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Contents

1 Biology of the voltage gated ionic channels 4

1.1 Introduction to the proteins . . . . . . . . . . . . . . 41.2 Ionic channels as a family of proteins . . . . . . . . . 81.3 Basic channels concepts and terminology . . . . . . . 101.4 Nomenclature and systematics of the ionic channels . 151.5 Details concerning the voltage gated potassium chan-

nels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5.1 Voltage gating and the propagation of nerve

impulses . . . . . . . . . . . . . . . . . . . . . 171.5.2 Functional parts of the voltage gated potas-

sium channels . . . . . . . . . . . . . . . . . . 201.5.3 Inactivation and the idea of the ball and chain

model . . . . . . . . . . . . . . . . . . . . . . 22

2 Techniques for measuring the electrical activity of the

voltage gated ionic channels 26

2.1 The idea of voltage clamp . . . . . . . . . . . . . . . 26

2.2 Gigaohm seal and the patch clamp technique . . . . . 292.3 Experimental data and its processing . . . . . . . . . 31

3 Diffusional approach to the ball and chain model 37

3.1 Introduction and basic assumptions . . . . . . . . . . 373.2 Random walk and its probabilistic description . . . . 39

3.2.1 Unrestricted noncorrelated random walk . . . 403.2.2 The unrestricted and correlated random walk 43

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3.3 Details of diffusional approach to ball and chain model 47

3.4 The parabolic diffusion . . . . . . . . . . . . . . . . . 483.4.1 Details concerning numerical solution of the

parabolic ball and chain . . . . . . . . . . . . 503.5 The hyperbolic diffusion . . . . . . . . . . . . . . . . 54

3.5.1 The absorbing boundary . . . . . . . . . . . . 553.5.2 Analytical solution for hyperbolic ball and chain

problem . . . . . . . . . . . . . . . . . . . . . 573.6 Comparison with experimental data . . . . . . . . . . 59

3.7 Calculation of the diffusion coefficient of the ball . . . 613.8 Concluding remarks . . . . . . . . . . . . . . . . . . . 63

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Preface

The purpose of this work is to give the physical interpretation andthe mathematical description of the phenomenon of inactivation,which is present in some of the voltage gated ionic channels i.e theproteins, which action underlays all the electrical signaling processesin each living organism.

Qualitative model of inactivation, termed as the ball and chain

model, was proposed by Armstrong and Benzanilla in 1960 [1], onthe basis of electrophysiological measurements, and molecular biol-ogy experiments. Model found also its later conrmation in struc-tural investigations of the ionic channels [2].

This work resolves the qualitative model by means of diffusion(parabolic and hyperbolic operators), gives quantitative results, andcompares them with experimental data.

It is worth to mention, that the part of this work, was presented at

the Marian Smoluchowski Symposium on Statistical Physics 2006in the form of the scientic poster. Yet, the essential concepts of this work are, arranged in the form of an article, are currently un-der editorial process in Acta Physica Polonica B (IF for 2005year=0.807).

Here, I would like to emphasize my honest gratitude, to prof. drhab. in z. Zbigniew Jan Grzywna, for his immeasurable help and

advice during the preparation of this work.

This document was created in the LATEXenvironment. All pro-grams for data processing and numerical solving, were written in theANSI C language. Graphs were prepared under Matlab 4 environ-ment. Additionally, each graphics was processed under GNUGIMPsoftware.

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Chapter 1

Biology of the voltage gated

ionic channels

This chapter is devoted to some basic biological and biophysicalfeatures of the ionic channels. It begins with the short introduc-tion to the proteins, that provides essential terminology for furtherdescription of the ionic channels. Further, to give a feeling of how

broad the channels family is, some basics of the channels system-atics are shown. Finally, detailed description of the structure andthe behavior of voltage gated ionic channels is shown.

1.1 Introduction to the proteins

Proteins (the name from the greek word protas, meaning of primary importance), together with the polysaccharides, lipids andnucleic acids, are the primary constituents of each living organism.From the chemical point of view, proteins are polymers, composed of -aminoacids, that are held together by the peptide bonds. Lengthof the protein chain varies from hundred, to about 27 thousands of aminoacid subunits. Therefore, the molar masses of a single pro-tein molecule can range from a few hundreds up to even 3 106 D1

1 D stands for Daltons the unit of mass equivalent to atomic mass unit, preferably usedin biochemistry and molecular biology

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[3]. Large sizes of proteins implies their complex shapes. Three

dimensional arrangement of the protein plays a central role in itsfunctionality. Therefore, a complex structure of the protein is de-scribed at four levels:

1. First order structure , which denes the aminoacids sequenceof the chain that builds the protein. It is dictated by the genesequence, and further modied in the process of alternativesplicing (thus one gene may code for different proteins). As-sembly of the rst order structure takes place at the rybosomes a small cellular organelles responsible for the translation of mRNA into aminoacid sequence.

2. Second order structure , which determines the three dimen-sional shape of a local segment of an aminoacid chain, usuallyarranged into the stable, repeatable motifs like -sheets or-helices, both shown in the Fig.(1.1). Sometimes polypeptidechain can also have local segments of unstable or not well de-ned shape. Second order structure results mostly from thehydrogen bonds interactions.

3. Third order structure denes the overall shape of the pro-tein. It is determined primarily by the hydrophobic/hydrophilicinteractions, presence of sulfur bridges (that is the covalentbonds), and in small extent by the hydrogen bonds.

4. Fourth order structure applies only to the protein com-plexes, created by at least two protein subunits (i.e. the singleaminoacid chains), kept together by hydrophobic/hydrophilicinteractions, or by hydrogen bonds. It is worth to notice, thatthe term subunits is sometimes used also to distinguish the longfragment of the aminoacid chain that creates a similar spacestructures (composed of a couple of -helices, or -sheets) in asingle protein.

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Figure 1.1: Examples of the second order structure taken by proteins. ( a ) -helix, (b ) -sheet (gure adapted from [4])

As mentioned previously, proteins rst order structure is deter-mined by the gene sequence. Process of its assembly, requires somespecial cellular organelles, as well as the input of energy. First orderstructure is almost always conserved through the whole life of the

protein. In contrast to that, the process of establishing second andthird order structure of a protein, called folding, is usually sponta-neous (under the physiological conditions temperature and pH),however sometimes, may be accompanied by some special chaperonproteins or the thermal shock proteins. Folding begins right withthe synthesis of the polypepetide chain in rybososmes, and com-pletes shortly after the whole protein is build. There is no one andunique second or third order structure assessed to the aminoacidsequence. As a results of their normal functioning, proteins takesdifferent shapes, called conformations

Traditionally, proteins are divided into following classes [5]:

Globular proteins , usually of spherelike shape, partially sol-uble in water, present in cytoplasm and intracellular space;.

Fibrous proteins responsible for the structural support,mostly insoluble in water.

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Membrane proteins subdivided into two classes: integral membrane proteins rmly attached to the mem-

brane, being their functional and structural elements, re-movable only with some strong detergents. They are fur-ther subdivided into the following classes:

transmembrane proteins, that span from the internal tothe external surface of the lipid bilayer in which theyare embedded;

membrane associated proteins located entirely in thecytosol, and attached to the cytosolic layer of the mem-brane;

lipid linked proteins, located entirely outside the bi-layer, on either inside or outside of the cell;

peripheral membrane proteins attached to the mem-brane indirectly, typically by binding to integral membraneproteins, or by noncovalent interactions with the lipidic,

polar head.In the living organisms, proteins accompany all biochemical pro-

cess. There is literally no single biochemical reaction that goes with-out the presence of proteins, either in the form of reagent, but pre-dominantly in the role of catalyst. Some examples, that shows thewide areas of proteins functioning are shown below:

structural issues proteins take a part in constituting the cellsbody, by creating the cytoskeleton. Good example is collagen,the most abundant protein in mammals, which is found almostin each tissue, and characterizes with the great tensile strength.

catalysis proteins as enzymes controll and allow to proceedliterally every reaction in the living organism. Their extraordi-nary specicity and activity that evaluated in the course of thenatural evolution, made them applicable for many industrial,biosynthetic processes.

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regulatory processes since each cell has a complete set of genes, proteins regulate their expression level adjusting thepresence of a given protein, depending upon needs. It is obvi-ous, that for example, the skin cell does not have to syntetisethe same proteins as need the liver cell. Thus the controlledexpression of genes is the basis for cellular differentiation, andby that, creation of tissued and organs.

signaling as hormones and their receptors, proteins governsthe stimulation or inhibition of growth, induction or suppres-sion of apoptosis, regulation of metabolism, etc. Intercellularsignaling is the basic feature that allowed multicellular organ-isms to evolve. Intracellular signaling is the skill, that assuresthe tight cowork between consecutive parts of the multicellularorganism. Signaling issues are performed also by ionic channels.

selective transport through the cellular membranes includingpassive (ionic channels) and active (some ATP supplied pumps)transport. Need for such transport results not only from themetabolic issues, but it also underlies all types of electricalsignaling (nerves, muscles, glandular tissue, etc).

immunologic reactions as antibody proteins used to mark andneutralize foreign objects like bacteria or viruses.

