Excitable cells In a sense all the specialized cells o - PDF document

In a sense all the specialized cells of a complexorganism like the human body are excitable sincethey all perform specific actions on receipt of ex-ternal signals. Here, however, we confine atten-tion to cells that perform their functions by ElectrocardiographyThe heart undergoes cyclic electrical activity whileperforming its pumping function. We infer this bymeasuring the electric and magnetic fields of the W. Einthoven Physics of the Human Body79Chapter 9Excitable cells1.Generally the signals are chemical, veloping a net electric charge. In fact the heartremains electrically neutral but exhibits an overalltime-varying elec-tric multipole mo-ment. Although allmultipoles are pre-sent in an asym-metric chargedistribution likethe heart, the di-pole moment dom-inates because ithas the longestrange.The pioneers of electrocardiography thereforemodeled the beating heart as an electric dipole that varies cyclically with time, both in mag-nitude and direction. This model was called “vec-tor cardiography”. Since we now know a lot moreabout hearts than these pioneers, we shall con-struct a simplified, but realistic, physical model ofa heart that produces an appropriately time-vary-ing dipole moment. To begin with we shall suppose the heart is spheri-cal and that its surface can be represented as alayer of electric dipoles, as shown below. We thencalculate the electric field from arbitrary distribu-tions of surface dipoles, so that we can relate theelectric potentials measured by electrodes at-tached to a patient’s skin to the time variations ofthe charge distribution at the surface of the heart.Of course the heart changes shape and size as thecoronary muscles contract and relax, but we as-sume this affects distant electric fields only in aminor way.Excitable cells (such as heart muscle) activelypump sodium ions from their interiors. This pro-duces a net polarization of charge at the surfacewith more negative charges inside the cell thanoutside. (These individual cell polarizations cancelout everywhere but at the heart’s exterior surfaceor pericardium.) Thus the heart begins its cycle ina state of polarization. The muscular contractionbegins at the sino-atrial (SA) node, and is accom-panied by a wave of depolarization that flowsthrought the atrial muscle tissue. Once the atriaare depolarized the atrio-ventricular (AV) node isstimulated and Purkinje cells cause the depolariza-tion of the ventricles from the bottom up. (Seebelow.) Electrocardiograph trace 80Chapter 9:Electrocardiographylike the cow in the Preface.5.More properly, the contraction is initiated by the depolarization of the cell membranes which initiates aflow of calcium ions that actually triggers the contraction of muscle. Our model of what happens during the heartbeatwill thus be that the upper hemisphere (repre-senting the atria) depolarizes from the North Poledownwards. Once the atria are depolarized, thesouthern hemisphere (the ventricles) depolarizesfrom the South Pole upward. The depolarized tis-sues become recharged for their next beat by aslow, diffusive process that nonuniformly repolar-izes the exterior of the sphere.The electrostatic potential arising from an isolatedelectric dipole at the origin is This is easily seen by calculating the potential froma point charge + located at0, and one of charge – located at0, as shown below.The resultant potential is so if is much smaller than we keep only theleading term. The dipole moment is the vectorwhose magnitude is and points in the directionfrom the negative to the positive charge.The potential resulting from a distribution of elec-tric dipoles is Obviously, from Gauss’s Law the potential outsidea neutral sphere of uniform surface dipoles van-ishes identically. So we may say the initial potential(when the heart has a uniform surface dipole dis-tribution) is zero or close to it, at any exterior pointof observation.Our next task is to calculate the potentials result-ing from fractions of the sphere, as seen by anexterior observer. To keep things simple we observefrom a point directly below the sphere. The surfacecharge multiplied by the surface thickness (thatis, the dipole moment per unit area) is related tothe potential across a capacitor. By Gauss’s Law,the electric field inside a parallel plate capacitor ofsurface charge is , so the potential across the capacitor is in other words the dipole moment per unit area isthe membrane potential when the polarization isi.e. just before the muscles contract.Hence the potential from the heart is sin cos Physics of the Human Body81Chapter 9Excitable cells where is the membrane potential. The angles and are functions of