Modelling Synchronised Calcium Oscillations in the

Liver

Complex MRes Case Essay

Tom Sumner

12th April 2006

Supervisors - James Hetherington and Marta Ela-ReyApprox Word Count - 4100

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1 Introduction

1.1 Function and Structure of the Liver - Role in Glucose Home-ostasis

The liver has a number of functions including the production of bile requiredfor digestion, synthesis of cholesterol and removal of certain blood toxins. Italso has a major role in glucose homeostasis, the regulation of blood glucoselevels.

Glucose is a major source of energy for the body and is converted into ATPvia the process of glycolysis. When the blood glucose level is high most cellsstore excess glucose as glycogen for subsequent use in cellular energy production.In the liver glycogen is primarily stored as a reserve for other cells and tissues.When the blood glucose level drops glucose is liberated from the glycogen storesin the liver (glycogenolysis) and in addition new glucose is synthesized fromlactate and amino acids (gluconeogenesis).

Changes in the blood glucose level can be detected directly by the liver or bythe release of the hormones insulin and glucagon from the pancreas. Insulin isreleased in response to high blood glucose and promotes glycogen synthesis andglycolysis. Glucagon increases glycogenolysis and gluconeogenesis in responseto falling glucose levels. Together, the levels of glucose, insulin and glucagonhelp regulate the processes of glycogen metabolism in the liver (see section 1.2).

The structure of the liver is best described in terms of its functional unit,the acinus (see figure 1) which is based on the blood supply. Nutrient rich bloodenters the liver in the periportal (PP) region via the portal vein and hepaticarteries. From here it enters the sinusoids and flows through the acinus to theperivenous (PV) region where it is drained via the central vein. The sinusoidsare surrounded by a single layer of hepatocytes, usually about 20 cells in length,which carry out the main functions of the liver.

The liver is functionally heterogeneous with different zones performing dif-ferent metabolic functions. The PP zone specializes in gluconeogenesis whileglycolysis and glycogen synthesis are predominately carried out in the PV re-gion. The metabolic zonation [12] results from a combination of the systematicheterogeneity amongst hepatocytes and the concentration gradient of hormonesand oxygen established by the blood flow through the acinus. The variationin hepatocytes consists of both gradients of protein expression, resulting in dif-ferent concentrations of membrane bound receptors, and differences in cell andorganelle structure. For example, the perivenous region is believed to be moreabundant in endoplasmic reticulum (ER)[15].

1.2 Glycogen Metabolism in the Liver

Glycogen synthesis and glycogenolysis are governed by the enzymes glycogensynthase and glycogen phosphorylase respectively (subsequently referred to assynthase and phosphorylase). Synthase produces glycogen by extending chainsof glucose units via α-1,4 bonds. The free 1 carbon (C-1) of a chain is then at-tached to the C-6 of a neighbouring chain (by branching enzyme) to form ”bush-like” glycogen particles. Glycogenolysis involves the removal of the branches

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Figure 1: The acinus is the functional unit of liver. Blood is delivered fromthe portal vein and hepatic arteries in the periportal zone and flows along thesinusoids to the central vein in the perivenous area

by debranching enzyme and the liberation of individual glucose units from thefree C-4 end of the chain by phosphorylase.

Synthase and phosphorylase both have an active (a) and inactive (b) form.Synthase is inactivated by phosphorylation and activated by dephosphorylationwhile the converse is true for phosphorylase. The changes in the phosphory-lation status of glycogen synthase and glycogen phosphorylase (and hence therates of glycogenolysis and glycogen synthesis) are controlled through the ac-tivation of protein kinases by cyclic AMP (cAMP) and cytosolic Ca2+. Thesemessenger molecules are produced in hepatocytes in response to hormonal stim-ulation by insulin, glucagon and adrenaline.

