Master Thesis. Modelling the Effects of Dofetilide on I Kr Channel Activation using a Markov Model Approach L. Ramekers

Transcription

1 Master Thesis Modelling the Effects of Dofetilide on I Kr Channel Activation using a Markov Model Approach L. Ramekers Supervisors: Dr. R.L. Westra J. Heijman MSc Maastricht University Faculty of Humanities and Sciences Master Operations Research April 2008

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3 Preface The objective of this thesis is to provide insight in how the effects of druginduced blocking of the I Kr ion channels on the channel activation can be modelled. Amongst others, this insight might eventually help in improving treatment for patients having a high risk of cardiac arrhythmias by detecting the potentially severe side effects of the drugs used for suppressing these arrhythmias. This master thesis was written to complete the master program Operations Research at Maastricht University. The research was conducted at the Department of Mathematics of the university and supervised by dr. Ronald Westra and Jordi Heijman MSc. I would like to thank both of them for their support, guidance and advice. Special thanks goes out to Jordi Heijman for his ideas on the mathematical aspects of the Markov models and to Ronald Westra for his interesting lectures, without which I would not even have considered a research topic in this area. Furthermore I would like to thank them for their comments on draft versions, which were very valuable in writing this thesis. i

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5 Abstract This study focusses on the behavioural dynamics of rapidly activating delayed rectifier potassium ion channels in rabbit ventricular myocytes, which are essential in normal cardiac functioning. The behaviour is modelled as a dynamic first order Markov model, employing fixed transition probabilities between the discrete conformal states of the I Kr channel. Three different physiological views on the blocking kinetics of the drug dofetilide in these I Kr channels are computationally modelled, all of which are based on the standard I Kr model of C. Clancy and Y. Rudy. Our Simple Model was developed to analyse the conjecture that the blocking effects of dofetilide depend solely on the added concentration. With the Cooperative Binding Model we analyse whether dofetilide exhibits cooperative binding, and with the 2-Phase model we test the conjecture that dofetilide contains both a slow and a rapid blocking component. The relevant parameters are estimated from an optimal fit with experimentally recorded activation data using a genetic algorithm. We proposed a computationally efficient approach for calculating these activation curves. A combination of the robustness of the model and the fit with the experimental data is employed to determine the best of the three considered models. We found that the 2-Phase model was the best, which confirms the conjecture that the drug dofetilide contains both a slow and a rapid blocking component. Integration of such extended models in ventricular cell and tissue models can facilitate the analysis of the effects of pharmacological compounds. Keywords: I Kr channel, computational analysis, dofetilide, block, Markov model. iii

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7 Contents 1 Introduction Research questions Outline Background Biophysical processes Ion channels Effects of ion channel deficiencies Effects of dofetilide Modelling the action potential Modelling the I Kr channel C. Clancy and Y. Rudy s I Kr model Block incorporation in the I Kr model Model I: Simple Model II: Cooperative Binding Model III: Two Phases Theoretical analysis of Markov models for ion channels Steady State Activation Curve Tail Current A method for model analysis Experimental data Optimization technique Experiments Optimization settings Population Size Best Percentage Mutation probabilities Mutation distribution Model performance Sensitivity to noise Various noise functions Various number of affected parameters v

8 5 Conclusions & Discussion Conclusions I Kr channel models Electrical activation Model comparison Modelling the effects of dofetilide Discussion Recommendations A I Kr model equations 37 A.1 Transition rates in C. Clancy and Y. Rudy s I Kr model A.2 Transition rate equations A.2.1 Shared equations A.2.2 Equations for model I: Simple A.2.3 Equations for model II: Cooperative Binding A.2.4 Equations for model III: Two Phases A.3 Estimated parameter values A.3.1 Parameter values for model I: Simple A.3.2 Parameter values for model II: Cooperative Binding A.3.3 Parameter values for model III: Two Phases B Performance of the genetic algorithm 41 B.1 Settings used in the experiments B.2 Results B.2.1 Population sizes B.2.2 Percentages of best individuals B.2.3 Mutation probabilities B.2.4 Mutation probability distributions vi

9 Chapter 1 Introduction The observed macroscopic electrical activation of the heart is the cumulative result of a vast number of various microscopic electrophysiological and chemical processes. These processes generate electric currents as a result of ion flow in and out of the heart cells. Perturbations in these electric currents, which can be caused by congenital conditions or chemical compounds in drugs, can potentially lead to uncontrollable, dangerous and sometimes fatal activation of the heart cells. There are numerous drugs available that induce antiarrhythmic effects by altering ion currents in order to treat patients who suffer from arrhythmias. However, since the exact effects of perturbations in these currents are as yet far from understood, the possible side effects of these drugs remain unpredictable. For example, many so-called class III drugs were originally used for their antiarrhythmic behaviour. However, these drugs have now been taken off the market, because at low heart rates they were discovered to initiate severe proarrhythmic side effects [16]. A better understanding of the exact physiological consequences of perturbations in the ion currents would facilitate improved clinical treatment, since it would allow for better targeted treatment with less adversary effects. Computational models may help to provide these insights. The presented research illustrates how such models can be used to simulate pharmalogical interventions. The main objective and prime focus of the research are described in Section 1.1. Section 1.2 gives an outline for the thesis and Section 1.3 provides a brief overview of background information that might help in understanding the research and its importance. 1.1 Research questions The foremost objective of this research is to contribute to the understanding of the exact consequences of perturbations in the ion currents by modelling the effects of drugs on the electrical activation of ion channels. The ion channel modelled in this research is the so-called rapidly activating delayed rectifier potassium channel, or I Kr channel, in a rabbit ventricular myocyte, a cell found in the ventricles of the heart. The drug of which the effects were modelled is the class III drug dofetilide, known to specifically block the I Kr channel [21]. 1