1.2 Ionic channels as a family of proteins

Ionic channels are specialized family of complexes of proteins.They are found to be immersed in the lipid cells membrane andthey span it from the internal to the external surface, in the mannershown in Fig.(1.2). Thus, according to previously given classica-tion, the ionic channels belong to the class of the transmembraneintegral proteins.

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Figure 1.2: Crossection of the ionic channel immersed in lipid bilayer. Channelspore is usually sealed with water. Shape of the channel is only approximate here(gure adapted from [3]).

Yet they sit in the lipid bilayer, ionic channels are additionally

anchored to the other membrane proteins, or to the intracellularcytoskeleton.The macromolecule of the channel consists usually of 1800 4000amino acids rests, with some hundreds of sugar residues covalentlylinked as the olygosaccharide chains to the aminoacids on the outerface [3]. Very little number of the ionic channels are literally singleproteins. Most frequently, they consist of four or more subunits (i.e.the single aminoacid chains), that are very similar in structure, and

are held together by the hydrophobic/hydrophilic interactions or bythe net of the hydrogen bonds.

The average density of the ionic channels at the surface of thecellular membrane is not constant. It depends upon types of thecells and types of the channels. Table (1.1) (adapted from [3]) givesa few examples.

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Origin Type of the channelChannel density

[channels/m 2]Frog node Potassium channel 570960Giant squid axon Potassium channel 30Mouse skeletal muscle Sodium channel 65Giant squid axon Sodium channel 330

Table 1.1: Examples of ionic channels density at different tissues

1.3 Basic channels concepts and terminology

The lipid bilayer, that forms the membrane of biological cells isin general weakly permeable for any substance present in the cellsinterior. Purely diffusive transport through the lipidic membraneitself is too slow, to provide a satisfactory uxes. Therefore, the cel-lular membrane is equipped with a several types of devices, mostlyproteins, to provide the possibility of fast, and easy-to-control trans-port of species between cells interior and the external space. Ionchannels are one type of such devices used for transport of ions.

Need for the exchange of some small inorganic ions between thecell and its surroundings originates not only from the metabolic orregulatory processes, that proceeds in the living cells. Exchange of ions is the basis of the electrical signaling, the technique, developedin the course of evolution by the multicellular organisms, used formonitoring, sensing, and coordinate responding for stimuli. Electri-cal signaling underlies the coordinated action of neurons, muscles orglandular tissue.

For the theory of electrical signaling viewed as the propagationof action potentials, Hodgkin and Huxley were awarded with theNobel Prize in 1963. Not going into details, it is enough to say, thatthe role of the ionic channels in the electrical signaling, comes downto the selective and triggered permeation of small inorganic ions sodium, potassium, calcium and chloride.As may be noticed from the Fig.(1.2), third and fourth order struc-ture of the channel creates a pore a passage for ions. Once the

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channel is open, transport of ions through its pore is spontaneous

and passive. It is dictated by the value of the transmembrane elec-trical potential, and by the concentration gradient for a given ion.This differs the ionic channels from the other transmembrane inte-gral proteins the ionic pumps, that facilitate the active transportof ions, at the cost of energy, that originates from an ATP molecule.

Therefore, one of the important measurable feature of the ionicchannel is the measure of its efficiency, expressed by the channelspermeability, i.e. the rate at which channel permeates ions. In most

cases, channels has high permeability for one type of ions only. Se-lectivity of the channel is not dictated directly by the size of an ion,but rather by the properties of hydrating sphere of water molecules,that surrounds it.

As stated in [3] passing an ion through the ionic channel is a verycomplicated process, inuenced by many factors including:

accessibility of ions to be permeated, limited by their diffusionfrom the surrounding solution to the channels mouth

crowd effects in the pore of the channel, caused by: mechanical interactions with water molecules

electrostatic repulsion by other ions in the pore

possible attraction to the pores walls,

necessity of removing some H 2O molecules from the hy-drating shell of an ion.

On the basis of the simple Ohms law, it is possible to estimateat least the order of magnitude of the ionic channels permeability.For the sake of derivation, we have to use a knowledge about theshape of a typical pore. At the most of its length, the pore canbe treated as a wide vestibule lled with water. However near itsend, pore constricts to the diameter of a few angstroms, forminga selectivity lter, that passes only a certain types of ions. Thisnarrow throat surely limits the channels permeability. Its length is

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estimated in [7] to be equal to 5 [A] and the radius to 3 [A], as given

in the Fig. (1.3a). The pore is bathed in a medium resembling the

Figure 1.3: Ideas for calculating the pores permeability. (a) model of the selec-tivity lter, used for calculations, together with its typical sizes. (b) componentsof the electrical resistance of the pore inner pore resistance together with twocomponents of access resistance. (c) illustration of the idea of a spherical sinkimmersed in the uniform medium. sink denotes the ux of mass absorbed inthe sink. Figure was adapted from [7]

typical, physiological solution surrounding the pore, characterizedby the resistivity = 1 104 [m]. Thus the electrical resistanceof the pore itself may be approximately estimated as for the simpleresistor:

R pore = la 2

(1.1)

where l is the length of the pore, and a is the radius of its crossection.After substitution, we get R pore = 1.8 [G], in the same way as in [8].In addition to the resistance within a pore, we have to include the

access resistance on both sides of the pore the resistance along theconvergent paths from the bulk medium to the mouths of the pore,as shown in the Fig.(1.3b). This requires to evaluate the spatialintegral at the limits from innity to the disklike mouth of thepore, what was in details considered at [9], and gives the accessresistance equal to

Raccess = 2a

(1.2)

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on both sides. Since the resistance of the pore and the access resis-

tance are in parallel, the total channels resistance reads as

Rchannel = R pore + Raccess = l + a

2 a 2

(1.3)

After substituting the numerical values, the resistance of the singlechannel is

Rchannel = 3.41 [G] (1.4)

and therefore the conductance

channel = 290 [ pS ]. (1.5)

From the treatment of ionic channel as a simple resistor, whatwas basically done in the above considerations, it follows, that anydesired ux of ions through the channel may be reached by apply-ing the proper potential difference across the selectivity lter. Thisis true, if the ux of ions through the channel is less, or equal tothe ux of some new ions arriving there from the bulk of solution.This ux will limit the maximum current, that a channel may pass.

Flux of the new molecules arriving to the mouth of the pore, may beestimated on the basis of simple diffusion (with some simplifying as-sumptions, like omitting any potential gradient in the surroundings,and treating the solution as an innite source of ions).

Problem considered here, has the same nature as calculations of the rate of the chemical reaction limited by diffusion. As stated in[3], method of solution for that, was proposed at 1916 by MarianSmoluchowski, who solved the diffusion equation for a spherical sinkof a radius a immersed in the innite medium with uniform initialcondition c, and absorbing boundary condition at the surface of thespherical sink, as shown in the Fig.(1.3c). The ux of a matter intothis sink, denoted by sink is

sink = 4aDc (1 + 1

a (Dt )) (1.6)

where D is the diffusion coefficient, and t is the time. For the suf-ciently high t, ux reaches its steady state value, obviously equal

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to:

sink = 4aDc (1.7)Since in our case, the sink is just hemispherical (we are interestedonly in the intracellular mouth of the channel), the ux of the ar-riving ions will be two times less than in (1.7). Taking the diffusioncoefficient for its typical value in water as D = 1.5 10 5[cm2/s ], weget after substitutions, that

mouth = 2aDc (1.8)

mouth = 2.0 108

[ions/second ] 33[ pA] (1.9)From Ohms law, 114 mV is needed to force the ow of 33pA throughthe selectivity lter of conductance equal to 290 pS. The well knownexample of such highly conductive channels are mammalian BKchannels (used as a source of experimental data in this work), selec-tive to potassium ions, which conductivity, according to [7] variesfrom 130 240 [pS]. However, most of the ionic channels do notreach this level of permeability at the transmembrane voltage equal

to about 110 mV. It is not surprising, taking into account the sim-plicity of the above calculations, and the fact that the above resultsare essentially the upper limits for the permeability. Yet, all theionic channels shows permeability in the range of picoamperes.