In the liver the rate of glycogenolysis and glycogen synthesis can also be reg-ulated by glucose. Glucose binds to phosphorylase a (the active form) inhibitingits activity and making it more susceptible to inactivation by dephosphoryla-tion. The presence of glucose, in the form of glucose-6-phosphate, also affectsthe activity of glycogen synthase promoting dephosphorylation of synthase b tothe active form.

A detailed review of the processes and regulation of glycogen metabolismin the liver can be found in [2].

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1.3 Role of Cytoplasmic Calcium

Calcium is an important messenger molecule in numerous physiological pro-cesses. In many cell types the main pathway from the stimulation of membranebound receptors to the activation of enzymes is via changes in cytoplasmic cal-cium. In the liver calcium is important in the regulation of bile secretion, bileflow and cell survival. Calcium is also involved in the regulation of glycogenmetabolism via several processes. The major effect seems to be the Ca2+ acti-vation of phosphorylase kinase which activates glycogen phosphorylase leadingto increased glycogenolysis. Calcium may also influence glycogen synthesis viathe inactivation of glycogen synthase by Ca2+ dependent kinases.

In muscle tissue this is crucial. Muscle contractions are mediated by anincrease in calcium which also serves to supply the cell with glucose to produceenergy required for the contractions.

In the liver the importance of calcium in glucose homeostasis is less well es-tablished. Physiological levels of glucagon and adrenaline can produce increasesin Ca2+ in hepatocytes [9] which would regulate the activity of phosphorylasekinase [7]. However it has also been argued that in the liver the activation ofphosphorylase kinase is largely via cAMP dependent pathways and that the ef-fect of calcium is negligible. Experiments to measure the rate of glycogenolysisin response to hormone stimulation while blocking the release of calcium frominternal stores (see section 2) may provide support for this argument.

The rest of this essay discusses the dynamics of cellular calcium concentra-tions in the liver and how mathematical modelling may be to used to investi-gate this behaviour. Particular emphasis is given to intercellular propagationof Ca2+ signals.

2 Calcium Dynamics

Changes in the cytoplasmic calcium concentration often have a distinct spa-tiotemporal pattern. In individual hepatocytes, and many other cell types, thetemporal variation consists of oscillations whose frequency is apparently depen-dent on the level of hormonal stimulation. In coupled cells these oscillationscan become synchronised via cell to cell communication providing a means ofcoordinating the cellular response to external stimulation.

2.1 What Causes Calcium Oscillations?

The changes in the cytoplasmic calcium concentration are a result of the fol-lowing processes: the influx of calcium from the extracellular medium throughCa2+ channels, the release of calcium from intracellular stores, the removal ofcalcium from the cell by membrane pumps and the reuptake of calcium intothe internal stores. In liver hepatocytes, the primary store of calcium is the en-doplasmic reticulum (ER). Calcium is also stored in other organelles includingthe mitochondria and, in muscle cells, the sarcoplasmic reticulum.

The release of calcium from the ER is regulated by inositol-1,4,5-trisphosphate(IP3). The production of IP3 is triggered by the binding of agonists to G-protein

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coupled receptors in the cell membrane which activate phospholipase C (PLC).This enzyme causes the formation of IP3 from phosphatidylinositol (PIP 2).Binding of IP3 to IP3 receptors (IP3-R) on the ER causes the opening of Ca2+

channels and the subsequent release of calcium. The IP3-R are further activatedby the increase in cytoplasmic calcium leading to calcium induced calcium re-lease (CICR). High calcium levels inhibit the IP3-R, although on a slower timescale, preventing calcium release. This sequential effect of calcium on the IP3-Ris thought to be the main cause of calcium oscillations [13].

The high calcium concentration in the ER (relative to the cytoplasm) ismaintained by the influx of Ca2+ from the extracellular medium which is se-questered into the ER by the action of the sarco/endoplasmic reticulum calciumATPase (SERCA).