10 The main research question can be formulated as: How can the effects of the drug dofetilide on the electrical activation of the I Kr channel in a rabbit ventricular myocyte be modelled? Which yields the following subquestions: 1. How can the behavioural kinetics and block of an I Kr channel be modelled? 2. How can an I Kr channel model be used to simulate the electrical activation? 3. What are the advantages and disadvantages of the different model structures? 1.2 Outline The remainder of this introductory chapter provides a brief overview of background information. In Chapter 2, the I Kr models under consideration are described and a theoretical analysis of Markov models is proposed, describing the mathematics that were involved in simplifying the model. Chapter 3 describes the methods that were used to test and compare the models, and the experiments that were performed and their results are shown in Chapter 4. Finally, the research questions are addressed and recommendations for further studies are discussed in Chapter Background In the introduction various terms were used that have not yet been explained. Therefore, the upcoming section clarifies some of the basic principles and terminology and provides background information to serve as a foundation for better understanding the research. The biophysical processes that are to be modelled are described in Section The role of ion channels in these processes are explained in Section and the consequences of deficiencies in the ion channels are described in Section The effects of dofetilide are discussed in Section and this chapter is concluded with a short historical overview of action potential modelling, given in Section Biophysical processes The flow of ions in and out of the heart cells is caused by a combination of electrophysiological and chemical processes [8]. Consider ions of a single type, for example positively charged potassium ions, K +, to be distributed unevenly between the intracellular and the extracellular space of the heart cells. In this case, there are two driving forces trying to redistribute the ions. The first one, a so-called concentration gradient, is established by the concentration difference between the intra and extracellular space and is therefore chemical in nature. The second driving force, the electrical gradient, is electrical in nature and is 2

11 caused by the difference in electrical charge. The driving forces result in the passage of the potassium ions through open ion channels, until both driving forces cancel each other and a dynamic equilibrium is reached. The corresponding voltage, or equilibrium potential E K, is given by the Nernst equation: E K = R T z K F ln[k+ ]o [K + ] i (1.1) where R is the gas constant, T the temperature, z K the valence of potassium, equal to +1, F is the Faraday constant and [K + ]o and [K + ] i represent the equilibrium potassium concentrations respectively outside and inside the cell. Since for single ion systems the voltage across the cell membrane, or the membrane potential V, is determined only by the distribution of the single ion type across the intracellular and extracellular space, a single ion system implies that the membrane potential V equals the equilibrium potential E. However, the behaviour of systems in which multiple ion types occur, such as heart cells, is more complex. The driving force for each ion can be calculated as the difference between its equilibrium potential E and the membrane potential V. The resulting current I X for an arbitrary ion X, is given by: I X = (V E X ) G X O X (1.2) where G X is the cell conductance and O X the percentage of ion channels allowing the passage of ion X residing in the open state. The combination of the electrochemical and physiological processes results in continuous driving forces influencing all ion types Ion channels The flow of ions in and out of the heart cells is caused by the chemical and electrical driving forces, but is regulated by ion channels. Each class of ion channels acts as a filter, allowing only specific types of ions to pass through the cell membrane. For example, the I Kr channels considered in this research, allow only the flow of potassium ions. Besides the class of the ion channel the transport of ions also depends on the internal state of the channel. Three distinct states can be distinguished for the I Kr channel: A closed state in which the ion channel is physically closed and hence prevents ions from passing through the cell membrane. An open state in which the channel is physically opened and potassium ions are allowed to flow through the channel. An inactivated state in which the ion channel is physically opened but an external part nonetheless blocks ion flow, a principle referred to as the ball and chain model. For more information on channel inactivation, see R. Aldrich s Fifty years of inactivation [2]. A visualization of the closed, open and inactivated states are given in Figures 1.1(a), 1.1(b) and 1.1(c). Moreover, what triggers a transition from one state to another depends on the gating class. For these, two classes can be distinguished: the ligand-gated ion channels, changing state as a reaction to changes in chemical 3

