K⁺ channel. These published findings are the basis for our computational analysis which can identify optimal compounds in the sense that the AP of the mutant CMs resembles essential biomarkers of the wild type CMs. Using recently developed insights regarding electrophysiological properties among myocytes from different species, we compute optimal drug combinations for hiPSC-CMs, rabbit ventricular CMs and adult human ventricular CMs with the SQT1 mutation. Since the ‘composition’ of ion channels that form the AP is different for the three types of myocytes under consideration, so is the composition of the optimal drug.

Poly-pharmacology (using multiple drugs to treat disease) has been proposed for improving cardiac anti-arrhythmic therapy for at least two decades. However, the specific arrhythmia contexts in which polytherapy is likely to be both safe and effective have remained elusive. Type 1 short QT syndrome (SQT1) is a rare form of cardiac arrhythmia that results from mutations to the human Ether-á-go-go Related Gene (hERG) potassium channel. Functionally, these mutations are remarkably consistent in that they permit the channel to open earlier during each heart beat. While hundreds of compounds are known to inhibit hERG channels, the specific effect of SQT1 mutations that allows for early channel opening also limits the ability of most of those compounds to correct SQT1 dysfunction. Here, we have applied a suite of ventricular cardiomyocyte computational models to ask whether polytherapy may offer a more effective therapeutic strategy in SQT1, and if so, what the likely characteristics of that strategy are. Our analyses suggest that simultaneous induction of late sodium current and partial hERG blockade offers a promising strategy. While no activators of late sodium current have been clinically approved, several experimental compounds are available and may provide a basis for interrogating this strategy. The method presented here can be used to compute optimal drug combinations provided that the effect of each drug on every relevant ion channel is known.

Computing Optimal Properties of Drugs Using Mathematical Models of Single Channel Dynamics

Single channel dynamics can be modeled using stochastic differential equations, and the dynamics of the state of the channel (e.g. open, closed, inactivated) can be represented using Markov models. Such models can also be used to represent the effect of mutations as well as the effect of drugs used to alleviate deleterious effects of mutations. Based on the Markov model and the stochastic models of the single channel, it is possible to derive deterministic partial differential equations (PDEs) giving the probability density functions (PDFs) of the states of the Markov model. In this study, we have analyzed PDEs modeling wild type (WT) channels, mutant channels (MT) and mutant channels for which a drug has been applied (MTD). Our aim is to show that it is possible to optimize the parameters of a given drug such that the solution of theMTD model is very close to that of the WT: the mutation’s effect is, theoretically, reduced significantly.We will present the mathematical framework underpinning this methodology and apply it to several examples. In particular, we will show that it is possible to use the method to, theoretically, improve the properties of some well-known existing drugs.

Defining candidate drug characteristics for Long-QT (LQT3) syndrome

Mutations of the SCN5A gene can significantly alter the function of cardiac myocyte sodium channels leading to increased risk of ventricular arrhythmia. Over the past decade, detailed Markov models of the action potential of cardiac cells have been developed. In such models, the effects of a drug can be treated as alterations in on- and off rates between open and inactivated states on one hand, and blocked states on the other hand. Our aim is to compute the rates specifying a drug in order to: (a) restore the steady-state open probability of the mutant channel to that of normal wild type channels; and (b) minimize the difference between whole cell currents in drugged mutant and wild type cells. The difference in the electrochemical state vector of the cell can be measured in a norm taking all components and their dynamical properties into account. Measured with this norm, the difference between the state of the mutant and wild-type cell was reduced by a factor of 36 after the drug was introduced and by factors of 4 over mexitiline and 25 over lidocaine. The results suggest the potential to synthesize more effective drugs based on mechanisms of action of existing compounds.