2.2 Why do Calcium Oscillations Occur?

The reasons for Ca2+ oscillations are not entirely understood. The frequency ofoscillations has been shown to depend on the level of agonist used to stimulatethe cell thus it has been proposed that the signal strength may be contained inthe frequency. This frequency encoding of the stimulus may provide a more re-liable signal than amplitude modulation particularly at low agonist dose wheresmall changes in the concentration would be hard to detect against random fluc-tuations [9]. An oscillatory signal may also protect the cell from the potentialdamage caused by prolonged exposure to high levels of Ca2+.

2.3 Synchronisation of Calcium Oscillations

Individual hepatocytes subjected to the same level of agonist display calciumoscillations at different frequencies. The variation in response is probably aresult of differences in the level of hormone receptor or in structural propertiesof the cell, specifically the ER content. However in preparations of multiplecells [22] and in the intact liver [8] oscillations in connected cells are seen to besynchronised at a common frequency despite the variation of gene expression,hormone uptake and ER level. This synchronisation may provide a mechanismby which the body can coordinate the response of a large number of heteroge-nous cells.

The mechanism by which the oscillations are synchronised has not beenfully determined but must require some form of intercellular communication.Communication between adjacent cells can occur by the diffusion of messengermolecules through the extracellular medium (paracrine pathways) or via gapjunctions (see section 2.4). In hepatocytes, it is commonly agreed that gapjunction signaling and not paracrine routes are responsible for the co-ordinationof calcium signals [4], [19]. Experimental evidence for this is provided by thework of Tordjmann et al. [22] who observed a loss of synchronisation betweencoupled cells when gap junctions were selectively blocked.

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2.4 Gap Junctions

Gap junctions are intercellular pathways via which ions and small moleculescan be exchanged between connected cells without leakage to the extracellularspace. Structurally, gap junctions consist of collections of channels each of whichis made up of of two hemi-channels or connexons, one provided by each cell.Each connexon is formed of six transmembrane proteins called connexins whichare members of a large protein family. Channels can be formed of two identicalhemichannels (homotypic) or from different hemichannels (heterotypic). Twoclasses of hemichannels can also exisit, those formed of one type of connexin(homomeric) and those with multiple connexin types (heteromeric). In the liverthree types of connexin, Cx26, Cx32 and Cx43 are expressed. Normal adultliver cells contain Cx26 and Cx32. Cx43 is mainly expressed in hepatoma andfetal cells and in regenerating tissue.

Gap junctions allow the passage of a range of molecules up to a molecu-lar weight of ∼1kDa including amino acids, sugars and cAMP. Large macro-molecules such as RNA and proteins can not pass through gap junctions. Saezet al. [18] showed that hepatocyte gap junctions are permeable to both IP3and calcium. Subsequently diffusion of both these messenger molecules hasbeen proposed as the mechanism via which calcium oscillations can becomesynchronised [4].

The permeability of gap junctions to specific molecules depends on the typeof connexins which form the channels. The permeability is not constant but isinfluenced by various environmental conditions. The primary regulators of per-meability are believed to be the trans-junctional voltage, the pH level and theconcentrations of ions and second messengers. In particular increased concen-trations of calcium have been shown to reduce the gap junction permeability[17]. However many experimental results have indicated that the concentra-tion of calcium required to block gap junctions exceeds those observed underphysiological conditions [21]. As a result the effect of calcium is often only con-sidered to be important under pathological conditions, for example when thecell membrane has been ruptured allowing high levels of Ca2+ to flow into thecell.

In contrast it has been proposed that in hepatoma (cancerous) cell linesexpressing Cx43, gap junctions can be blocked by physiological calcium con-centrations as low as 500nM [14]. This is interesting as it raises the possibilitythat hepatocytes that express Cx43 may be less well coordinated by intercellu-lar calcium signalling than normally functioning cells. More recently the use ofnon invasive measurement techniques has also suggested that much lower levelsof calcium than previously thought may be required to inhibit gap junctioncoupling in normal liver cells [5].