As was mentioned at the beginning of this section, the role of ionic channels comes down to selective and triggered permeationof ions. Indeed, channels are usually permeable for one type of ionsonly. Triggering shows itself in the fact, that for a given kind of ion,

channels permeability is not constant, and may uctuate in time,in response to different stimuli. Process of modulation of the perme-ability of the ionic channel is called gating. Gating occurs throughthe induced changes in the channels structure. It follows than, thatone conformation of the channels protein is permeable for ions, andother is not. Ability of gating is the basis for all regulatory mecha-nisms in the living organisms (like endocrine secretion), connectedwith controlled exchange of chemical species between cells body

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and the surroundings. Examples of different gating possibilities are

given below.

ligand gating when activity of the channel is modulated bythe addition of a some chemical compounds, collectively knownas the neurotransmitters. Most often, their primary role isthe initiation of action potentials. Ligand gated channels arefound to be responsible for the initiation of the nerve impulses.A prototypic channel for this group is a sodium permeatingchannel, gated by acetylcholine.

voltage gating activity of the channel is modulated by thevalue of the transmembrane electric potential. Voltage gatedionic channels primarily participate in the nerve impulses con-duction (nerves, muscle tissue, etc.) by the propagation of action potentials. Famous example is a Giant Squid K + ionicchannel, used by Hodgkin and Huxley in their studies on theelectrical signaling phenomena.

mechanosensitive and cell volume regulated gating ac-tivity of the channel is regulated by the changes in osmotic pres-sure, membrane curvature, or tension exerted on the cell. Thosechannels are necessary for the sense of touch and hearing. Oneof the most intensively studied mechanosensitive ionic channelis isolated from the sensory brittle neuron of the Drosophila y.

Worth of notice is the fact, that quite often an ionic channel has fewgating mechanisms. Good example are some of the voltage gated K +

channels, which activity is modulated also by the present of Ca2+

ions.

1.4 Nomenclature and systematics of the ionic

channels

Yet, there is more than 150 kinds of ionic channels identied. Thenaming of the ionic channels have not yet been systematic. Usually,

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channel is named by its most important permeant. Additionally,

channels are also named after the anatomical regions, from whichthey were isolated. Sometimes, naming comes from the channelsprimary inhibitors, specic neurotransmitters, or other species thatmodulates their activity [3].

As stated in [10], very useful in classication of the channelswere the methods of molecular genetics, that allowed to obtain thecomplete sequence of genes that codes for the particular channels.On that basis there were distinguished a families of homologous

channel proteins, that have evolved by the processes of successivegene duplication, mutations, and natural selection from commonancestor. In the Fig.(1.4), adapted from [21] the schematic viewof the family of homologous voltage gated ionic channels is given.Every branch is the separate kind of ionic channel.It is seen, thatall of the voltage gated ionic channels have a one common ancestor.It may also be noticed, that the channels that have the commonpermeating ion, have also a common precursor.

1.5 Details concerning the voltage gated potas-

sium channels

Among many families of the voltage gated ion channels, potas-sium channels are of special interest in this work. In literature, theyare used to be consider as prototypical voltage gated channels, andtogether with Na + and Ca 2+ channels, they constitute a family of

structurally related integral membrane proteins. K+

channels re-semble all basic features of voltage gated channels family, and areone of the most extensively studied group of channels having there-fore vast amount of literature references available. As stated in [11],family of potassium channels is coded by at least 22 different genes,with additional variety given by alternative splicing. Though, as fornow, it is estimated, that the family is made of more than thirtykinds of channels. What is more important, most of the potassium

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Figure 1.4: The family tree of the voltage gated ionic channels. Different colorsat the graph, denote the subfamilies, which differ in the most permeable ion.Labels around the circular graph, denotes different channels names, the namesof the most important permeant, or the substances, that additionally inuencesthe gating. (Figure was adapted from [21])

channels, including the BK channels, that are the source for ex-perimental data here, undergoes the voltage independent process of inactivation the temporal plugging of their pore. The mechanisticmodel of inactivation, called the ball and chain will be concernedin details in next section of this work.

1.5.1 Voltage gating and the propagation of nerve im-

pulses

Chemical composition of the solution that lls the living cells isusually entirely different than the chemical composition of a sur-rounding medium. Differences are noted not only in the concentra-tions of dissolved organic species like sugars, or proteins, but alsoin the concentration of some of the inorganic species, particularly

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ions. Table (1.2), that bases on data found in [3], gives an example

of extracellular and intracellular concentrations of couple of inor-ganic ions, measured at the vicinity of the cellular membrane of themammalian skeletal muscle at rest.

IonExtracellular concentration Intracellular concentration [Ion ]o

[Ion ]i[mM] [mM]Na+ 145 12 12K+ 4 155 0.026

Ca 2+ 1.5 10 7 15000Cl 123 4.2 29

Table 1.2: Concentrations of some of the ions for the mammalian muscle at rest.

Resting concentrations of ions are kept thanks not only to theimpermeability of the membrane, but also thanks to the presenceof ionic pumps, that are a class of proteins involved in the activetransport of ions. Gradients in concentrations of ionized speciesconstitutes a voltage cell, and therefore causes the voltage differenceto be present across the membrane.

Since the transmembrane resistance for potassium ions is the low-est from those mentioned above (due to existence of the leakagechannels), the transmembrane voltage is mostly dictated by the ratioof the intracellular and the extracellular potassium ions concentra-tions. The approximate transmembrane voltage may be estimatedon the basis of the Nernst equation:

E rest = RT z K + F

ln[cK +out ][cK +in ]

(1.10)

where T is the absolute temperature, R is the gas constant, z isthe charge of potassium ions, and F is the Faraday constant. Orderin the ratio of concentrations in (1.10) is taken according to thephysiological conventions. After substituting all values to (1.10),with T =293K, and ratio of concentrations as in Table (1.2), we getthe estimation of the resting potential as equal to:

E rest = 98.41[mV ] (1.11)18

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Such a transmembrane potential value is established, when the ex-

citable cell is at rest it does not conduct any signals, since all theirvoltage gated ionic channels are closed.

A nerve impulse is the local disturbance in the resting potentialof the excitable cell, that propagates along the cells body withthe constant speed and the constant amplitude. Mechanism of thepropagation of the action potentials bases on the phenomenon of voltage gating.

Nerve impulse is created by the presence of a certain stimuli

usually detected by the mechanosensitive or by the ligand gatedsodium channels, that under the inuence of stimuli get open, al-lowing sodium ions to ow into the cell (direction of sodium ionsow may be deduced from Table (1.2)). This changes the trans-membrane voltage, to be more positive. Once the potential rises,voltage gated sodium channels open, and the rate of the depolariza-tion becomes faster, causing even more sodium channels to open, inthe larger vicinity of the point, where the impulse was generated.

Once the transmembrane potential still grows, voltage gated potas-sium channels begin to open, and K + starts to ow into the cell,what results in decrease of transmembrane voltage back to about-60 [mV], especially, that short after opening, Na + channels becomeinactive (do not conduct ions), and cannot open by about 2 [ms].Inactivation is essential to assure that the impulse is conducted onlyat the one direction. Details concerning the inactivation, which isthe main topic of this work, will be given in the later section.

The initial, resting distribution of the sodium and potassium ionsis reestablished by the action of sodium/potassium pump, the inte-gral membrane protein, that at the expense of energy, pumps outK + and pumps in Na+ ions. Short after the action potential, theneuron membrane reaches its resting potential and becomes readyfor the next conduction of the nerve impulse. Illustration for theidea of the action potential, taken from [12], is in the Fig. (1.5).

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Figure 1.5: The idea of the action potential. This gure shows the plot of the transmembrane voltage, denoted as E M , together with underlying openingof sodium and potassium channels, with respect to the time. The peak of theaction potential is close to the equilibrium potential, dictated by the sodium ions(E Na ), while the resting potential is close to the equilibrium potential of the

potassium ions ( E K ). Maximum number of opened sodium channels is presentunder 15 [mV] of transmembrane potential, while the maximum number of opened potassium channels is present under -35 [mV] of transmembrane voltageafter the peak of action potential (gure adapted from [12]).

1.5.2 Functional parts of the voltage gated potassium chan-

nels

In general, the voltage gated K + channels are homotetramericstructures, with each subunit containing a voltage sensor and sur-rounding the central pore [14]. All mers (subunits) have identical(or very similar) rst and second order structures. Each subunitis composed of six transmembrane regions that also show similari-ties in the structure and function. Schematic view of one subunit,adapted from [14], is shown in the Fig. (1.6). The ion selectivepart of the channels pore is formed by the link between S5 and S6

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(1.6). This subunit is the highly conserved part among the voltage

gated channels. Every third position in its rst order structure isoccupied by the positively charged arginine or lysine. Replacementof those aminoacids by others electrically neutral, had complicatedor made impossible the effects of the voltage gating. Translocationof S4 charges from one side of the membrane to the other side,when voltage across the membrane changes, was conrmed directlyin many experiments. Therefore, the S4 segment is thought to playthe role of a voltage probe that senses the transmembrane voltage.

The S4 segment is mainly composed of helical structures, giv-ing therefore a regular pattern of positive charges arrangement.This lead to the conception of possible mechanism of conforma-tional movement of this segment, termed helical screw motion.According to this mechanism described in [2] the S4 would advancescrew-like, to move each charged aminoacid to the position of thenext one, maintaining thus the charge distribution in the core, wileproducing the overall translocation of the segment. Direct proof for

the movements of the charges from the S4 segment some changesof the transmembrane voltage, was given in [15], where the currentassociated with movements of those charges was measured.

To conclude, movements of the S4 segment can be thereforetreated as conformational move, which changes the third and fourthorder structure of the channels pore, making it impermeable forions.