12 (a) An I Kr channel in closed state. (b) An I Kr channel in open state. (c) An I Kr channel in inactivated state. Figure 1.1: Different states in which an ion channel can reside. Figures from F. Ashcroft et. al. [10], modified to illustrate the ball and chain model. (a) An I Kr channel in the closed state, physically blocking the passage of potassium ions. (b) The channel in the open state, allowing the passage of potassium ions. (c) An inactivated I Kr channel. Although the channel is opened, the ball-like particle blocking the channel prevents the passage of ions. 4

13 signals, and the group of voltage-gated ion channels, to which the I Kr channel belongs, responding to changes in membrane potential [4]. As mentioned earlier, the ion channel considered in this research is the I Kr channel. It is also known as the rapidly activating delayed rectifier potassium ion channel, because of its transports of potassium or K + ions through the cell membrane with a rapid activation rate. Furthermore, it can also be referred to as the herg channel, since the gene encoding the ion channel is the so-called human ether-a-go-go related gene or herg gene. The I Kr channel was not discovered until 1990, when research indicated that the I K channel known at the time consisted not of one, as assumed, but of two components, a rapid I Kr and a slow I Ks component [27] Effects of ion channel deficiencies The combination of the electrophysiological and chemical processes described in Section and the ion channel kinetics described in Section results in a continuous flow of ions in and out of the heart cells, causing various ion channels to generate electric currents. Each of these individual currents contributes to an electric current defining the cardiac action potential (AP), which in turn defines the heart rhythm. A regular cardiac action potential and the corresponding heart rhythm are shown in Figures 1.3(a) and 1.3(c) respectively. For the cardiac action potential five distinct phases can be distinguished, each of which are indicated in Figure 1.2. In the rapid depolarisation phase, or phase 0, the fast Na + channels open, causing the cell membrane to depolarise. Then in phase 1 the fast Na + channels inactivate and the transient outward I T o1 and I T o2 channels open, causing the notch. In phase 2, the plateau phase, a balance is established between the inward Ca 2+ and outward K + ions, until in phase 3 the Ca 2+ channels close while the I Ks channels allowing the passage of the K + ions outwards remain open, causing repolarisation. This in turn causes both the I Kr and I K1 channels to open. Finally in the resting membrane potential phase, phase 4, the I Kr channel closes while the I K1 remain open. The membrane potential stays at rest until it reacts to electrical stimuli from, for example, adjacent heart cells and goes back into the rapid depolarisation phase. In this context, the I Kr current is an important factor in timing the electrical repolarisation of the action potential [31]. Perturbations in the current may Figure 1.2: A cardiac action potential recorded from a single ventricular myocyte. The five different phases are indicated by the numerals 0 to 4. The ion channels that contribute most to each phase are displayed next to the corresponding phases. Image modified from [1]. 5

14 (a) Regular action potential. (b) Prolonged action potential. (c) Regular QT interval. (d) Prolonged QT interval. Figure 1.3: Cardiac action potentials and QT intervals. Figures modified from F. Ashcroft et. al. [10]. (a) A regular cardiac action potential recorded from a single ventricular myocyte. (b) A prolonged action potential from a patient with LQTS. The action potential is represented by the solid line. LQTS increases the risk for early after-depolarisations, indicated by the dashed line and arrow. (c) A regular heartbeat corresponding with the regular action potential from (a). (d) An irregular heartbeat in a patient with LQTS. The QT interval is prolonged, resulting from the irregular action potential shown in Figure (b). lead to a disturbed action potential, resulting in an abnormal heart rhythm. An example of a disorder caused by perturbations in the I Kr current is LQT2, a specific type of long QT syndrome or LQTS. It can be either congenital, resulting from mutations in the herg gene which encodes the I Kr ion channel, or it can be drug induced [11, 30]. The abnormalities in the I Kr channel of a patient with LQT2 lead to decreased repolarising currents, resulting in a prolonged action potential and increasing the tendency for early after-depolarisations. The action potential of a patient suffering from LQTS is shown in Figure 1.3(b), an early after-depolarisation is indicated by the dashed line. The abnormal action potential results in an abnormal heart rhythm, portrayed in Figure 1.3(d). As the name LQTS suggests, the QT interval is longer than in regular heart rhythms. These abnormalities pose a higher risk to, for example, a ventricular arrhythmia known as Torsades de pointes or to ventricular fibrillation, of which the ECG s are depicted in Figures 1.4 and 1.5 respectively. 6