These experimental results raise a number of interesting questions. Firstly,what is the mechanism by which calcium oscillations are synchronised via gapjunctions. Secondly, do the calcium oscillations affect gap junction permeabilityand finally, what effect might this have on the level of synchronisation. Theremainder of this essay discusses mathematical models which could help answerthese questions.

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3 Mathematical Modelling of Calcium Oscillations

Numerous mathematical models of intra- and intercellular calcium dynamicshave been produced. For a review of the various types of model see [20]. Gen-erally these models incorporate some combination of the following dynamicvariables: the concentration of IP3, the fraction of active IP3-R, the concentra-tion of cytoplasmic, ER and mitochondrial calcium and the binding of calciumto buffering molecules. These process are shown in figure 2. Models whichdescribe the behaviour of coupled cells are based on single cell models joinedby gap junctional flux terms.

Figure 2: Reproduced from [20]. The essential processes in calcium oscillations.vin is the influx of calcium from the extracellular space, vout is the pumpingof calcium out of the cell. vrel and vSERCA are the release and reuptake ofcalcium by the ER. vmi and vmo describe the calcium fluxes across the mito-chondrial membrane and vb is the buffering of calcium. Finally vplc and vd arethe formation and destruction of IP3.

3.1 A Simple Model of Intracellular Calcium Oscillations

The first step in modeling the calcium dynamics of coupled hepatocytes is toconstruct a model which reproduces the behaviour of a single cell. Here I discusssuch a model presented by Thomas Hofer [11].

The model includes two dynamic variables, the cytoplasmic calcium concen-tration, x and the total cellular calcium concentration, z. Assuming a spatiallyuniform concentration the changes in the cytoplasmic and total Ca2+ concen-

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trations are given by:

dx

dt=

APM

VC(Jin − Jout) +

AER

VC(Jrel − JSERCA) (1)

dz

dt=

APM

VC(Jin − Jout) (2)

where Jin, Jout, Jrel and JSERCA are the fluxes as described in figure 3 andAPM and AER are the total areas of the plasma and ER membranes. VC is theeffective cytosolic volume which takes into account the buffering of calcium.

The influx of calcium from the extracellular medium, Jin is driven by theconcentration gradient which exists across the cell membrane. The flux equationincludes two terms, a constant leak and a variable flow rate through calciumchannels. Various mechanisms have been suggested for the control of the cal-cium channels including a secondary effect of the IP3 activating hormones [20].Hofer takes the channels to be regulated by the emptying of the internal calciumstores termed ”calcium-release activated calcium entry” [11]. This is modeledby assuming that the average value of z decreases with IP3 concentration hencethe calcium influx is an increasing function of IP3:

Jin = vleak + vinIP

Kin + IP(3)

where vleak is the background calcium leakage through the membrane, vin isthe maximum rate of the membrane channel and Kin is the threshold value ofthe IP3 concentration for the channel. The IP3 concentration (IP) is assumedto be constant

The removal of calcium from the cytoplasm by the membrane and ER pumpsis modeled as a pair of Hill functions with coefficient 2. The pump rates andthreshold values are vout and vSERCA and Kout and KSERCA.

Jout = voutx2

K2out + x2

(4)

JSERCA = vSERCAx2

K2SERCA + x2

(5)

The release of calcium from the ER is given by

Jrel = (k1(m∞(IP, x)w∞(IP, x))3 + k2)(y − x) (6)

where k1 is the maximum rate of release via IP3-R channels and k2 is the rate ofleak from the ER. As for Jin the flux from the ER is driven by the concentrationgradient across the ER membrane, y−x where y is the ER calcium concentrationgiven by y = VC/VER(z − x) and VER is the effective volume of the ER.

The model assumes that the IP3-R consists of three subunits which operateindependently. Each subunit has three binding sites, two activating (for IP3 andCa2+) and one inactivating (for Ca2+). The activation processes are assumed to

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be fast in comparison to the inactivation by calcium and are therefore assumedto be in a quasi steady state given by m∞:

m∞(IP, x) =IP

dp + IP

x

dx + x(7)

where dp and dx are the thresholds for the IP3 and Ca2+ induced opening ofthe channels.