time, in our model given , 0 0 , t  2 and , 3 2  t , t  2 The symmetry makes it possible to perform theintegral in closed form, giving cos cos .The predicted potential, as a function of time, isshown below:With a resting membrane potential of 60–100 mV,and taking =5 cm, =15 cm, the predicted mag-nitude of the spike is about 1.5–2.5 mV. This istypically what is measured in electrocardiograms. NeuronsThe neurons are cells specialized to transmit sig-nals rapidly over long distances. A simple experi-ment yields a lower bound on the speed of neuraltransmission: one person drops a dollar bill (heldvertically just above a second person’s hand) andthe second person tries to catch it between histhumb and forefinger. If the bill is dropped withoutwarning, the second person is likely to just misscatching it. Since the time for an object to fall 6inches starting from rest is 0.17 seconds, we deducethat the nerve impulse to the fingers had to travelfrom the brain—a distance of about 1 m—in atime no greater than 0.17 sec. That is, the speedof nerve impulses is at least 6 m/sec. We can dobetter than this, since in order to decide to moveour fingers we must see the dollar bill falling.Moreover, we know from the way a motion picturefools the eye that we can process visual informa-tion no faster than 10 frames per second (the filmspeed at which we see a flickering rather than acontinuously moving image). Thus if we subtract0.1 seconds to allow for visual processing, we esti-mate that the nerve impulse must travel 14 m/sec.In fact, Helmholtz measured the speed of the im-pulse in a frog’s sciatic nerve in 1850, finding avalue of 27 m/sec.A typical neuron consists of a cell body, with fiberscalled dendrites that bring signals to the cell, anda long fiber called the axon that carries the outputsignal. The neuron illustrated on the followingpage is a motor neuron—that is, it controls musclefibers.The axon may be modeled as an electrical circuit(a kind of transmission line) as also shown on thefollowing page. If we think of the axon as a longtube surrounded by cell membrane material andfilled with electrically conductive stuff, there willbe electrical resistance along its length, as well asfrom the outside in. The membrane, being aninsulator between two conductive regions actssomewhat as a capacitor. The longitudinal resis- 82Chapter 9:Neurons tance is modeled by the resistors , whereas theinward resistance is modeled by the resistors The membrane capacitance is modeled by thecapacitors . If we consider a voltage measured ata node between two longitudinal resistors, it obeysthe differential/difference equation where and are time constants of the system.Now it is possible to show that under certainconditions on the numbers the second differ-ence 2is a negative operator. Thus we expect that thetime derivative of will be negative and an initialnon-zero value will fall to zero. That is, the net-work as shown is dissipative. However, if there isanother element in the circuit in parallel with theresistor , that can act like a battery—that is, if itdepends on the voltage across it in a certain non-linear manner—then the voltage can grow withtime. In fact, if we take the limit of the preceding equa-tion as the spacing between lumped elements be-comes small, the equation turns into a partialdifferential equation of the form  2Vx2  Physics of the Human Body83Chapter 9Excitable cellsin the sense that an appropriately defined inner product ()is always negative. which is a nonlinear diffusion equation. Suchequations are known to possess solutions in theform of travelling pulses, such as shown below. Sensory cellsClosely related to neurons are the specialized cellsthat transform external stimuli into nerve im-pulses. These include the taste buds and olfactorycells, that respond to particular chemical stimuli;the rod and cone cells of the retina, that respondto light; the cells of the basilar membrane of thecochlea (inner ear), that respond to sound vibra-tion; and various pressure and thermal transducersin our skin, that provide our tactile sensations.They differ in their trigger mechanisms, but theygenerate action potentials by the same mechanismas the axon—namely depolarization of an initialseparated ionic charge. Membrane potentialsWe can estimate the electrostatic potential acrossan excitable cell’s membrane as follows: we recallthat the thermodynamic force per unit volume,opposing a concentration gradient of some solute if the solute is an ion such as Na, if there were anelectric field opposing the thermodynamic force(and thereby creating the concentration gradient)we would have, in equilibrium, Since we may relate an electric field to a voltagegradient, we require a voltage