CELL MODELLING
 

     INTRODUCTORY NOTES

     Cell models for different cardiomyocyte processes, such as electrical activation, calcium dynamics, mechanical contraction, signal transduction and metabolism exist. These models can be combined to create a composite model that will represent the function of the cardiomyocyte as a whole. This is illustrated in the article byHunter, P. J., E. J. Crampin, et al. (2008) Brief Bioinform 9(4): 333-343 in figure 3.

     A  reference to an electrophysiology cell model will follow after a brief introduction to electrophysiology below.



     BRIEF INTRODUCTION TO ELECTROPHYSIOLOGY

     Resting membrane potential

     The membrane potential is generated by the difference in charge at the two sides of the cell membrane (termed sarcolemma for the cardiomyocyte). Different ion species with their charges are responsible for this, and the degree of movement of ions via channels, pumps and transporters will determine the contribution of each ion to the membrane potential. Potassium and sodium are considered the most important contributors of the membrane potential at rest. In particular, most potassium channels are in the open state at rest and therefore the sarcolemma is relatively highly permeable to potassium ions, in contrast to sodium ions.

     Here is an example from Opie, L. H. (1998),The Heart, Physiology, from Cell to Circulation (Lippincott-Raven Publishers) which illustrates this concept :
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     "Let 400.000 Na+ ions be pumped outward and 400.000 K+ ions be pumped inward (per millisecond per unit area) (Woodbury 1963). Suppose that from the 400.000 K+ pumped inward, 200 K+ will diffuse outward. Because of the lower conductance of the sarcolemma to sodium than that of potassium, suppose that from the 400.000 Na+ pumped outward, only 4 Na+ will diffuse inward. This creates a charge difference of 196 negative ions on the inside of the sarcolemma. It is this difference that causes the electric potential.

     The actual contribution of each ion to the membrane potential can be calculated from the Nernst equation. For instance, for the potassium equilibrium potential, or EK:

     EK =-61,5 log Ki/Ko

     where Ko is the external potassium ion concentration and Ki is the internal potassium concentration.

     If the external potassium concentration were 4mmoL/L and the internal value 80mmoL/L, the calculated equilibrium potential for potassium would be:

     EK = -61,5 log Ki/Ko= -61,5 log (80/4)= -61,5 log20 = -61,5 * 1,3 = -80mV

     Similarly, the equilibrium potential for sodium is calculated to be ENa= +60mV "

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     The above relations mean that the movement of potassium will tend to bring the membrane potential to the value of -80mV, while that of sodium to the value of +60mV. In order to find the "average behaviour" formed from these two contributions, we have to "weigh" them "in a mathematical manner"  taking into account the size of the ion movement (using for example a virtual old-fashioned scale). If the degree of those two movements was equal, then the membrane potential would be at the half of the 140 mV difference between -80mV and +60mV, that is at -10mV. However, the permeability of the membrane to sodium is 5% or less than that to potassium, therefore the membrane potential will be at the 5% of that 140 mV difference and to the side of potassium. As 5% * 140 mV=7mV, the membrane potential will be -80mV minus 7mV, that is -73mV.

     A formal expression of the rational above, introducing the notion of "weighted average" is mentioned below from wikipedia.

     "In a more formal notation, the membrane potential is the weighted average of each contributing ion's equilibrium potential (Goldman equation). The size of each weight is the relative permeability of each ion. In the normal case, where three ions contribute to the membrane potential:
where

• Em is the membrane potential, measured in volts
• EX is the equilibrium potential for ion X, also in volts
• PX is the relative permeability of ion X in arbitrary units (e.g. siemens for electrical conductance)
• Ptot is the total permeability of all permeant ions, in this case PK+ + PNa+ + PCl"

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     Action potential

     An online tutorial describing the action potential of the cardiomyocyte is mentioned at this link.

Figure 1Screen capture of the tutorial available at this link.

 

     The ionic currents that are responsible for the changes in the membrane voltage are mentioned in the slide below from Dr. Peter Hunter's presentation available at this link.

Figure 2:  Currents that participate in depolarization and repolarization (Dr. Peter's Hunter slide available at the following link).