Hofer also assumes that because the time constant for the inactivation ofthe IP3 receptor is a lot shorter than the period of calcium oscillations (∼1scompared to >30s) it can also be assumed to be in quasi-equilibrium. Theinactivation is then given by:

w∞(IP, x) =Q(IP )

Q(IP ) + x(8)

Q(IP ) = d2IP + d1

IP + d3(9)

where d1, d2 and d3 are constants.A version of this model has been implemented in Mathematica (see Ap-

pendix A) using parameter values taken from [11] (see table 1). The modelreproduces calcium oscillations which are in agreement with experimental ob-servations. Most importantly, the frequency of the oscillations in a single cellincreases with increasing agonist level, modelled as an increase in the value ofthe IP3 concentration. Figure 3 shows the output for the simulation with 3different levels of IP3.

3.2 Intercellular Synchronisation via Calcium Diffusion

As mentioned previously, diffusion of both calcium and IP3 through gap junc-tions has been proposed as the mechanism via which calcium oscillations becomesynchronised. Here we look at the possibility that the synchronisation is a re-sult of calcium diffusion by considering a pair of cells i and j. Incorporatinga gap junctional flux of calcium between the cells equations 1 and 2 for cell ibecome [11]:

dxi

dt=

AiPM

V iC

(J iin − J i

out) +Ai

ER

V iC

(J irel − J i

SERCA) + J ijG (10)

dzi

dt=

AiPM

V iC

(J iin − J i

out) + J ijG (11)

where AiG is the area of the gap junction connection. The gap junctional flux,

J ijG from cell j to cell i is given by:

J ijG =

pAijG

V ijC

(xj − xi) (12)

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Parameter Value

!= AER /APM 2

"=VER/VC 0.1

#= APM /VC 0.02 µm-1

vleak 0.2 µMs-1

vin 4.0 µMs-1

Kin 4.0 µM

vout 3.6 µMs-1

Kout 0.12 µM

vSERCA 9.0 µMs-1

KSERCA 0.12 µM

k1 40.0 s-1

k2 0.02 s-1

dp 0.2 µM

dx 0.4 µM

d1 0.3 µM

d2 0.4 µM

d3 0.2 µM

Table 1: Parameters used in the implementation of the model

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500 550 600 650 700 750 800 850 900 950 1000

Time (s)

IP3=5

IP3=3

IP3=1.5

Figure 3: Osillations showing the increase in frequency with increased IP3concentration. Plots have been shifted vertically on the y axis for clarity.

where p is the gap junction permeability which is assumed to be symmetricali.e. pij = pji. The cells are assumed to be at the same potential so that theflux depends only on the chemical gradient between the cells. This approachhas been used to investigate two cells which are structurally heterogeneous (e.g.V i

ER 6= V jER etc.) but produce the same level of IP3 [11]. It is found that when

uncoupled (γ = pAG/VC = 0) each cell oscillates at its own frequency. As thecoupling strength γ is increased the cells move from asynchronous oscillationsthrough 2:1 phase locking and finally to synchronous oscillations indicating thatcalcium diffusion may play a role in synchronisation. This is shown in figures4-6. These figures were produced using the implementation of the Hofer modelin Mathematica.

4 A Gap Junction Model

The model described above, and most other published models of calcium oscil-lations e.g. [6], [23] assume the diffusion of a messenger molecule (either Ca2+

or IP3) that is governed by a gap junctional flux term of the form J = γ(xj−xi)where γ is assumed to be some constant coupling term. However as discussed insection 2.4 the permeability of gap junctions is dependent on the chemical andelectrical conditions of the intracellular environment including calcium concen-tration. Therefore it is possible that the coupling strength of the junction maychange during the course of the calcium oscillations.