1.5.3 Inactivation and the idea of the ball and chain model

On the basis of previous considerations devoted to the mecha-nism of gating by electric eld, one might think, that the positivetransmembrane voltage should actually constrain the channel in itsopen state as long as the voltage is maintained. However, such be-havior was not observed in experiments. With depolarization, chan-nel opened by transition of the S4 region, but soon after it fall intolong-living nonconducting state even then, when the transmembrane

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voltage was held constant. In other words, channel inactivated it-

self it was not opened, since it was not passing ions, but it wasalso not closed in the sense that S4 region was still in position, thatshould kept the channels pore open. As give in [2], in the course of the more detailed studies following properties of inactivation werefound:

No component of the gating current was measured, that couldbe associated with the inactivation, as if inactivation would notinvolved any charged particles.

Application of the proteolytic agents, that could easily removesome accessible parts of the channels protein, completely turnedoff the inactivation phenomenon. This happened only, whenthe proteolytic agents were supplied at the intracellular part of the channel.

In the channels treated with proteolytic agents, inactivationwas restored, by addition of some of the pharmacological com-

pounds (like tetraethyl ammonium) to the intracellular face of the channel.

It has to be stressed, that at the electrophysiological recordings, nocomponent of inactivating current was found, assuring thus, that theinactivation has nothing to do with movement of charged particles,and was not directly induced by the transmembrane voltage.

These facts lead Armstrong and Benzanilla to propose a mecha-nism for inactivation called the ball and chain. On its simplest,

it may be illustrated by the sketch shown in Fig. (1.7), that wasadapted from [2]. Basically, the model assumes, that a movable partof the channels protein, is able to occlude the pore, when it becomesopen, switching off the ionic current.

Ball and chain model was proposed in the times, where the ad-vanced techniques of the genetic engineering were unavailable, andthus it was based only on the electrophysiological and pharmacologi-cal data. Therefore, at the beginning it was also a kind of convenient

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Figure 1.7: Schematic presentation of ball and chain model of inactivation. Inclosed state, ball, tethered on the chain, wanders at the cytoplasmic side of thechannel. Once the channel opens by the proper transmembrane voltage, thereceptor for the ball becomes available. Once the ball hits the receptor its blockthe channel and stops the permeation of ions (taken from [2]).

hypothesis, that tted to the experimental data, but still lacked arm structural conrmations. Nowadays, the ball and chain inacti-vation is documented and structurally conrmed feature of the ionicchannels. At the late eighties, when the Shaker 2gene was isolatedand sequenced, functional roles of ball and chain have been assignedto the parts of the channels protein. It was found, that the NH 2-terminal aminoacids are involved in the inactivation, constitutingthe inactivation gate. The terminal amonoacids can be matched intotwo functional regions. Mutations in the rst region, constituted bythe rst 20 aminoacids, slowed or even removed the inactivation. Asstated in [2], deletions in the second region, extending from 29 to

83 aminoacid position, speeded up the inactivation, while insertionsslowed it down. Therefore, it follows, that rst twenty aminoacidscompose the ball, that binds to its receptor and causes the inacti-vation. Any modication of the balls structure, makes its ttingto the receptor worse, thus partially or totally eliminating the in-

2 Mutations in this gene, in the Drosophila melanogaster, results in some aberrant move-ments of the animals legs thus the name Shaker .

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activation. On the other hand, aminoacids from the second region,

constitute the chain, that links the ball to the channels body. Itis obvious, that the shortening of the chain makes the inactivationprocess faster, since than the ball wanders closer to the channel, andwill hit the receptor faster.

At the late nineties, some structural models of the parts of thechannels protein constituting ball and chain became available. Itappears, that for different potassium or sodium channels, balls mayhave different structure (in the sense of rst, second and third order

structure), but they all share similar spatial distribution of charge,and position of hydrophobic surface domains. To conclude, Fig.(1.8), taken from [2], gives an overview of the three dimensionalstructure of ionic channel, with well marked ball and chain parts.

Figure 1.8: Model at atomic scale level of the voltage gated Shaker potassium

ionic channels with well visible ball and chain. For simplicity, only one of thefour domains is shown (gure taken from [2]).

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Chapter 2

Techniques for measuring

the electrical activity of the

voltage gated ionic channels

The purpose of this chapter is to describe the electrophysiologi-cal methods, that are widely applied in the studies of the electrical

activity of the ionic channels. Once applied, they give a result in theform of a time series of a transchannel current vs. time, which is ob-viously the measure of the channels permeability. Together with thedirect techniques of structural investigations (including the Xraydiffraction, the DNA recombinant technology and some methods of genetic engineering), electrophysiological methods became success-ful in establishing a link between the structure, and the functioningof the particular parts of the ionic channel.

2.1 The idea of voltage clamp

The intensive progress in the studies on the permeability of ionicchannels has begun in the late forties of the twentieth century, withthe development of a new experimental procedure, worked out byMarmont in [25], Cole in [16], and Hodkgin, Huxley and Katz mostlyin [24]. The group of methods basing on the idea of voltage clamp,

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has been the best biophysical technique for the study of ionic chan-

nels for over 40 years. To voltage clamp means to controll thepotential across the cells membrane. This controll is the key ideain investigations of the ionic channels permeabilities. The constanttransmembrane potential eliminates the capacity currents, and thespread of the local circuit currents. Thus it minimizes the unwanted,uncontrolled changes in the conditions of experiment, so that therecorded, transmembrane current is the direct measure of the ionicmovements across the membrane at its known, uniform potential.

Holding a constant potential across the cells membrane is not aneasy task. First of all, an electrode has to be introduced to the in-tracellular space, which is difficult, having in mind the average cellssize. Further, application of the simple voltage source is not enough,because of unpredictable local voltage drops at the electrodes and inthe surrounding medium. Therefore, the more sophisticated appa-ratus has to be used. The one described further, with two impaledelectrodes and third reference electrode is very instructive for illus-

tration of the idea of voltage clamping. Its simplied scheme isshown in Fig. (2.1), adapted from [3]. As may bee seen in the

Figure 2.1: The idea of voltage clamp with metallic electrodes (gure from [3]).

Fig. (2.1), the device has two intracellular electrodes. ElectrodeE measures the potential across the membrane with respect to thereference electrode usually immersed at the solution that surrounds

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the cell. E is connected to, so called, follower circuit, denoted at the

gure by 1. The follower circuit increases the current that staysbehind the signal it receives, and keeps the voltage of the signal un-changed. It thus makes the signal convenient for further handlingby the electronics of the voltage clamp set.

The amplitude of the signal is then compared with the commandpotential by the feedback amplier circuit, denoted in the Fig. (2.1)as FBA . Command potential is the value of the desired transmem-brane voltage, that may be adjusted according to the specied ex-

periments conditions. FBA circuit amplies the difference betweenthe actual transmembrane voltage recorded by E , and the commandpotential. It applies a certain current through I to minimize therecorded difference. Measurement of the transmembrane currenttakes place at bathgrounding electrode (denoted as I ).

As stated in [17], the electrodes used in this voltage clamp setupare usually made of a thin (20 in diameter) silver wires. Theyare placed inside the cell, or sometimes inside the cells cut, as in

the case of some long neurons, by the simple impalement. Theadvantage of such electrodes is, that they allow to keep the spaceclamp condition the spatial uniformity of the potential in the longcells, just by inserting the electrodes of the proper length.

The classical voltage clamp with wirelike electrodes is suitablefor examination of the electrical properties of relatively big cells,because of the size of the electrodes, that have to be impaled inside,to the cells body. The typical workhorse for this technique istherefore the giant squid axon, that takes from 0.5 up to 2 mmin diameter. Many smaller cells, including most of the mammalianexcitable cells, are beyond the capabilities of the voltage clamp withwirelike electrodes.

The big size of the investigated cells brings up another limitationto the measurement. Big cells have obviously high area of the mem-brane, what causes its low overall resistance. It implies that in themeasurement of current, considerable amount of the thermal noise

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(so called Johnson Nyquist noise) will be present, according to the

following formula, derived in details in [18]:

I = 4kT f R (2.1)where I is the root mean square deviation of the measured current,k is the Boltzmanns constant, T is the absolute temperature, f represents the frequency at which the signal is measured, and R isthe resistance of the noise generator. This formula was derived byNyquits for the noise generator to be a simple resistor, over whichno net macroscopic current was owing. For the case with the cellsmembrane, the noise generator is obviously the membrane itself for which the above formula may be treated as a rough estimation.As may be noticed on the basis of (2.1), the low overall resistance of the membrane of a big cell implies the high noise level at the highfrequencies of collected data. Therefore, the necessity of lteringthe collected data with a lowpass lter arises. This leads to usuallypoor time resolution of the recorded values of current, and unables

to record small currents, that are at the level of noise.To conclude with, voltage clamping with silver microelectrodes

allows generally to perform experiments only on relatively big cells.Measured transmembrane current is therefore a result of the per-meability of multitude of ionic channels, and does not give a usefulcharacteristic for the single channel. It has to be mentioned how-ever, that in the 1972 Hladky at the [19] shown, that it is possibleto indirectly estimate a single channel current, on the basis of the

voltage clamp with impaled electrodes. This was however just arough estimation with the considerable level of uncertainty.