Computing the stability of steady-state solutions of mathematical models of the electrical activity in the heart

Instabilities in the electro-chemical resting state of the heart can generate ectopic waves that in turn can initiate arrhythmias. We derive methods for computing the resting state for mathematical models of the electro-chemical process underpinning a heartbeat, and we estimate the stability of the resting state by invoking the largest real part of the eigenvalues of a linearized model. The implementation of the methods is described and a number of numerical experiments illustrate the feasibility of the methods. In particular, we test the methods for problems where we can compare the solutions with analytical results, and problems where we have solutions computed by independent software. The software is also tested for a fairly realistic 3D model.

Systems approaches and algorithms for discovery of combinatorial therapies

Effective therapy of complex diseases requires control of highly nonlinear complex networks that remain incompletely characterized. In particular, drug intervention can be seen as control of cellular network activity. Identification of control parameters presents an extreme challenge due to the combinatorial explosion of control possibilities in combination therapy and to the incomplete knowledge of the systems biology of cells. In this review paper, we describe the main current and proposed approaches to the design of combinatorial therapies, including the heuristic methods used now by clinicians and alternative approaches suggested recently by several authors. New approaches for designing combinations arising from systems biology are described. We discuss in special detail the design of algorithms that identify optimal control parameters in cellular networks based on a quantitative characterization of control landscapes, maximizing utilization of incomplete knowledge of the state and structure of intracellular networks. The use of new technology for high-throughput measurements is key to these new approaches to combination therapy and essential for the characterization of control landscapes and implementation of the algorithms. Combinatorial optimization in medical therapy is also compared with the combinatorial optimization of engineering and materials science and similarities and differences are delineated.

Modeling the effect of Kv1.5 block on the canine action potential

A wide range of ion channels have been considered as potential targets for pharmacological treatment of atrial fibrillation. The Kv1.5 channel, carrying the I(Kur) current, has received special attention because it contributes to repolarization in the atria but is absent or weakly expressed in ventricular tissue. The dog serves as an important animal model for electrophysiological studies of the heart and mathematical models of the canine atrial action potential (CAAP) have been developed to study the interplay between ionic currents. To enable more-realistic studies on the effects of Kv1.5 blockers on the CAAP in silico, two continuous-time Markov models of the guarded receptor type were formulated for Kv1.5 and subsequently inserted into the Ramirez-Nattel-Courtemanche model of the CAAP. The main findings were: 1), time- and state-dependent Markov models of open-channel Kv1.5 block gave significantly different results compared to a time- and state-independent model with a downscaled conductance; 2), the outcome of Kv1.5 block on the macroscopic system variable APD(90) was dependent on the precise mechanism of block; and 3), open-channel block produced a reverse use-dependent prolongation of APD(90). This study suggests that more-complex ion-channel models are a prerequisite for quantitative modeling of drug effects.

TIde: a software for the systematic scanning of drug targets in kinetic network models

BACKGROUND

During the stages of the development of a potent drug candidate compounds can fail for several reasons. One of them, the efficacy of a candidate, can be estimated in silico if an appropriate ordinary differential equation model of the affected pathway is available. With such a model at hand it is also possible to detect reactions having a large effect on a certain variable such as a substance concentration.

RESULTS

We show an algorithm that systematically tests the influence of activators and inhibitors of different type and strength acting at different positions in the network. The effect on a quantity to be selected (e.g. a steady state flux or concentration) is calculated. Moreover, combinations of two inhibitors or one inhibitor and one activator targeting different network positions are analysed. Furthermore, we present TIde (Target Identification), an open source, platform independent tool to investigate ordinary differential equation models in the common systems biology markup language format. It automatically assigns the respectively altered kinetics to the inhibited or activated reactions, performs the necessary calculations, and provides a graphical output of the analysis results. For illustration, TIde is used to detect optimal inhibitor positions in simple branched networks, a signalling pathway, and a well studied model of glycolysis in Trypanosoma brucei.

CONCLUSION

Using TIde, we show in the branched models under which conditions inhibitions in a certain pathway can affect a molecule concentrations in a different. In the signalling pathway we illuminate which inhibitions have an effect on the signalling characteristics of the last active kinase. Finally, we compare our set of best targets in the glycolysis model with a similar analysis showing the applicability of our tool.