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0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500 600 700 800 900 1000

Figure 4: Unsynchronised oscillations in a pair of coupled cells. γ = 0.001

This could be investigated theoretically by incorporating a model of a gapjunction into the model of intercellular calcium oscillations described above. Apossible starting point is the model of a gap junction presented by Baignet etal. [1].

The model assumes the general case of two cells of fixed volume V, linkedby a gap junction. Each cell contains n species of molecule which are uniformlydistributed throughout the cell. The intracellular concentration of a speicesl in cell i is defined as Xi

l and Xi = (Xi1, ........., X

in) is the vector of species

concentrations in cell i. The dynamics of the concentration of species Xl in celli is given by:

V i dX il

dt= (J i

l )j + (J il )m + (J i

l )p (13)

where the J’s are the junction flux (j), passive membrane flux (m) and mem-brane pump flux (p) of the species l.

The purpose of studying this model is to derive a suitable junctional fluxterm to be included in a model of calcium dynamics therefore only the detailsof the junction flux will be discussed. It is also assumed that calcium is the onlyspecies of interest and, more importantly, the only one that influences gap junc-tion permeability therefore Xi can be reduced to xi, the calcium concentrationin cell i.

The junction flux between cell 1 and 2 is taken to be a function of thecalcium concentration x in each cell, the intracellular potentials φi and thegating state, s of the junction. It is also assumed to be symmetrical so that:

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0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500 600 700 800 900 1000

Figure 5: 2:1 synchronised oscillations in a pair of coupled cells. γ = 0.01

(J1)j = (J1)j(x1, x2, φ1, φ2, s) = −(J2)j (14)

For simplicity it is assumed that 1) φ1 = φ2 = constant so that any de-pendence on the electrical potentials can be ignored, and 2) the junctional fluxdepends only on the concentration gradient of calcium across the junction andnot on the absolute concentrations so that (J1)j = (x2−x1)ρ(s) where ρ is thepermeability of the junction determined by the gating state.

4.1 Gap Junction Gating

A gap junction can be visualised as an array of channels which are controlledby some gating mechanism. Each channel can exist in a finite number of states,determined by the gating mechanism, with transitions between these statescontrolled by the chemical and electrical properties of the environment. Eachstate has a different permeability so that the total permeability of the gapjunction is determined by the fraction of channels in each state.

This can be expressed mathematically by assuming each channel to haveN possible gating states with each state, k having a permeability ρk. Thefraction of channels in each state is defined as sk such that

∑Nk=1 sk = 1. s =

(s1, ........., sN ) is the vector of the fraction of channels in each of the gatingstates. The overall permeability of the channel is given by:

ρ =N∑

k=1

ρksk (15)

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0

0.1

0.2

0.3

0.4

0.5

0.6

0 100 200 300 400 500 600 700 800 900 1000

Figure 6: 1:1 synchronised oscillations in a pair of coupled cells. γ = 0.07

It is also necessary to define the temporal dynamics of the population ofeach gating state. If the probability of a transition from state i to state j perunit time is defined by αij = αij(x1, x2) then the change in the fraction ofchannels in state k is:

dsk

dt=

N∑j=1

αjksj − (N∑

j=1

αkj)sk (16)

This assumes that the transitions are only dependent on the calcium concen-tration. To produce a useable model it is necessary to define the possible statesof the channel, the transition probabilities and the permeability of each statehowever the method by which calcium, or any other chemical, can gate a gapjunction channel is unclear. It has been proposed that the Ca2+ effect on gatingis mediated by an intermediate substance specifically calmodulin (CaM) [16].Calcium binding to CaM induces a conformational change in the protein thatexposes two hydrophobic areas.Ca2+-CaM can then bind to a receptor domainon another protein via hydrophobic interactions. It has been shown that Ca2+

activated CaM can bind to certain connexins and suggested that this may phys-ically block the gap junction channel providing a means for Ca2+ to regulategap junction permeability.

Due to the lack of information on the mechanism of chemical gating it isdifficult to suggest a mathematical model to describe the process. However,loosely based on the hypothesis of CaM binding the following is suggested.