2.2 Gigaohm seal and the patch clamp technique

As stated previously, while performing the voltage clamp withthe microelectrodes, it is generally not possible to register the timeseries of activity of a single channel, because of structure and size of

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the electrodes. The task of inventing the technique, that would allow

to record a current originated from just a single ionic channel, wasaccomplished at the late seventeens of twentieth century by ErwinNeher and Bert Sakmann (for what they received a Nobel Prize at1991). The new technique bases on the idea of isolation of a smallpatch of the cellular membrane with the aid of a thin glassmadepipette, that is lled with conducting solution, and used also as anelectrode. Isolated membranes patch has a very small size (radiusof about 10 m), and thus is characterized by the impedance at the

order of magnitude equal to 10 [G].As stated in [17], very important aspect of this technique is the

establishment of a tight seal between the pipette, and the mem-branes patch, because, as stated in [17], an incomplete seal is seenby the measuring equipment in parallel to the patch, and its noise issuperimposed onto the patch signal. By the proper adjusting of theshape of the pipettes tip and also by applying a gentle suction, theseal establishes its impedance at the level of a few gigaohms (that

is why the name gigaseal), which is enough to measure a currentat the order of picoampers. The idea of gigaseal is illustrated in theFig. (2.2), that was adapted from [17]

If an electrode is attached to the cell in a manner shown in Fig.(2.2), it is likely, that the patch of the membrane under the electrodewill contain only one ionic channel. It is than possible to record acurrent from that single ionic channel, under physiological condition(the interior of the cell remains unchanged since the membrane isnot damaged).

As stated in [3], patch clamp setup uses exactly the same ampli-er as the voltage clamp with silver microelectrodes with only onedifference voltage measuring and current passing circuits are con-nected directly to each other, as shown on the simplied schematicin the Fig.(2.3).

It follows, that the setup uses the same electrode to measure thepotential and to deliver the compensating current, what leads to

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Figure 2.2: The illustration of gigaseal idea. Size of the cell and the suctionpipette are not comparable on this picture (gure taken from [17]).

some technical difficulties in establishing the voltage clamp condi-tion. However, present, highly automated patch clamp setups, usu-ally controlled by the computer, compensate most of the technical

traps and supply reliable, unbiased measurements.The, so called, on cell measurements, presented in the Fig.

(2.3), when the pipette is tightly bounded to the surface of themembrane, is just one of the many available congurations, at whichpatch clamp may work.

For example, tight sealing between the pipette and the cellularmembrane membrane allows to completely remove the patch of amembrane from the cell simply by withdrawing the pipette. This

is known as excised patch technique. It allows to freely manipulatewith the composition of external and internal media that surroundsthe excised patch.

2.3 Experimental data and its processing

Potassium BK channel, was the source of experimental record-ings used in this work. According to [20], where the same exper-

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Figure 2.3: The schematic view of the patch clamp setup, working in the modeof on cell measurements. The follower circuit was omitted for simplicity.Transmembrane current is recorded by I , and the command potential is suppliedby the wire denoted with hat (taken from [3])

imental data were used, recordings were taken from the cell at-

tached patches of adult locust (Schistocerca gregaria) extensot tibiaemuscle bers. The muscle preparation solution was bathed in 180mM NaCl, 10mM KCl, 2mM CaCl 2, 10mM 4-(2-hydroxyethyl)-1-piperazineethenesulphonic acid (HEPES 1), pH 6.8, and the patchpipettes contained 10mM NaCl, 180 mM KCl, 2 mM CaCl 2, 10mMHEPES, pH 6.8.

As given in [20], channel current was recorded using a List EPC-7 Patch-Clamp amplier. Output was low pass ltered at 10 kHz,

and digitized at 22kHz. Level of ltering was dictated by the noiseremoval issue, while frequency of digitizing, by the NyquistShannonsampling theorem. Data were nally saved on the computer withsampling rate at the level of 10 kHz. Since the whole measurementwas 25s long, the time series of measurement has about 25 thousandspoints, at which the transchannel current was recorded. Thanks to

1 HEPES is an organic chemical buffer, that is widely used in cell culture to maintainphysiological pH.

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the previously described composition of the external and the internal

media, the transmembrane potential was held positive, what causedthe voltage gate to be permanently open. Thus the channel couldswitch only between opened and ball-and-chain-induced inactivatedstate. An example of recorded data is shown in the Fig.(2.5).

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.012

4

6

8

10

12

14

16

time [s]

c u r r e n t

[ p A ]

Figure 2.4: A sample from the measurement of the single channel current. Plotshows one hundred experimental points, connected with straight lines. Currentis at the order of picoampers.

Shape of the signal, as given in the Fig. (2.4), is quite difficultto interpret. Especially, that channel was kept in the medium, thatcaused the difference of potential across the patch of the membraneto be positive keeping thus the voltage gate open, and allowingthe channel to uctuate between two states opened (conductive) andinactivated (nonconductive) by the ball and chain mechanism.

One could expect, that the current characteristic of the ionicchannel, that uctuates between opened and inactivated state, shouldhave the binary current characteristic of a simple switch turnedoff/turned on. Unfortunately, as may be seen, the current char-acteristic is more complicated, since it is being shaped not onlyby the uctuations of the channel between opened and inactivatedstates. As usually, the signal bears this deterministic componentof the gating (which itself occurs stochasticly, but its inuence on

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the signal is deterministic, since it turns off or on). But also, signal

contains the stochastic component resulting, from thermal motions,conformational changes of the channel, etc.

To gain the information only about uctuations between openedand inactivated states, experimental data have to be ltered onceagain in the quite drastic way. This ltering bases on setting thethreshold value of the measured current, and stating:

if the current value at a certain time is above the threshold, itindicated that the channel is open and conduct ions

if the current value at a certain time is below the threshold, itindicates, that the channel is closed and does not conduct ions thus in such situation any nonzero current is treated as noise.

The assumption that we have to take, when applying the thresholdltering is, that the shape of the signal is dominated by the switchingof the channel, and not by some background noise. The idea of threshold ltering is shown in Fig. (2.5)

As my be seen in Fig. (2.5), during only 0.01 second of measure-ment, channel inactivated and activated a dozen of times. It maybe observed, that the time, that took the channel to inactivate, wasdifferent in each inactivation act. Since the measurement is about25 second long, there are about 25000 of inactivations registered. Itis enough to try to estimate the probability density distribution, of times needed for the channels inactivation to take place.

The simplest method to do this, is to count how many timeschannel was inactivated in 0.1 ms, 0.2 ms, 0.3 ms, etc. Numberof such counts for each time, divided by the total number of inac-tivations, will be the measure of probability, that the inactivationwill proceed in that particular time. Such probabilities collectedtogether will nally give the shape of the probability density curve.

Further, one can estimate the cumulative distribution for thisprobability density, using the well known relation

F inactiv (t) = t

0 p(t) (2.2)

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Figure 2.5: The idea of ltering with the threshold. Original signal, markedwith red line, and superimposed on it ltered signal marked with blue bars.As may be seen ltering returns essentially a type of binary signal constituted

of two values opened or closed.

where the probability density is denoted by p(t) and the cumulativedistribution by F inactiv (t). Therefore F inactiv (t) might be interpretedas the probability, that at the certain time, channel is still inacti-vated. For some conventional reasons, we will use further the proba-bility that at the certain time, channel is still activated (open). Thismay be calculated simply by:

F activ = 1 F inactiv (2.3)which is true, assuming, that the channel may exist only in two

states. Probability density and cumulative distribution are showntogether in Fig. (2.6). The probability density distribution wasrescaled, to t to the picture.

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Figure 2.6: Probability density distribution of the channels closure time, to-gether with the probability of surviving in the opened state. As may be notice,both lines are not smooth, what is caused by too short measurement series.It may be noticed, that the whole inactivation, occurs within the millisecondsrange.

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Figure 3.1: Initial and boundary conditions for the ball and chain model (guretaken from [2]).

logical measurements shows, that the inactivation process is voltageindependent, and therefore, that the ball does not move under theinuence of the external, electric eld present across the membrane.

If the electric eld does not induce the balls motion, than what

force pushes the ball to its receptor causing inactivation? The an-swer is, that there is no such force, that moves the inactivating ballright to its receptor. The only reasonable justication is, that theball hits the receptor accidentally, when it performs the randomwalk, which is of the same nature, as that performed by the pollengrains suspended in water, observed by the rst time by RobertBrown in 1827. Inactivating ball stays therefore the in ceaselessmotion, until it hits the receptor and becomes absorbed. Therefore,the most important conclusion to be drawn from here, is that theinactivating ball behaves like the Brownian particle.