15 Figure 1.4: An ECG recording of the ventricular arrhythmia Torsades de Pointes, showing the characteristic twisting of the peaks around the baseline. Image from M. Keating and M. Sanguinetti [17]. Figure 1.5: An ECG recording showing ventricular fibrillation. Image from M. Keating and M. Sanguinetti [17] Effects of dofetilide The drug analysed in this research is named dofetilide. It belongs to a group of anti-arrhythmic drugs called class III drugs, all of which are known to predominantly block the I Kr channel and thereby prolonging repolarisation. Since dofetilide is found to have pure class III properties, i.e., it blocks only the I Kr channel, it allows the effects of alterations in the I Kr current to be studied without the current being influenced by side effects. Dofetilide affects the I Kr current by blocking the passage of K + ions through the I Kr channels. This blocking occurs when dofetilide molecules physically bind to certain ion channel molecules known as drug binding sites, resulting in the dofetilide molecules physically blocking the passage of ions. The severe side effects discussed in the introduction can be triggered by dofetilide. It exhibits a property known as reverse use dependency, which means that the blocking effect of dofetilide increases at low heart rates. Dofetilide, originally used for its antiarrhythmic behaviour, can then lead to undesired proarrhythmic behaviour which in some cases can even lead to sudden death. For more on dofetilide, refer to [21] Modelling the action potential The effects of different ionic currents on the action potential were first modelled by A. Hodgkin and A. Huxley in 1952 [15]. Although this model replicated the action potential of a squid axon, taking no more than 3 different ion currents into account, it was the foundation to which action potentials are still modelled today. Later studies showed the Hodgkin Huxley models were not able to accurately describe ion channel behaviour [18, 26] and a second generation of action potential models emerged when D. DiFrancesco modelled Purkinje Fibers [9], incorporating different states explicitly into the model. In 1991, C. Luo and Y. Rudy modelled a ventricular myocyte with their LRd model [19], which is still adapted frequently to incorporate new findings [20, 25]. The LRd model formed the basis for the rabbit ventricular model by J. 7

16 Puglisi and D. Bers [22] depicted in Figure The I Kr models studied in this research can be incorporated in Puglisi and Bers s ventricular cell model to study the effects of drug induced blocking on the ventricular action potential. Figure 1.6: A schematic diagram of the rabbit ventricular cell model by J. Puglisi and D.Bers [22]. The I Kr current is marked by the dashed box. 8

17 Chapter 2 Modelling the I Kr channel The I Kr channel plays a pivotal role in the speedy dynamics of potassium ions of the myocyte cell membrane. In this chapter we analyse to some detail how to mathematically model the essential features of the kinetics of the I Kr channel. The I Kr current is determined by the fraction of I Kr channels residing in the open state of the channel. Our aim here is to determine this fraction in time. For a model containing all ion channel states and known transition rates from one state to another, the fraction of open states can be determined at any given point in time, as this is determined by the transition rates. The fraction of open states can then be used to model the I Kr current. In order to achieve this, C. Clancy and Y. Rudy proposed a Markov model [6] for the I Kr channel kinetics in which closed, open and inactivated states are taken into account. This model will form the basis for three models that each incorporate drug-induced blockage in a different manner, based on different theories on the blocking mechanisms of dofetilide. A description of C. Clancy and Y. Rudy s model is given in Section 2.1. Sections 2.2.1, and introduce the three different extensions to this model. A full overview of the equations corresponding to all four models is given in Appendix A. A theoretical analysis of ion channel models in general is given in Section C. Clancy and Y. Rudy s I Kr model The Markov model proposed by C. Clancy and Y. Rudy models the cardiac I Kr current on a cellular level in time. Their widely used model is supported by various studies and is therefore used as a foundation for the three models developed in this research. Based on the research of S. Wang et. al. [33], suggesting an activation model for the I Kr channel containing three closed states to account for both the time and voltage dependencies in the activation curve, C. Clancy and Y. Rudy proposed the cardiac I Kr channel model depicted in Figure 2.1(a). The model contains three closed states indicated as C 1, C 2 and C 3. As suggested by S. Wang et al., the transition rates between C 1 and C 2, denoted as α in and β in, are voltage independent. Furthermore, the model contains a single inactivated state I and one open state O. The transition rates α i and β i between these inactivated 9

18 (a) C. Clancy & Y. Rudy s model... (b)...extended with a single blocked state. (c)...extended with two blocked states. Figure 2.1: Markov Models for the cellular cardiac I Kr channel gating kinetics. (a) The model proposed by C. Clancy and Y. Rudy [6]. (b) The Simple Model and the Cooperative Binding Model: An adaptation to the model, incorporating a single drug binding state B. (c) The 2-Phase Model: A different adaptation, incorporating two drug binding states B r and B s, representing a rapid and a slow blockage component. and open states depend on the extracellular potassium concentrations denoted as [K + ]o. A transition from C 1 to O is, according to S. Wang et. al., equally likely as a transition from C 1 to I, which are therefore both indicated by αα. To satisfy one of the properties of a Markov model called microscopic reversibility, it must hold that the sum of transition rates in one direction equals the sum of transition rates in the opposite direction, implying that one of the transition rates, µ, is merely a function of all others. The experimentally found transition rates corresponding to the model can be found in Appendix A, equations A.1 to A Block incorporation in the I Kr model In order to explicitly incorporate block of the physical current into this I Kr channel model, the approach of C. Clancy et al. [7] was followed. In their theoretical analysis on incorporating drug binding states in a Markov model for sodium channels, the models were extended by adding one or more blocked states to the model, each one being accessible by either a single open, closed or inactivated state. Assuming dofetilide is an open channel blocker [29], combined with C. Clancy and Y. Rudy s model containing only one open state, yields a series of models in which the open state is connected with a blocked state. Biological constraints and observations have contributed to the construction of the three models analysed in this research. The models and theories on which they are based are discussed in the three upcoming subsections. The model equations for all three models are based on the equations found by C. Clancy and Y. Rudy. Since the parameter values differ per biological species and experimental settings, their parameter values were substituted by the variables γ and δ and have to be estimated by fitting the models to known 10