     In summary, the series of events that are responsible for the action potential (depolarization) and the subsequent repolarization is as follows (relevant link) (Opie, L. H. 1998):

1. Sodium influx through the sodium channel: Sodium channels open very quickly and by allowing increased sodium (positive) ion flow inside the cell. They close immediately. The negative potential of the cell becomes very quickly less negative, then 0, then positive (depolarization) reaching a value of +47mV.

2. Calcium influx through the calcium channel: At about -20mV, calcium channels of L-type open very quickly and start closing slowly. Calcium (positive ions) flow inside the cell.

3. Potassium and chloride efflux carried by the Ito1 and Ito2 currents, respectively.

4. Potassium efflux through the slow delayed rectifier potassium channels, IKs.

5. Potassium (net) efflux through the rapid delayed rectifier K+ channels (IKr) and the inwardly rectifying K+ current, IK1

This net outward, positive current (equal to loss of positive charge from the cell) causes the cell to repolarize.

 

 


     ELECTROPHYSIOLOGY MODELLING

     Electrophysiology models

     The best known electrophysiology models are:

1.  Hodgin-Huxley model: 3 currents (INa, I­K, IL) and 3 gating variables (m,h,n).
2.  DiFranscesco-Noble model of the Purkinje fiber cell: 12 currents (If, IK, IK1, It0, IbNa, IbCa, Ip, INaCa, INa, ICaf, ICas, Ipulse ) and 7 gating variables (y, x, r, m, h, d, f).
3.  Beeler-Reuter ventricular cell model: 3 currents (INa, IS, Ix1, IK1) and 6 gating variables (m, h, j, d, f, x1).
4.  Luo-Rudy mammalian ventricular cell model. The most recent version of the Luo-Rudy II model contains 14 currents (INa, ICa, ICaNa , ICaK , IK , INaCa , IK1 , IKp , Ip(Ca), INab ,ICab , INaK , InsNa , InsK ), 4 fluxes (Irel , Iup , Ileak , Itr ), and 11 gating variables (m, h, j, d, f, fCa, X, X i , K1 , Kp , f NaK ).
5.  Noble model of the guinea pig ventricular cell.

     A comprehensive review on the Noble electrophysiology models is the following: Nickerson, D. P. and P. J. Hunter (2006). "The Noble cardiac ventricular electrophysiology models in CellML." Prog Biophys Mol Biol 90(1-3): 346-359. 

 



     Presentation of an electrophysiological model - Prediction of therapeutic approaches via electrophysiological modelling

     Let us consider an electrophysiological model via an example of how electrophysiological modelling can provide therapeutic approaches with pharamaceutical compounds. 

     Some diseases are associated with specific well-defined profiles of differential expression of ion channels and transporters. For example, in congestive heart failure as mentioned in the study by Noble, D. and T.J. Colatsky (2000) Expert Opin Ther Targets. 2000 4(1):39-49 four major transporter proteins have altered levels of expression: the transient outward potassium current, Ito1, the inward rectifier potassium current, IK1 and the calcium pump in the sarcoplasmic reticulum (SERCA) or Iup are all downregulated, while the sodium-calcium exchanger is upregulated. 

     If we introduce these changes in electrophysiology cell models, we can reproduce the action potential prolongation and the early after depolarizations observed in cells from CHF patients. Also, if these pathologic cell models are incorporated in whole ventricle models, an arrhythmogenic behaviour is obtained (termed "Torsades de Pointes").

     Specific drugs exist that can affect different aspects of the behavior of different channels. Is it possible to predict a combination of drugs that can correct the disease profile? Also can novel drugs be tested?

     As mentioned in the article by Garny, A., D. Noble, et al. (2009) Philos Transact A Math Phys Eng Sci 367(1895): 1885-1905, in section 4d "Case study", it is estimated that approximately 40 per cent of all drugs have side effects on the rapid delayed rectifier K current (IKr ; HERG channel), and therefore testing of this is a fundamental requirement for drug safety evaluation. IKr is one of the main contributors to repolarization, the process by which the voltage of the membrane of the cell is returned to control levels.

     Let us consider the example of a specific compound, BRL32872, which had been for a long time a leader compound in the treatment of arrhythmias. For reference, it has been demonstrated that a successful strategy in order to treat a specific type of arrhythmias is the prolongation of the action potential. As the main factor that repolarizes the membrane and ends the action potential is the potassium rectifier current, it is reasonable to adopt approaches that partially block this current.