The molecular mechanism and statistical description of the phe-nomenon, called after the discoverers name: Brownian motion,was not know until the work of Smoluchowski published at 1906.Smoluchowski connected ceaseless and direction-changing motion of a Browns particle with the action of surroundings. According toSmoluchowski, comparatively big Brownian particle, is bombarded

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with molecules of surroundings, what effects in creation of the net

force, which changes the direction of motion of Brownians parti-cle. This corresponds to the situation in ball and chain. Compar-atively big inactivating ball, constituted by twenty aminoacids, issurrounded mostly by small molecules of water, that exerts the netforce on it, making the ball wander in the random, unpredictableway, until it hits its receptor at the surface of the ionic channel.The equation of motion of a Brownian particle was introduced byLangevin in 1908. It is the stochastic differential equation, that

takes into account the viscous resistance of the surroundings, andthe stochastic force representing the effect of a continuous series of collisions between Brownian particle, and surrounding molecules.On the other hand Smoluchowski derived the equation for the prob-ability density distribution of the Brownian particle position, thatwas further generalized by Goldstein in [29].

In this work, the probabilistic description of the balls positionis chosen to be the proper one. Further it will be shown, that the

probabilistic approach gives in result the quantities, that are easilycomparable with experimental measurements of the transchannelionic current.

3.2 Random walk and its probabilistic descrip-

tion

Connection between random walk, and the probability density

distribution of the Brownian particles position is not straightfor-ward. This section justies this link, by the formal derivation of two differential equations the parabolic and the hyperbolic one,that govern the spatial and the temporal evolution of the probabilitydensity distribution of the position of Brownian particle.

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3.2.1 Unrestricted noncorrelated random walk

Mean and variance in one dimensional unrestricted random walk

Suppose we observe the particle for some measurable interval of time T . Observation is repeated, and let us nd the average dis-placement of a particle, denoted as c . Spread of the measured dis-placements let us also nd the variance, denoted as D . Now, wewould like to construct the model, which stays in agreement withmacroscopic properties ( D and c) and gives an explanation for thebehavior of the Brownian particle.

For the sake of derivation, problem will be reduced spatially toonedimensional case. This reduction is plausible, as we consider themovement that in each direction is noncorrelated with movementin other direction. This reects the isotropy of surroundings.

First, let us discretize the position ( x variable) and the time ( t variable) used in description of phenomenon. Therefore, the particlemay move along xaxis (both directions) only through unit jumpsof length . Time of one such jump is equal to . Probability, thatparticle will jump from its current position to the left is equal to q .Jump in right direction has its probability p (particle may not stayin place, therefore p+ g = 1). We may introduce a random variable,which will describe jumps:

x(i) = , jump to the left with probability q , jump to the right with probability p

(3.1)

Position of a particle after n jumps is given as:

X (n) =n

i=0(xi ) (3.2)

Position of the particle is therefore a random variable as well. Wecan nd its average value and variance. From the properties of theaverage and the variance, in view of (3.1) and (3.2) it can be derived:

E (X n ) = ( pq )n (3.3)

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with variance:

V (X n ) = 4 pq 2n (3.4)

Since we want to model the continuous movement of a particle weshould take the following limits: 0 and n . and n shouldtend to their limits in such a way, that(3.3) and (3.4) should havemacroscopic, nite, and in general non-zero quantities:

E (X n ) = ( pq )n c (3.5)V (X n ) = 4 pq 2n D (3.6)

It is impossible to achieve that, assuming that p and q are constant(unless in trivial case, when p and q are equal to 0.5, what leadsto ordinary diffusion and will be shown furhther). Having n thattends to the nonzero, nite value, we get 2n that tends to zero. Onthe other hand, taking 2n that tends to the nite, nonzero value,we get n that tends to innity. In order to avoid this, we mayassume the following denitions of probabilities p and q :

p = 0.5(1 + b) q = 0.5(1 b) (3.7)where b is a constant. Substituting (3.7) to (3.5) and (3.6) we get:

2nb c (3.8) 2n D (3.9)

Therefore, b constant is equal to c/D .Above consideration may be concluded in the statement: when 0 and n

, with the denitions of probabilities given by (3.7)

with b = c/D , we have:

n( p q ) c (3.10)4 pq 2n D (3.11)

At a unit time, number of jumps n is equal to 1/ , where is thetime that takes the particle to perform one jump. Therefore (3.10),

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and (3.11) may be rewritten as follows:

( pq ) c (3.12)

4 pq 2

D (3.13)These results will be useful in construction of the Smoluchowskiequation in next section.

Derivation of the parabolic diffusion equation

Smoluchowski equation governs the probability density of parti-cles position, which is dened as:

v(x, t ) = P (X n = x) (3.14)

where t = n. Probability of nding the particle at the position x and time t+ is equal to:

v(x, t + ) = pv(x , t) + qv(x + , t) (3.15)Since we are interested in nding the form of v(x, t ), functionspresent in the equation (3.15) will be expanded to the Taylor se-ries, as follows:

v(x, t + ) = v(x, t ) + vt (x, t ) + O( 2) (3.16)

v(x , t) = v(x, t ) vx (x, t ) + 12

2vxx(x, t ) + O( 3)

Truncation of both series is done as such, due to stability require-ments. Substitution of (3.17) to (3.15) with some simple rearrange-ments leads to to following equation:

v(x, t )t

= (q p)

v(x, t )x

+ 0 .5 2

2v(x, t )

x 2 (3.17)

Term (q p) tends to c as in (3.12). Term 2

tends to D be-casue of (3.13). Final form of the equation that governs the densitydistribution of the Brownians particle position is given as:

v(x, t )t

= cv(x, t )

x + 0 .5D

2v(x, t )x 2

(3.18)

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subsequent time steps. To motivate this idea, it is quite reasonable

to think, that a particle traveling at one direction, will have a biggertendency to keep it rather than to reverse it.

Derivation of hyperbolic diffusion equation

We begin by assumption, that a particle starts its movement fromthe origin. The density distribution of its position denoted as v(x, t )takes the following denition: v(x, t ) = (x, t ) + (x, t ) where:

(x, t ) gives the probability of nding a particle at position x and time t moving to the right

(x, t ) gives the probability of nding a particle at position x and time t moving to the left

We also specify the probability p of continuing motion in given di-rection and the probability q of reversing the direction of motion.Those probabilities are position and time independent (explained byhomogeneity of surroundings), and its sum is equal to one (Brownian

particle may not stand still).Time and one dimensional space are made discrete in the way

that denotes the length of a unit jump, while denotes the timeneeded to perform it. Evolution of (x, t ) and (x, t ) is describedby the following coupled system of equations:

(x, t + ) = p(x , t) + q (x , t) (3.20) (x, t + ) = p (x + , t) + q(x + , t) (3.21)

Equation (3.20) may be explained as follows: probability that par-ticle is at (x, t + ) and moves to the right is equal to the sum of probabilities:

p(x , t) that a particle was moving to the right when hasbeen at (x , t), and continuing its motion, fell into position(x, )

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q (x , t) that particle was moving to the left when has beenat ( x , t), and reversing its motion, fell into position ( x, )

Similar explanation may be given for 3.21.Since we are interested in nding (x, t ) and (x, t ), (3.20) and(3.21) are will be expanded into Taylor series:

(x, t ) + t

= p(x, t ) p x

+ q (x, t ) q x

(3.22)

(x, t ) + t

= p (x, t ) p x

+ q (x, t ) q x

(3.23)

Adding to the (3.22) q(x, t ) and to (3.23) q (x, t ), and using p+ q =1 we may write:

q(x, t ) + t

= p x

+ q (x, t ) q x

(3.24)

q (x, t ) + t

= p x

+ q (x, t ) q x

(3.25)

Now a word of comment on p and q . It is intuitive that as 0correlation of movements direction between consecutive jumps goes

to one. Therefore, at the the limit, p 1 and q 0. Thus forsmall it should be sufficient to take only the rst elements of Taylor series expansion of p( ) and q ( ):

p( ) = 1 + O( 2)q ( ) = + O( 2)

where is the rate of reversal of directions. Substituting followingdenitions to (3.24) and (3.25), and dividing both equations by

(/ = where is the particles speed), we get:

(x, t ) + (x, t )

t =

(x, t )x

+ (x, t )

x + (x, t )

(x, t )x

(3.26)

(x, t ) + (x, t )

t = +

(x, t )x

(x, t )x

+ (x, t ) + (x, t )

x (3.27)

Since we want to describe the continuous motion of a particle, wetake the following limit: 0 and 0 (but still / = whichis nite and nonzero). Some elements from (3.26) and (3.27) that

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tends to zero are ommitable. We therefore have:

t

+ x

= + (3.28) t

+ x

= + (3.29)Since we are interested in v(x, t ) = (x, t ) + (x, t ), we add andsubstract above formulas, to get after rearrangements:

( + )t

+

x = 0 (3.30)

( )t

+ +

x =

2 (

) (3.31)

Differentiating (3.30) with respect to time, and (3.31) with respectto position, well be able, after addition, to rule out the unwanted( ) term:

2( + )t 2

+ 2( )

xt = 0 (3.32)

2( )t 2

+ 2( + )

xt = 2

( )x

(3.33)

Multiplying (3.33) by , substituting 2 ( )

x by 2

( + )t on the

basis of (3.30) and adding both equations, we nally get: 2v(x, t )

t 2 2 2v(x, t )

x 2 + 2

v(x, t )t

= 0 (3.34)

where v(x, t ) is the probability density distribution of the particlesposition, is the speed of a particle, and the rate of reversal of particles traveling direction. Dividing (3.34) by 2 we get:

12

2v(x, t )t 2

2

2 2v(x, t )

x 2 +

v(x, t )t

= 0 (3.35)

Quantity 1 / 2 is the correlation time (the average time after whichparticle will reverse its direction of motion). Quantity 2/ (2) isequal to the diffusion coefficient. Dening the correlation time with and diffusion coefficient as D we nally get the ultimate form of the differential equation, that governs the evolution of probabilitydensity distribution of particle that undergoes persistent randomwalk:

2v(x, t )

t 2 D 2v(x, t )

x 2 +

v(x, t )t

= 0 (3.36)

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3.3 Details of diffusional approach to ball and

chain model

The diffusional approach to the ball and chain model bases onthe following assumptions:

reduction to one spatial dimension position of the ball, at the moment of opening of the channel, is

given between 0 and L, where L is the length of the polypeptide

chain

every collision with the channels inlet is effective once theball strikes the channel it blocks it

length of the chain generates the length of the region at whichthe ball may wander; chain has no any other inuence on theballs movement

It uses the following differential equations:

Parabolic equation of diffusionAlthough it implies the innite speed of propagation, parabolicequation of diffusion is easily solvable analytically and behaveswell under numerical studies. It produces results, that are intu-itive and easy to interpret; for long times, its parabolic defect(i.e. innite speed of density propagation) is omitable. It maybe used as a sort of reference for other approaches.

Hyperbolic equation of diffusion

This equation implies nite speed of propagation; it is the gen-eralization on short times of parabolic diffusion equation[28].For sufficiently long times, both hyperbolic and parabolic equa-tions give the same results. This equation is more difficult inanalytical and numerical treatment than the parabolic one;

Diffusional approach to ball and chain model, allows to estimatethe probability density distribution of the balls position as a func-

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tion of time and space. Based on that the following quantities, easily

comparable with electrophysiological data, may be derived:

Survival probability dened as:S (t) =

L

0f (x, t )dx (3.37)

which indicates the probability that the ball is still in regionx (0, L) (where L is the length of the polypeptide chain),and therefore, that the channel remains open (survives at itsopened state); plot of S vs. time shows the shape of relation of

average ionic trans-channel current vs. time, when the channelis open and inactivates;

First passage time dened as:F (t) =

d(1 S (t))dt

= d(S (t))

dt (3.38)

which is the probability density distribution for times of chan-nels closure. The above formula my be justied by noting that1 S (t) indicates the probability if the time t was bigger orequal to the time that ball needed to reach the channels inlet.Therefore 1 S (t) is the cumulative distribution for so calledrst passage time.To get its probability density distribution,1 S (t) has to be differentiated with respect to time.

To stay in agreement with models assumptions the initial and bound-ary conditions have to be imposed as specied in the Fig. (3.2):

Important parameter in the diffusional approach is the lengthof the chain, denoted by L. Its value is estimated on the basis of

aminoacid sequence of the chain. According to [31] it is equal to2.1 10 8[m].

3.4 The parabolic diffusion

The following IBVs problem was posed

D 2f (x, t )

x 2 +

f (x, t )t

= 0, x (0, L), t (0, + )(3.39)48

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While the solution obtained through reection and superposition reads

f (x, t ) =

n =00.5L sin

(2n + 1) 2L

x0 sin(2n + 1)

2L x exp

(2n + 1)2L

2Dt (3.44)

Series (3.43) shows fast convergence for short times (only a few terms are needed

to get satisfactory result), while series (3.44) shows excellent convergence for

long times.

3.4.1 Details concerning numerical solution of the parabolic

ball and chain

In a many problems involving the differential equations, an an-alytical solution is very hard to nd. It happens usually, when thedifferential equation is nonlinear, or when the boundary conditionsare time dependent (so called moving boundary problems). More-over, hard difficulties may arise in multidimensional problems, espe-cially, when the spatial region is characterized by some complicatedgeometry, what is frequently met in the engineering applications.At such situations, the numerical method of solution is the only

possible choice.Fortunately, the parabolic ball and chain problem, does not in-

volve the nonlinear operator, and uses rather standard boundaryconditions. Thus it is quite easily solvable, by some standard meth-ods, like separation of variables. Therefore, the numerical solutionplays only an auxiliary role here. It helps to verify the correctnessof the analytical solution, especially that the verication by directsubstitution is not so straightforward, since the analytical solution

is given in a form of an innite series.What is more fortunate, parabolic diffusion equation behaves

very well under the numerical treatment. First of all, there are somenumerical schemes developed for it, that are stable and consistent.Moreover, parabolic problems are easily scalable, in a sense, that thenumerical solution obtained for a non-dimensional variables, may beeasily scaled, to cover cases with the different diffusion constant, orwith the different length of the spatial region.

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Having the problem posed as in (3.393.42), it is convenient to

introduce the following new, rescaled, dimensionless variables X andT :

X = xL

, T = DtL2

(3.45)

Spatial and time derivatives from (3.39), rewritten with respect tothe new variables, take the following form:

f (x, t )x

= f (X, T )

X dX dx

= f (X, T )

X 1L

2f (x, t )x 2

= f (X, T )

X 1

L2(3.46)

f (x, t )

t =

f (X, T )

T

dT

dt =

f (X, T )

T

D

L2 (3.47)

Thus, the parabolic ball and chain problem, becomes:

D 2f (X, T )

X 2 +

f (X, T )T

= 0, X (0, 1), T (0, + )(3.48)

f (X, T = 0) = (X X 0) (3.49)

f (X, T )|X =0 = 0 (3.50)

f (X, T )X X =1

= 0 (3.51)

where the spatial variable X belong to the interval X (0, 1).Normalized ball and chain problem, may be therefore easily treatednumerically. For this purpose, the Crank-Nicolson scheme may bechosen. As an implicit scheme, at the expense of the computationaltime needed for simultaneous solving of the set of linear algebraic

equations, it allows to choose the time steps, that are bigger than inany explicit method reducing thus the total volume of calculations.The scheme is second order correct correct with respect to time andspace.

Application of the Crank-Nicolson method, begins with discretiza-tion of time and space variables, as shown in Fig. (3.3)

Next step in the implementation of the Crank-Nicolson method,is the construction of the nite difference representations of the

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Figure 3.3: The idea of discretization of space and time variables. i indexes theconsecutive space points, while j the time levels. k is the length of the timestep, while h is the length of the spatial step of the mesh. Each consecutivetime level is evaluated on the basis of the set of the linear algebraic equation,constructed on the basis the previous time row. The rst time row is obviouslyknown on the basis of initial condition.

derivatives, present in (3.48). According to the [22], it they takethe following form:

df (X, T )dT

= F i,j +1 F i,j 1

2k (3.52)

d2f (X, T )dX 2

= (F i+1 ,j +1 2F i,j +1 + F i 1,j +1 )

2h2 +

(F i+1 ,j 2F i,j + F i 1,j )2h2

(3.53)

where F is the value of numerical solution at the given node of thenet, and the rest of the signs follows the convention shown in theFig. (3.3). After substitution of (3.52) and (3.53) to (3.48), withsome simple rearrangements, we get the following linear, algebraicequation, with three unknowns:

rF i 1,j +1 +(2+2 r )F i,j +1 rF i+1 ,j +1 = rF i 1,j +(2 2r )F i,j + rF i+1 ,j(3.54)

since F i 1,j ,F i,j and F i+1 ,j are known for j = 0 from the initialcondition, or from the previous calculation for any different j . Inthe (3.54), r = k/h 2.

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Taking equations (3.54) for each i (2, 1/h ) at the given timelevel j , the set of N 2 linear equations with the N unknows isformed1. To make such set of equations to have a unique solution,two supplemental equations have to be added, to equate the numberof equations and the number of unknowns. Fortunately, on thebasis of the two boundary conditions, namely (3.50) and (3.51), twomissing equations may be formed. For (3.50):

F i=1 ,j = 0 (3.55)

and for the (3.51)F i= N 1,j F i= N,j = 0 (3.56)

The last equation is derived on the basis of the difference quotient,that approximates the derivative present in the equation (3.51). Foreach time level, we have now a system of N linear equations withN unknowns, which may be arranged in the form of the tridiagionalmatrix, with the rst (i=1) and the last (i=N) equations dictatedby the boundary conditions, and the remaining equations dictated

by the form of 3.54. Such set of equations, can be nicely arranged inthe matrix notation. For example, for N = 5, we have the followingset of equations:

b1 c1 0 0 0 F 1 z 1a2 b2 c2 0 0 F 2 z 20 a3 b3 c3 0 F 3 = z 30 0 a4 b4 c4 F 4 z 40 0 0 a5 b5 F 5 z 5

(3.57)

First and last equations are dictated by the boundary conditions.Therefore, according to (3.55), b1 = 1, c1 = 0 and r1 = 0. Also,according to the (3.56), a5 = 1, b5 = 1 and r 5 = 0. Simmilary, onthe basis of (3.54), remaining (that is for i (2, N 1), ai , bi , ciand z i coefficients, are equal to:

a i = r (3.58)1 Where N is the number of spatial nodes in each time step ( N = 1 /h + 1).