difference ln!."n2n1 to create a difference in concentration,across—say—a semipermeable membrane. Thuswe may associate with such a concentration differ-ence (in the absence of an external electric field)the (thermodynamic) potential energy difference ln Assuming an ion pumping mechanism, that main-tains a concentration difference across a perme-able cell membrane, we expect to find voltagedifferences. Suppose the concentration differenceis a factor 20 ; then at room temperature, 300 , the potential across the cell mem-brane is about 340 V 75 mV. This is indeed inthe range of actually measured potentials.We may ask how much power is required tomaintain an ionic concentration gradient acrossa membrane. An electric current densitymoving against a voltage difference requirespower n2  n1 ln!."n2n1 #/$ . 84Chapter 9:Sensory cells where is the thickness of the membrane and its surface area. Now, for (valence 1) ions of (num-ber) density the charge density is and the(electric) current is then 2 n from which we can express the mobility (that is, theviscous drag coefficient) in terms of the electricalconductivity The pumping power is thus n ."n2n1 Consider a cell of radius 10 and membrane thick-ness 50 nm, whose electrical conductivity is thatof seawater, mho , and taking the factor n ."n2n1 20 20 20 5.4we find the ohmic power dissipation to be 2.7 Watts.This is enormous—enough to boil the cell in lessthan 0.01 second! What could be wrong? The onlyfactor where there is some play is the electricalconductivity. Manifestly, to bring the power dissi-pation down to something a cell can handle wemust assume the conductivity is some 10 smallerthan that of seawater. In fact, a typical averagecurrent during a single nerve pulse is about 1.6 pA.The pulses last about 1 ms, so the power is about64 pW, or about 16 times the average basal meta-bolism of a mammalian cell. Energetics of muscle tissueMuscle tissue consists of basic contractile unitscalled sarcomeres, as shown below. The sarcomeresare attached end to end, with the demarcationsmarked by Z-plates. The contractile unit is some-thing like a linear motor or ratcheting jack, thatcan move only one way. It contains interleavedfibers: actin fibers attached to the Z-plates, andmyosin fibers between them. Each myosin fiber has about 100 cross-bridges, thatrepresent the active elements. During contractionthese links move along the actin fibers, draggingthe Z-plates closer together and thereby shorten-ing the sarcomere. The maximum longitudinalcontraction is about 20%. A cross-bridge, or myosin “head”, is a tiny ratchet-ing motor that adheres to successive sites on theactin fiber. (It is thought that these sites are actu-ally turns of the helix.) This is shown schematicallybelow. Note that the head is actually a double Physics of the Human Body85Chapter 9Excitable cells Shown at the right is an electron micrograph of amyosin head. If one uses one’s imagination liberallyone can see the helical structure of the actin fiber(around the green line), as well as the two projec-tions (below the cyan lines). The adhesive force of a myosin head is about5.3 Nt. To see what this means in energeticterms, imagine that the force is exerted over adistance of 40 Å; the energy is then about 2 10 J, or 0.1 eV. Since typical chemical bonds in stablemolecules have much larger energies (of order1 eV or greater), we see the biochemical machin-ery that has evolved to enable motility employsforces weak enough that parts can attach anddetach repeatedly without damaging the mole-cules themselves. It is basically the same principleas “sticky notes”: the mucilage forms weak bondsbetween surfaces, that can be formed and pulledapart without tearing the paper.On the other hand, the energies associated withthe attachment of the ratcheting machinery mustbe substantially larger than , or else the ratch-ets would be so buffeted by random thermal mo-tions that they could not function to provideunidirectional motion. Changes in electron density, or of dielectric con-stant (at the business ends of the molecules) pro-mote the attachment and detachment of themyosin heads. It is believed the double structurealternates attachments in order to “walk” alongthe axin helix.Presumably, successive attachments are fueled byATP molecules, as with other endoergic processesat the cellular level. The trigger for muscle contraction and relaxationis the transport of calcium into and out of the cell.The precise mechanism is not known at the timeof this writing.The 100 cross-bridges on a myosin fiber exert acombined tension of 5.3 Nt; with a fiberpacking density of about 5.7, muscle tis-sue can exert a tension of 3 Nt/m Actin fiber 86Chapter 9:Energetics of muscle tissue