19 data. The equations for the modified models are given in Section A.2. Note that in α i and β i the rate dependency of the extracellular potassium concentration [K + ]o is no longer explicitly contained. Since the potassium concentration used in the experimental recordings remains constant throughout the experiment, it is implicitly defined by γ αi and γ βi Model I: Simple The first model, which we call the Simple Model, was developed with a famous quote of Albert Einstein in mind: Everything should be made as simple as possible, but no simpler. To incorporate block as simple as possible a single state B was added and connected to the open state O. The resulting Markov model, hereafter referred to as model I, is shown in Figure 2.1(b). By adding the new component two new transitions are introduced and therefore two extra equations are added to the equation set. Research suggesting that neither the blockage of dofetilide, nor the recovery from this blockage, is voltage-dependent [32], leads to the following equations for the transition rates α b and β b : α b = γ αb [Dof] β b = γ βb Model II: Cooperative Binding The second model, model II or the Cooperative Binding Model, is based on the principle of cooperative binding, a biochemical principle comparable to autocatalysis in which the probability for a drug molecule to bind to the ion channel depends on the number of drug molecules that have already bound. For more on this subject, we refer to [24]. If dofetilide exhibits cooperative binding, it can be explained by a simplified formulation resembling the findings of J. Gerhardt and A. Pardee in their study of the enzyme aspartate transcarbamylase [12], using Figures 2.2(a) and 2.2(b). (a) An I Kr channel (b) An I Kr channel blocked by a drug molecule Figure 2.2: An I Kr channel. Figures modified from F. Ashcroft et. al.[10]. (a) A channel in the open state. (b) The channel blocked by chemical compounds in drugs, changing the physical shape of the ion channel. 11

20 The probability for a drug molecule to bind to an ion channel depends on the physical shape of the channel. J. Gerhardt and A. Pardee found that the binding of ligands, in this case dofetilide molecules, can change the physical shape of the binding site, in this case the ion channel. Figure 2.2(a) shows an I Kr channel in the open state. The possibility of a drug molecule blocking the ion channel is a function of the size of the opening of the ion channel. Figure 2.2(b) shows the I Kr channel where a drug molecule bound and changed the shape of the ion channel, thereby increasing the possibility for other drug molecules to bind to the ion channel. It is well-known [14] that this physical condition can be accurately described by the Hill-equation. This equation indicates that the effects of cooperative binding are of an exponential nature. The structure of this Cooperative Binding Model is equal to that of the Simple Model, depicted in Figure 2.1(b), but the equations for the transition rates differ. The effect of cooperative binding is implemented by defining the transition rate from the open to the blocked state as an exponential function with base [Dof], where δ αb indicates the degree of cooperation: α b = γ αb [Dof] δα b The transition rate β b from the blocked state B to the open state O is defined in the same way as for the Simple Model: β b = γ βb Model III: Two Phases The third model, hereafter referred to as the 2-Phase Model, is based on the observations by E. Carmeliet [5] that dofetilide block occurs in two separate phases, the first one being a fast phase and the second a slow phase. This might indicate the existence of two separate block states, which is incorporated in model III by adding a rapid block state B r and a slow block state B s. A diagram is shown in Figure 2.1(c). The equations are once again kept as simple as possible. Hence, α Br and α Bs have similar structures as transition rate α B in model I while β Br and β Bs have similar structures as to model I s β B : α Br = γ αbr [Dof] β Br = γ βbr α Bs = γ αbs [Dof] β Bs = γ βbs 2.3 Theoretical analysis of Markov models for ion channels The set of transition rates defines the fraction of channels in each state for an arbitrary ion channel. The approach commonly used for determining these fractions at time t is by numerically integrating this system of ordinary differential 12