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bi = 2 + 2 r (3.59)

ci = r (3.60)z i = rF i 1,j + (2 2r )F i,j + rF i+1 ,j (3.61)

In practice, number of the equations N is always much larger thanve. In such case, all the matrices from (3.57) are of course larger.

As the method of solution for the tridiagonal set of equations(3.57), the standard Gauss elimination method with pivoting waschosen. It is described in details in [23]. This is not the fastest

method available for solving the sparse system of linear equations,but it is easy to implement, in most cases stable, and especially forthe tridiagonal systems, requires small amount of computer opera-tional memory.

Numerical solution with Dirac delta as the initial condition, mighthave stability troubles at early times, when deltas peak is sharp.To avoid this, we initially used at early time stages analytical so-lution for corresponding Cauchy problem (given in [27]), and onceboundaries became important, we switched to pure numerical scheme.

3.5 The hyperbolic diffusion

In this case the following IBVs problem is posed

2f (x, t )

t 2 D 2f (x, t )

x 2 +

f (x, t )t

= 0, x (0, L), t (0, + )(3.62)

f (x, t = 0) = (x x0), f (x, t )t t=0 = 0 (3.63)

2 f (x, t )

t x=0= 2 v

f (x, t )x x=0 f (0, t ) (3.64)

f (x, t )x x= L

= 0 (3.65)

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where D stands for the diffusion coefficient, for the correlation

time, v for the mean speed of the inactivating ball, and L for thelength of the chain.

It the case of hyperbolic equation, formulation of absorbing andreecting boundary condition requires some comments.

At the reecting boundary condition, we demand to have uxequal to zero. In case of hyperbolic diffusion the ux is given by thefollowing relation:

J hip (x, t ) =

D

f (x, t )

x

J hip (x, t )

t (3.66)

Since at x = L, J (L, t ) = 0 for all t, its time derivative is alsozero. Therefore (3.66) reduces effectively to the form given in theformulation of the problem.

The absorbing boundary condition for hyperbolic operator is to-tally different in comparison with that used in parabolic diffusion.Posing f (x, t )|x=0 = 0 in hyperbolic case, causes the solution to takethe negative values, which is unacceptable, since probability density

must be positive denite. Correct formulation of absorbing bound-ary may be justied by considering correlated random walk fromwhich hyperbolic equation of diffusion may be derived.

3.5.1 The absorbing boundary

In this derivation, one usually considers a particle moving to theleft or right, for which, we may dene the following probabilities:

(x, t ) which is the probability that at time t and position x,we my nd particle that arrived there from left;

(x, t ) which is the probability that at time t and position x,we my nd particle that arrived there from right;

Discretizing time and space, we make particle to move at jumps (of length and time T ). After each jump, particle may change itsdirection of motion (with probability q) or keep it (with probabilityp). Since particle may not stand still, p + q = 1.

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Evolution of (x, t ) and (x, t ) may be described by the following

set of equations:

(x, t + T ) = p(x , t) + q (x , t) (3.67) (x, t + T ) = p (x + , t) + q(x + , t) (3.68)

In our boundary condition, there is no particle, that arrived therefrom the left side (from position x ), therefore (x, t + T ) isequal to zero. Further, (x + , t) has to be also equal to zero,since at point x + there is no particle that arrived there from

the left (because particle can not escape from absorbing boundarycondition). Therefore the above set of equations reduces to:

(x, t + T ) = p (x + , t) (3.69)

Expanding (x, t + T ) and (x + , t) into the Taylor series, with p = 1 T , where is the rate of reversal of direction of particlesmovement we get:

(x, t ) + (x, t )

t T = (x, t ) +

(x, t )

x T (x, t )

T

(x, t )

x(3.70)In the limit, when T 0 and 0, such that /T v, where

v is the nite speed of a particle, (3.70) reduces to:

(x, t )t

= (x, t )

x v (x, t ) (3.71)

Since, as mentioned previously, at the absorbing boundary onlythe particle that travel to the left is present, we have (x, t ) =f (x, t )

|x=0 . Moreover, from the derivation of hyperbolic equation

we have: = 12 where is the correlation time (the mean timeafter which particle will reverse its direction of motion). Rewriting(3.71) we get:

2 f (x, t )

t x=0= 2 v

f (x, t )x x=0 f (0, t ) (3.72)

which is the absorbing boundary condition posed at the formulationof the problem. It is worth to notice, that at the limit, when 0,

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this symmetry, the following equality holds:

f 1(x, t )x x= L

= f 2(x, t )

x x= L(3.75)

If we dene the function f (x, t ) as

f (x, t ) = f 1(x, t ) + f 2(x, t ) (3.76)

we may notice, thatf (x, t )

x x= L= 0 (3.77)

Thus the f (x, t ) reads:

is the solution of a linear hyperbolic diffusion equation, for itis the sum of two solutions of that equation.

reproduces the initial condition of the ball and chain problem,because the second Dirac delta peak from (3.74)is outside theregion (0, L) at t = 0.

keeps the absorbing boundary condition at x = 0, since bothf 1(x, t ) and f 2(x, t ) also keep it.

keeps the reecting boundary condition at x = L, as shownabove.

Therefore, f (x, t ) is the solution to hyperbolic ball and chain prob-lem.

3.6 Comparison with experimental data

Our theoretical results can be compared with electrophysiologicaldata 2. Experimental data were sampled with the time resolutionequal to 0.1 [ms ] and the whole experiment was 25 [s] long. Duringwhole measurement channel activated and deactivated many times,thus we were able to nd mean experimental rst passage time of inactivating ball.

2 thanks to the courtesy of Peter N. R. Usherwood, University of Nothingam, UK

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Based on solution of parabolic equation, with aid of denitions

(3.37, 3.38), theoretical rst passage time was found. Fig. (3.5)

Figure 3.5: Experimental (adult locust muscle K + channels) and theoretical(parabolic diffusion) rst passage time

shows the rst passage time calculated from experiment, with thecorresponding rst passage time obtained theoretically, from theparabolic diffusion equation. For initial position x0 = 0.4L (84 10 10[m]) with diffusion coefficient D = 1.210 9[cm2/s ] theoreticalcurve was best tted to experimental data.

In Fig.(3.6) the experimental rst passage time is compared withthe one calculated from the hyperbolic diffusion approach. Theo-

retical curve was tted to experimental data, for the same initialposition of the ball (x0 = 0.4L) as in the case of parabolic tting.For diffusion coefficient D = 1.3 10 9[cm2/s ] and correlation time = 2 10 5[s] we get the best agreement. As may be seen for thelong times the hyperbolic rst passage time behaves similarly toparabolic one. Noticeable difference is observed for short times.

Hyperbolic rst passage time predicts that the probability of blocking the channel at the time t (0, x0/ D/ ) is equal to zero

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Figure 3.6: Experimental (adult locust muscle K + channels) and theoretical(hyperbolic diffusion) rst passage time

due to the nite speed of the ball which is dened as v = D/ .As may be seen in Fig.(3.6), too low resolution of experimental datamade no possibility to verify if such a lag appears also in themeasurements (experimental points were denoted with circles).

3.7 Calculation of the diffusion coefficient of the

ball

Knowing the diffusion coefficient for a single aminoacid in water,we are able to estimate value of diffusion coefficient for inactivatingball, taking the corrections due to size and mass of the ball.

Diffusion coefficient for a single aminoacid rest is of the order of 10 5 [cm2/s ] (as given in [31]). An inactivating ball is character-ized by different diffusion coefficient, since it has larger mass (whatinuences average velocity) and different area (and thus differentcollisions frequency with the molecules of surroundings) comparing

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to aminoacid rest.

Microscopically, diffusion coefficient for an inactivating ball, is equalto:

D = v , (3.78)where is mean free path, and v is the average velocity of the ball.Mean free path may be dened as:

= vk

(3.79)

where k is the frequency of collisions with the molecules of surround-ings. Therefore, reformulating (3.78) we get:

D = v2

k (3.80)

Frequency of collisions k is linearly proportional to the surface areaof the ball, that scales with balls mass as follows:

k area mass 2/ 3 (3.81)Average velocity of the ball, denoted as v, scales with balls massas:

v m 1/ 2 (3.82)

(providing that the kinetic energy does not change) Concluding, wehave the diffusion coefficient, that scales with the mass of the ballas:

D m 5/ 3 (3.83)