Each connexon (gate) can exist in two states, open or blocked (closed).Blocking of either (or both) hemichannel(s) does not completely inhibit the

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junction thus there are 4 possible states for the channel:

S0: gate 1 open, gate 2 openS1: gate 1 open, gate 2 closedS2: gate 1 closed, gate 2 openS3: gate 1 closed, gate 2 closed

To simplify the transition equations it is assumed that both gates cannot changestate simultaneously. So, for example, transitions from state S1 to state S2 cannot occur without going via state S0 or S3.

ds0

dt= α10s1 + α20s2 − (α01 + α02)s0 (17)

ds1

dt= α01s0 + α31s3 − (α10 + α13)s1 (18)

ds2

dt= α02s0 + α32s3 − (α20 + α23)s2 (19)

ds3

dt= α13s1 + α23s2 − (α31 + α32)s3 (20)

The transition probabilities could be assumed to be functions of the abso-lute calcium concentration in the cell being modified. Two forms of transitionprobability need to be defined, one for the closing and one for the opening of achannel. The exact form of these probabilities is not discussed but should beincreasing and decreasing functions of the calcium concentration respectively.

5 Conclusions

Calcium oscillations in hepatocytes may regulate a number of physiologicalprocesses including the activity of phosphorylase kinase and synthase kinaseswhich influence the rates of glycogen synthesis and breakdown. In preparationsof multiple cells and in the intact liver these oscillations can become synchro-nised. This may act as a method of coordinating the response of heterogeneouscells in response to external stimuli. It is clear that gap junctions play a rolein coordinating the oscillations however the mechanism by which this occurs isnot obvious. Diffusion of both calcium and IP3 has been proposed.

Mathematical models have been used to show that the synchronisation ofcalcium oscillations in hepatocytes could be controlled by the diffusion of cal-cium through gap junctions [11]. Similar models have also indicated a rolefor IP3 diffusion in this process [6]. These models assume that the couplingstrength of the cells is constant.

A possible extension of the existing models is to consider the effects ofcalcium (or other agents) on gap junction permeability. A general model of agap junction [1] was used to try and derive a model for the gap junction fluxof calcium regulated by calcium. The proposed model is far from complete asvery little is known about the Ca2+ gating mechanism of gap junctions (here it

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is assumed that the gating is mediated by an intermediate protein, calmodulin)and it would be interesting to try to develop it further.

However before investigating the effect of calcium on gap junctional perme-ability further it is important to consider the biological relevance. While theinhibition of gap junctions by calcium has been demonstrated experimentallyit is not clear whether physiological levels of calcium are sufficient to affect thepermeability. If this is the case, modelling of this effect may still be relevantbut in the context of pathological rather than physiological conditions.

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[13] J. Keener and J. Sneyd (1998) Mathematical Physiology. Springer-Verlag,New York

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[15] T. Oinonen and K.O. Lindros (1998) Zonation of hepatic cytochrome P-450expression and regulation, Biochem. J. 329 17-35

[16] C. Peracchia (2004) Chemical gating of gap junction channels: roles ofcalcium, pH and calmodulin, Biochimica et Biophysica Acta 1662 61-80

[17] F. Ramon, G. A. Zampighi and A. Rivera (1985) Control of junctionalpermeability. In Gap Junctions (M. Bennet and D. Spray, Eds.) 155-166.Cold Spring Harbor Laboratory, New York.