21 equations to determine the change in channel state occupancy. This can be a computationally challenging approach and a method to analytically determine the activation curve is therefore preferred. This section presents a method for algebraically determining the steady state given a set of transition rates and formulates a function X(t) representing the activation curve under voltage clamp conditions Steady State In a Markov model for an ion channel, the fraction of ion channels per state at time t depends on the initial state occupancy. In experimental settings this initial state is referred to as the steady state, in which the ion channel state occupancy no longer changes over time. The fraction of ion channels per state in this steady state can be calculated using the approach proposed in this section. For an arbitrary Markov Model for an ion channel, let X = (X 1, X 2,..., X n ) T be a vector representing the fraction of channels in every state X i. Because of this definition it holds that X i [0, 1] i 1...n (2.1) n X i = 1 (2.2) i=1 Let R(θ) be the n n matrix with transition rates. That is: R ij (θ) is a function that maps the experimental settings θ(t) to the transfer rate from state i to state j. Channels residing in the same state do not affect the state occupancy. Hence, transition rates from one state to itself are not taken into account. Therefore, R ii (θ) = 0 i 1...n, θ(t). The change in channel occupancy is determined solely by the current state occupancy and the incoming and outgoing rates defined by R(θ), a characteristic known as the first order Markov Property: dx dt = RT (θ)x diag(r(θ) 1)X (2.3) where 1 denotes an n 1 vector of ones. This is equivalent to dx dt = T (θ)x (2.4) with T (θ) = R T (θ) diag(r(θ) 1) (2.5) If this model is analysed under suitable experimental voltage clamp conditions then, in general, θ is constant for every step in the voltage protocol; for which it holds that T (θ) = T is a constant, real n n matrix. Furthermore, because of the conservation of channels it holds that 1 T T (θ) = 0 θ (2.6) so, clearly, T cannot have n independent rows and therefore does not have full rank. However, combined with constraint 2.2, which can be formulated as 1 T X = 1, this can be used to find the equilibrium state occupancy X of this Markov model under a fixed θ = θ, provided T ( θ) has rank n-1, by solving the linear system: 13

22 T ( θ) X =. 0 1 (2.7) Activation Curve For computing the activation curve we continue with equation 2.4 given in the previous section. In general, the solutions to an ordinary differential equation system of this form are given by X(t) = e T ( θ)t X 0 (2.8) where e denotes the matrix exponent. Using the eigenpairs of matrix T ( θ), defined by the transition matrix R( θ) as described in the previous section, this system of equations can be rewritten using the following procedure. Determine all real eigenvalues λ 1,..., λ p and corresponding real eigenvectors u 1,..., u p of T ( θ). Use these eigenpairs to determine α 1,..., α p by solving [u 1,, u p ] α 1. α p = X 0 (2.9) where X 0 is the state occupancy at the beginning of the step in the voltage protocol corresponding to T ( θ). Using a Mathworld procedure [35] for solving homogeneous systems of differential equations by spectral decomposition, the solution to equation 2.4, defining the activation curve X(t), can now be written as X(t) = p α i u i e λit (2.10) i=1 which no longer involves the matrix exponent Tail Current The formulations for the steady state occupancy and for the activation curve enable us to determine the electrical current for fixed transition rates. However, in experimental settings the cell s membrane potential is regulated and deliberately changed, resulting in a change in transition rates and an abrupt change in the resulting current. This changed current, known as the tail current, is used to study the relationship between membrane potential and electrical current. External factors, such as congenital conditions or drug induced channel alterations, can result in alterations in this relationship. The combination of the equations explained in Sections and yields the following general procedure for finding the tail current for any three-step voltage clamp protocol as shown in Figure

23 Figure 2.3: An arbitrary voltage clamp protocol. First a voltage of v 1 mv is applied for a long period of time, in order for the ion channels to reach the equilibrium state occupancy. Then a voltage pulse of v 2 mv is applied for t milliseconds and finally the channel is clamped to v 3 mv. Note that the applied voltage v 1 does not need to be equal to voltage v 3. First, determine the steady state channel occupancy at the end of the first voltage step, using the steady state formulation. 1. Using the definition of R(θ), determine the transition matrix R(θ 1 ) for voltage v Determine the corresponding T (θ 1 ), using equation Solve equation 2.7 for T (θ 1 ) to find the fraction of channels per state at the end of the first voltage step. Second, determine the fraction of ion channels per state at the end of the second voltage step, using the activation curve. 4. Determine the transition matrix R(θ 2 ) for voltage v Determine the corresponding T (θ 2 ). 6. For T (θ 2 ), find the real eigenvalues λ 1,..., λ p and corresponding real eigenvectors u 1,..., u p. 7. Solve equation 2.9 to find α, using the eigenpairs found in step 6. and using the state occupancy determined in step 3. as the initial occupancy. 8. Using the formulation in equation 2.10, determine the state occupancy at time t, indicating the end of the second step of the voltage clamp. Last, the tail current can be found using an approach similar to that in steps 4. to Determine R(θ 3 ) and T (θ 3 ) for voltage v Find the real eigenpairs for T (θ 3 ). 11. To determine α, solve equation 2.9 using the real eigenpairs found in step 10. and the fraction of ion channels per state found in step The fraction of ion channels per state in the tail current is now given by equation 2.10, where the values for λ, u and α are those found in steps 10 and Using the fraction of channels in the open state, the tail current is given by equation