      It has been determined that this compound has a dual action:
a. It mediates a 70% block of the IKr current: This effect alone would tend to increase the time for repolarization i.e. prolongate the action potential; but simultaneously it would result in incomplete repolarization and would lead to spontaneous reactivation of the IcaL. As a result the membrane voltage would oscillate around the 0 mV value.
b. It mediates a 20% block of the IcaL current. This allieviates the above mentioned effect of IcaL which constitutes the principal mediator of spontaneous membrane voltage oscillations.

     The sum of the two effects is the net prolongation of the action potential. 

     We can actually "test" this drug ourselves, understand its mode of action and grasp the strategy that can be implemented to treat a condition by using different compounds, each having different properties. We will folow the example of the study by Garny, A., D. Noble, et al. (2009) Philos Transact A Math Phys Eng Sci367(1895): 1885-1905 for the BRL32872 compound, which is mentioned in section 4d and we will reproduce figure 8a following the instructions below (kindly provided by Dr. Alan Garny).

     For this purpose, we will obtain an electrophysiology model from the repository and we will initially simulate it in normal conditions. Then, in order to treat the arrhythmias, we will try to apply interventions that will increase the duration of the action potential, similarly to the mode of action of the therapeutic compound. We will first block the potassium rectifier current by 70% and we will examine the result. Then, we will block the calcium L-type channel by 20% and we will evaluate the final result.

     We will work with the Noble 2K electrophysiology model in the CellML standard. The model is represented in the image below (from the CellML site).

Figure 3Schematic representation of the Noble 2K guinea pig ventricular model (from the CellML site)        

 

     We will simulate the model with the instructions that follow that were kindly provided by Dr. Alan Garny (Note: The instructions were slightly modified by the author of the site and comments were added as well).
 

  • The link to the Noble 2000 (2K) guinea pig ventricular model at the CellML repository is here. From this link, from the tab "Downloads" download "Complete archive as .tgz". Save "noble_2000.tar" and extract files (e.g. with 7-zip).
  • In order to simulate the model, download a simulator e.g. COR or OpenCell (the tools of the CellML standard are referenced athttp://www.cellml.org/tools/modeling-environments). For this example, download COR from http://cor.physiol.ox.ac.uk/

 

  • After installation, start the software and open file "noble_2000_a" (from tab "File").
  • At the tab "Run" press "Compile and Run" (or F9) in order to switch from editorial to computational mode.
  • We would like to use a specific protocol where the duration of the stimulus current is 2ms. For this purpose, at the left hand side panel, under "Intergator" it is recommended to set  "Maximum time step" to 2 ms.
  • To obtain the simulation result for the voltage of the membrane, under 'membrane' header select parameter V.
  • Create a new graph panel (by clicking on the G+ icon or the Tools|Add Graph Panel menu item);
  • Under 'L_Type_Ca_channel' header select the i_Ca_L parameter;
  • Create a third graph panel;
  • Under the ‘rapid_delayed_rectifier_potassium_current’ header select the i_Kr parameter ;
  • Simulate the model (by pressing the F9 key or selecting the Run|Run menu item). This will give the first set of traces for publication figure 8a, as shown in figure 4 below.

Figure 4Simulation result for normal conditions. The upper pannel represents parameter V (membrane voltage), the middle panel parameter  i_Ca_L (L-type calcium current) and the bottom panel parameter  i_Kr (rapid delayer rectifier potassium current).

 

  • Reset the variable values (by clicking on the recycle icon or selecting the Tools|Reset Variables|All menu item);
  • Block IKr by 70%, i.e. set g_Kr1 and g_Kr2 (under the ‘rapid_delayed_rectifier_potassium_current’ header) to 0.00084 mS and 0.00051 mS, respectively (as shown in figure 5);

Figure 5Blocking the rapid delayed rectifier potassium current by 80% by entering the above values (80% of the initial values).

 

 

  • Simulate the model (by pressing the F9 key or selecting the Run|Run menu item). This will give the second set of traces for publication figure 8a, as shown in figure 6 below.

Figure 6Simulation result for 80% blockade of the  i_Kr (rapid delayer rectifier potassium current). The upper pannel represents parameter V (membrane voltage), the middle panel parameter i_Ca_L (L-type calcium current) and the bottom panel parameter i_Kr (rapid delayer rectifier potassium current).