[18] J.C. Saez, J.A. Connor, D.C. Spray and M. L. Bennet (1989) Hepato-cyte gap junctions are permeable to the second messenger, inositol 1,4,5-triphosphate, and to calcium ions, PNAS 86 2708-2712

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A Mathematica Code

(*Based on Hofer et al 1999 - two cells*)(*Parameters*)(*Plasma Membrane influx parameters*)v0 = 0.2; (*Calcium channel leak*)vc = 4; (*Channel rate*)K0 = 4.0; (*Threshold*)(*Plasma membrane pump parameters*)v4 = 3.6; (*Pump rate*)K4 = 0.12; (*Threshold*)(*ER pump parameters*)v3 = 9; (*Pump rate*)K3 = 0.12; (*Threshold*)(*ER release parameters*)da = 0.4; (*Calcium opening threshold*)dp = 0.2; (*IP3 opening threshold*)k1 = 40; (*ER channel rate*)k2 = 0.02; (*ER channel leak*)d1 = 0.3;d2 = 0.4;d3 = 0.2;(*Structural Parameters*)alpha1 = 2; (*Ratio of ER membrane area to plasma membrane area*)beta1 = 0.1; (*Ratio of cytoplasmic volume to ER volume*)rho1 = 0.02; (*Ratio of plasma membrane area to cytoplasmic volume*)alpha2 = 2; (*cells assumed to be strcuturally identical*)beta2 = 0.1;rho2 = 0.02;(*Functions*)hillfunc[x,xT,hc] := (xhc)/((xT hc) + (xhc))mp1 := v4 ∗ hillfunc[Cc1[t],K4, 2] (*membrane pump*)ep1 := v3 ∗ hillfunc[Cc1[t],K3, 2] (*ER pump*)mc1 := v0 + vc ∗ (hillfunc[P1,K0, 1]) (*membrane channel*)Q1 := d2 ∗ (P1 + d1)/(P1 + d3) (*ER release*)U1 := hillfunc[P1, dp, 1] ∗ hillfunc[Cc1[t], da, 1] ∗ hillfunc[Q1, Cc1[t], 1]ec1 := (1/beta1) ∗ (Tc1[t]− (1 + beta1) ∗ Cc1[t]) ∗ (k1 ∗ (U13) + k2)j1 := GapC ∗ (Cc2[t]− Cc1[t]) (*junctional flux*)mp2 := v4 ∗ hillfunc[Cc2[t],K4, 2]ep2 := v3 ∗ hillfunc[Cc2[t],K3, 2]mc2 := v0 + vc ∗ (hillfunc[P2,K0, 1])Q2 := d2 ∗ (P2 + d1)/(P2 + d3)U2 := hillfunc[P2, dp, 1] ∗ hillfunc[Cc2[t], da, 1] ∗ hillfunc[Q2, Cc2[t], 1]ec2 := (1/beta2) ∗ (Tc2[t]− (1 + beta2) ∗ Cc2[t]) ∗ (k1 ∗ (U23) + k2)j2 := GapC ∗ (Cc1[t]− Cc2[t])(*define the differential equations*)eqnCc1 := Cc1′[t] == rho1 ∗ (mc1−mp1 + alpha1 ∗ (ec1− ep1)) + j1

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eqnTc1 := Tc1′[t] == rho1 ∗ (mc1−mp1) + j1eqnCc2 := Cc2′[t] == rho2 ∗ (mc2−mp2 + alpha2 ∗ (ec2− ep2)) + j2eqnTc2 := Tc2′[t] == rho2 ∗ (mc2−mp2) + j2(*Solve and plot for values of IP3, P1 and P2*)P1 = 3;P2 = 1.5;GapC = 0.07; (*coupling strength*)solution = NDSolve[eqnCc1, eqnTc1, eqnCc2, eqnTc2,Cc1[0] == 0.05, T c1[0] == 5, Cc2[0] == 0.05, T c2[0] == 5, Cc1[t], T c1[t], Cc2[t], T c2[t], t, 0, 2000];Plot[Cc1[t]/.solution, Cc2[t]/.solution, t, 0, 1500, AxesLabel− > ”time”, ”Cc”,P lotRange− > 0, 0.6, P lotStyle− > RGBColor[1, 0, 0], RGBColor[0, 1, 0]];(*output data to CSV file*)Export[”Calciumdata003.csv”, Table[t, Cc1[t]/.solution, Cc2[t]/.solution, t, 0, 2000]];

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