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25 Chapter 3 A method for model analysis Up until this point, the models proposed in Chapter 2 have not yet been compared with experimentally recorded data. The methods used to test the accuracy of the models, that is, how well they are able to recreate known behaviour, are described in this chapter. 3.1 Experimental data To test the level of accuracy of the models, the outputs generated by the three models were compared to experimental data found in an article by E. Carmeliet [5], describing the effects of I Kr blockage caused by dofetilide on the action potential in a rabbit ventricular myocyte. Experiments were performed by adding different doses of dofetilide to a solution in which a single ventricular myocyte was placed, ranging from the absence of drugs to a dose of 4.5 x 10 9 M. A twoelectrode voltage clamp was placed on the myocyte, allowing the membrane potential to be regulated. The voltage clamp protocol that was used is shown in Figure 3.1. The membrane potential was held at a set level of -50 mv, until the current amplitude reached the steady state. Then a voltage pulse is applied and the membrane Figure 3.1: The voltage clamp protocol used in the experiment. First a voltage of -50 mv is applied for a long period of time, in order for the ion channels to reach the equilibrium state occupancy. Then a voltage pulse is applied for 4s and finally the channel is clamped back to -50 mv. This protocol was used for a series of experiments, each having a different voltage pulse. The pulses ranged from -30 to +10 mv with intervals of 5 mv. 17

26 (a) Activation curve (b) Concentration response curve Figure 3.2: The blue and red data points in (a) and the data points in the concentration response curve in (b) were combined to produce the graph depicted in (a). The combined data points are used to estimate the model parameters. The blue and red dots in Figure (a) indicate the relative amplitudes of tail currents. A trendline resulting from a sigmoid fit is portrayed by the solid line. The amplitudes are scaled by taking the activation without the addition of dofetilide at +10 mv as 100%. Figure (b) shows the concentration response curve. The relative block was measured after applying a voltage pulse of 0 mv. The activation without the addition of dofetilide was scaled to 100%. Again, the measured data points are indicated by the dots whilst the solid line represents a sigmoid trendline. potential was held at pulse level for 4000 ms, after which the membrane potential was clamped back to -50 mv. This series was repeated several times, varying the voltage pulses between -30 to +10 mv with intervals of 5 mv. The maximum channel activation in the third phase of the voltage clamp protocol, known as the peak tail current, was measured and used to generate both the activation curve and the concentration response curve shown in Figures 3.2(a) and 3.2(b). The combination of these curves provides a data set indicating the activation of the I Kr current at various voltages for different doses of dofetilide. This combined data set, shown in Figure 3.2, serves as a target output for the three models. 3.2 Optimization technique The models proposed in this research are developed to recreate known behaviour. Since the model output depends on its transition rates, an optimization technique is used to find the best transition rate parameters resulting in the optimal resemblance between the known experimental data and the model output. A genetic algorithm, or GA, was used to find the optimal fit. An advantage of genetic algorithms over other heuristic techniques is that the genetic algorithm uses a population of possible solutions rather than a single solution. An initial guess of parameters is therefore not necessary. Furthermore, genetic algorithms are guaranteed to find the optimal solution, if it exists within the search space. Where traditional techniques can converge to a local optimum, a GA can escape local optima by pure chance and will therefore eventually find 18

27 the global optimum. Genetic algorithms are based on the concept of biological evolution. Individuals that perform well within a population have a better chance of reproducing than others, and therefore good properties are propagated through the population for generation after generation. This implies that the population gets fitter with each generation. For genetic algorithms, individuals are determined by a set of parameters, analogue to a DNA sequence determining individuals in biological evolution. Their fitness is measured as a function of how well the parameters perform in a specified environment. For datafitting this environment is a model in which the parameters are used as input and in which the output determines the fitness. The implementation details for the genetic algorithm used in this research are based on the approach followed by M. Gurkiewicz and A. Korngreen [13]. For each of the 16, 17 or 18 parameters, for the Simple Model, Cooperative Binding Model and the 2-Phase Model respectively, a lower- and upperbound are determined to limit the search space. The algorithm consists of three distinct steps. 1. Creating an initial population The first step in the algorithm is to form an initial population by randomly generating multiple individuals. Each individual has a parameter set that is randomly drawn from the search space. 2. Evaluating the individual performances The second step is to determine the fitness for each individual by calculating the difference between the output generated by the model and the desired output. A quadratic criterion is used, implying that outliers contribute more to the error than regular data points and are therefore less likely to evolve. The so-called sum of squares function can be formulated as: error = n i=1 j=1 m (O i,j T i,j ) 2 where O is an n m output matrix generated by the model and T is the n m target matrix, containing the data found in literature. The error obviously denotes the inverse of the fitness, implying that a lower error results in a higher fitness level. Once the fitness is determined for each individual, the algorithm continues to the next step. 3. Populating the next generation The third step in the algorithm is to populate the next generation via reproduction. The new population can be divided into three groups of individuals: the copied individuals, the new individuals and the offspring. The first group populates a small part of the next generation. The fittest individuals of the previous generation are copied directly to the next generation, in order to preserve their good genes and prevent a genetic drift. 19