 

  • Comment: We observe that the membrane voltage now oscillates. We would expect that the blockage of the current that is mainly responsible for repolarizing the membrane would tend to increase the time for repolarization i.e. prolongate the action potential; this trend is evident in the first oscillation. However, this resulted in incomplete repolarization and led to spontaneous reactivation of the IcaL. As a result the membrane voltage oscillates around the 0 mV value.
  • We will now proceed with double blockade of the IKr and ICaL. Reset the variable values;
  • Reblock IKr by 70%;
  • Block ICaL by 20%, i.e. set P_Ca_L (under the ‘L_type_Ca_channel’ header) to 0.8 nA/mM as shown in figure 7 below;

Figure 7Blocking the L-type calcium current by 20% by entering the above value (20% of the initial value which was 1).

 

  • Simulate the model (by pressing the F9 key or selecting the Run|Run menu item). This will give the third set of traces for publication figure 8a, and thus complete figure 8a, as shown in figure 8 below.

Figure 8Simulation result for 80% blockade of the i_Kr (rapid delayer rectifier potassium current) and 20% block of the IcaL current. The upper pannel represents parameter V (membrane voltage), the middle panel parameter i_Ca_L (L-type calcium current) and the bottom panel parameter i_Kr (rapid delayer rectifier potassium current).

 

  • Comment: We observe that by blocking the IcaL current by 20%, in addition to the IKr current, we allieviate the above mentioned effect of IcaL which constitutes the principal mediator of spontaneous membrane voltage oscillations. As a result, the rhythm of the cardiomyocyte is corrected.

 

 

 

 

     Basic Research and Systems Biology meet Applied Science and the Pharmaceutical Industry

     

     Part of an article from "The Economist" available at this link is pasted below.

     "In contrast to the two years it takes to assess the effects of a new compound using conventional research methods, Dr.Young's approach takes an average of just two weeks. "

 

"The pharmaceutical industry stands to gain much from this approach. Around 40% of the compounds that drug companies test cause arrhythmia, a disturbance to the normal heart rate. Drugs such as the anti-inflammatory medicine Vioxx and the diabetes treatment Avandia have been linked with an increased risk of heart disease. The result is that billions have been wiped off their makers' share prices. 

Not surprisingly, the pharmaceutical industry has sought out Denis Noble of Oxford University, the creator of the beating-heart model, to help. Dr Noble is now part of a consortium involving four drug firms—Roche, Novartis, GlaxoSmithKline and AstraZeneca—that is trying to unravel how new drugs may affect the heart. Virtual drugs are introduced into the model and researchers monitor the changes they cause just as if the medicines were being applied to a real heart. The production of some proteins increases while others are throttled back; these changes affect the flow of blood and electrical activity. The drugs can then be tweaked in order to boost the beneficial effects and reduce the harmful ones. 

Systems biology thus speeds up the drug-testing process. Malcolm Young is the head of a firm called e-Therapeutics, which is based in Newcastle upon Tyne. Using databases of tens of thousands of interactions between the components of a cell, his company claims to have developed the world's fastest drug-profiling system. In contrast to the two years it takes to assess the effects of a new compound using conventional research methods, Dr Young's approach takes an average of just two weeks. Moreover, the company has been looking at drugs known to have damaging side effects and has found that its method would have predicted them".

 

References

Hunter, P. J., E. J. Crampin, et al. (2008). "Bioinformatics, multiscale modeling and the IUPS Physiome Project." Brief Bioinform 9(4): 333-343

Nickerson, D. P. and P. J. Hunter (2006). "The Noble cardiac ventricular electrophysiology models in CellML." Prog Biophys Mol Biol 90(1-3): 346-359.

Noble, D. and T.J. Colatsky (2000) “A return to rational drug discovery: computer-based models of cells, organs and systems in drug target identification” Expert Opin Ther Targets. 2000 4(1):39-49

Garny, A., D. Noble, et al. (2009). "CELLULAR OPEN RESOURCE (COR): current status and future directions." Philos Transact A Math Phys Eng Sci 367(1895): 1885-1905.

Opie, L. H. (1998),The Heart, Physiology, from Cell to Circulation (Lippincott-Raven Publishers)

article from "The Economist"