28 The second group is a group of newly created individuals. To explore the parameter space, the group consists of individuals having a parameter set randomly drawn from the search space. The third group is filled with offspring from the previous population. Based on biological evolution, the fitter individuals have a higher probability of reproducing than individuals that are less fit. The offspring is generated according to the following three steps: 1. The first is to determine which individuals are allowed to generate offspring, by a procedure called tournament selection [3]. Two individuals are randomly selected from the population and their fitness is compared. The individual with the highest fitness, or lowest error, is selected for generating offspring. The tournament selection is repeated until a predefined number of individuals is selected. 2. The second step in reproducing is creating offspring. Two parents are randomly selected from the pool of tournament winners. Each parameter in the offspring s parameter set has the usual probability of 50% to be copied from the first parent and obviously the same probability of being copied from the second parent. 3. The last step in generating offspring is the mutaton of parameter values. Two types of mutation can be distinguished, one in which a new parameter value is randomly drawn from the search space, hereafter referred to as type-1, while the other mutation type, type-2, draws a parameter value from a normal distribution centred around the current parameter value. Each parameter value of each child has a small probability of having either type mutation. Once the next generation is populated, the process of evaluating individuals and creating a new population is repeated until a termination criterion is reached, and hence the population has converged. 20

29 Chapter 4 Experiments In this research, three types of experiments were conducted. In the first series, discussed in Section 4.1, various settings for the genetic algorithm are compared in order to find optimal settings for the algorithm. These optimal settings were used to fit the models to the data in the second series of experiments, discussed in Section 4.2. In Section 4.3 a sensitivity analysis was performed to analyse the robustness of the three models. 4.1 Optimization settings The solution to the optimization problem can in principle be found by the genetic algorithm. However, how likely it is that the optimal solution is found within a finite time frame depends on the settings of the algorithm. A large population size and a high mutation probability cause the algorithm to explore the search space rather than focusing on promising areas, converging slowly but steady to the optimal parameter set. A small population size and low mutation probabilities on the other hand exploit the fittest individuals and converge relatively quickly to an optimum, but the probabilities of it being a local instead of a global optimum increase as the exploitation increases. The aim of the experiments described in this section is to find a balance between exploration and exploitation. The optimal settings for the algorithm are determined by examining the effects of four settings: Population size: The total number of individuals in each generation. Best percentage: The percentage of best individuals to transfer unchanged from one generation to the next. Mutation probability: The probability for a mutation to occur. Mutation distribution: The probability for a type-1 mutation versus a type-2 mutation to occur, given that a mutation occurs. For each model four series of experiments were run, each varying only one of four settings while using default values for the other three settings. A full overview of the settings used in each experiment is given in Table B.1. 21

30 An initial population was determined for each of the three models, which was used in all four series of experiments for the model. Every experiment lasted 500 generations. The results of the experiments are analysed to find settings that perform well in all three models. These settings will be used in the experiments in Section 4.2, in which the models parameters are estimated. A single optimal value is defined for each setting rather than defining optimal values per model, to eliminate the exploration versus exploitation differences caused by the different settings. For example, if one model uses settings that result in exploration behaviour, while the other model uses settings resulting in exploitation, a comparison between the performances might not yield an accurate result. The models are compared using the parameters found after a fixed number of generations. The exploitation model then might lead to a better fit than the exploration model, while in the long run the exploration model would have produced the best results. If the models use equal settings, these effects are non-existent because the global pattern of error convergence is equal and therefore one model cannot outperform another due to its algorithm settings only Population Size M. Gurkiewicz and A. Korngreen suggested to use a population size of 20 times the number of parameters [13]. The analysed population sizes, all within this order of magnitude, are 50, 100, 200 and 500 individuals. The time needed for a population size of 200 individuals to produce 500 generations is taken as a termination criterion for the other experiments. The resulting graphs, shown in Figures B.1, B.2 and B.3, globally show the expected behaviour: the larger the population size, the slower in time the error is reduced in the first 50 generations. After 500 generations the larger populations are catching up with the smaller populations. The population size of 200 individuals managed to outperform the other sizes in the Cooperative Binding Model and 2-Phase Model. In the Simple Model the population size of 50 is best after 500 generations, while the 200 individuals perform second best, perhaps due to the smaller number of parameters in the Simple Model. Because the population size of 200 leads to the best overall results, this setting is used for datafitting Best Percentage The percentages of best individuals that are copied directly from one generation to the next usually lies around 1% or 2%. For larger percentages, evolution is more withstanded because the population changes less, while for smaller percentages, the population can drift away from good individuals. We analysed the progression of the error using percentages of 0,5%, 1%, 2% and 5%, the graphs are shown in Figures B.4, B.5 and B.6. In the first 50 generations the error progression shows only a random pattern. In generations 50 to 500 it becomes clear that a percentage of 1% outperforms the other percentages in the Simple and the Cooperative Binding Model. In the 2-Phase Model the curve representing 2% performs best, but there is only a small difference in the performance of 1% and 2%. Therefore, based on the 22