A mechanical model of muscle mechanics

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Lorenzo Marcucci

To cite this version:

Lorenzo Marcucci. A mechanical model of muscle mechanics. Life Sciences [q-bio]. Ecole

Polytech-nique X, 2009. English. �pastel-00004880�

1 Mus le physiology and early modeling 1 1.1 Mus le physiology. . . 1 1.2 Me hani alexperiments . . . 7 1.3 Me hani almodeling . . . 14 1.3.1 Hill1938 model . . . 14 1.3.2 Huxley 1957 model . . . 19

1.3.3 Huxley and Simmons1971 model . . . 22

2 Power Stroke 29 2.1 Introdu tion . . . 29

2.2 Re ent Models. . . 30

2.3 New model of a power stroke. . . 38

2.3.1 Generalshape of the energy . . . 38

2.3.2 Double paraboli approximation . . . 41

2.4 Deterministi ase: globalminimum . . . 43

2.4.1 One ross-bridge . . . 43

2.4.2

N

ross-bridges . . . 45

2.5 Sto hasti ase:

N

ross-bridges atnite temperature . . . 48

2.6 Variable power stroke size . . . 50

2.7 Negativeslope of the

T

2

(δ)

urve . . . 57

2.7.1 Inhomogeneity in a hain of sar omeres . . . 62

2.7.2 Distributionof the atta hment positions . . . 73

2.8 Rate of fast tensionre overy . . . 76

2.9 Dis ussion . . . 85

3 The atta hment-deta hment pro ess 87 3.1 Introdu tion . . . 87

3.3.2 Prost et al. model . . . 98

3.3.3 Cooperative and non- ooperative motors . . . 101

3.4 Dire t simulationof a set of sto hasti equations . . . 107

3.5 Thermalrat het . . . 109

3.6 Cooperative Magnas omodel . . . 111

3.6.1 Governing equations . . . 111

3.6.2 Ben hmark problem:

K = 0

. . . 113

3.6.3 Cooperative Magnas omodel with

K

6= 0

. . . 116

4 Full ross-bridge y le 123 4.1 Introdu tion . . . 123

4.2 Numeri al implementation of the powerstroke model . . . 125

4.2.1 Isometri ase . . . 125

4.2.2 Length lamp devi e . . . 127

4.3 Whole y le models . . . 130

4.3.1 Extended Huxley and Simmonspotential . . . 130

4.3.2 Chemi al-Spring-Motormodel (CSM) . . . 133

4.3.3 Spring-Chemi al-Motormodel (SCM) . . . 136

4.3.4 Spring-Chemi al-Motormodel with a ba kbone (SCM1) . . 140

4.3.5 Dis ussion . . . 148

5 Con lusions 151 A Appendix 155 A.1 Brownianmotion . . . 155

A.2 Probability . . . 155

A.3 The Langevin Equation. . . 157

A.4 Diusion of aparti le ina uid . . . 159

A.5 The Fokker-Plank equation. . . 161

A.6 High fri tionlimit . . . 163

A.6.1 The Fokker-Plank equation inthe high-fri tion limit. . . 163

A.6.2 Canoni al distribution . . . 165

A.6.3 The First passage time . . . 166

A.7 Kramers' approximation . . . 167

Thisthesiswasdevelopedatthe Laboratoirede Mé aniquedes Solides(LMS)and

mainly funded by the Marie Curie Resear h Training Network and partially by

So iété Fx-Conseil. Many people have ontributed in dierent ways to this work,

I would liketo thank all ofthem and inparti ular:

My advisor Prof. Lev Truskinovsky for the dis ussions whi h we have had

duringallthese threeyears andforhisen ouragingtorea hastronger understand

of the mathemati alte hniques used in this work.

Dr. Jean-Mar Allain for the interest expressed on the development of this

work and for hishelpful suggestions.

Prof. ChaouquiMisbahfora epting tobethe Presidentof my thesis

ommit-teeaswellasProf. FrançoisGalletandProf. Vin enzoLombardiwhi hhavefound

the time to read and report on my thesis do ument. I would also like to thank

Prof. Marie-Christine Ho Ba Tho, Prof. Cé ile Sykes and Prof. Pas al Martin

for a epting to parti ipate to my thesis ommittee and for their suggestions to

improve the nal versionof the thesis.

All the people of the Laboratorie de Mé anique des Solides where I spent the

lastthree years. I met many ni e persons, with ahigh professionality. My spe ial

thanksgotoDire torBernardHalphenforhissupportbeforeandafterthedefense

of the thesis.

Prof. Grégoire Allaire, team leader at the E ole Polyte hnique of the

Multi-s ale modellingand hara terisation for phase transformationsin advan ed

mate-rials(MULTIMAT) proje t inthe Marie CurieResear h Training Network.

I would like to express my gratitude to Prof. Vin enzo Lombardi and Prof.

GabriellaPiazzesi for having shared with me part of their tremendous amount of

knowledgeonmus le me hani sfromthe very beginningofthis work. Iremember

espe ially our rst meeting and the time whi h they have given to my questions

despite the dieren es in our respe tive approa hes to the same subje t. I have

Skeletal mus le ontra tion is a broad domain of s ien e that overs many areas,

from biophysi s and hemistry to me hani s. The foundations of the theory of

mus le ontra tion were built 50 years ago, when it was understood that it is

myosin  ross-bridge,linking adja ent myosin and a tin laments,that generates

for e and motion. Sin e that time many experimentaladvan es have been made.

Theseadvan eshavenot been alwaysmat hed byimprovementsinthe buildingof

mathemati almodels.

Mathemati al approa hes tomus le ontra tion are mainlybased onthe ideas

proposed inthe Huxley 1957 model[4℄ andHuxley and Simmons1971model[10℄,

that dominated the eld for the past half entury. Althoughthey do not a ount

for all observed phenomena, these models still represent the paradigm of hoi e.

The two models of Huxley an be seen as omplementary sin e the Huxley 1957

model des ribes the atta hment-deta hmentpro ess and the events related tothe

slow time s ale, while the Huxley and Simmons 1971 model des ribes the power

stroke pro ess and the events related tothe fasttime s ale.

In this Thesis we shall follow some re ent insight and explore the possibility

to bring together these two type of pro esses and to obtain a unied model that

is able to des ribe the whole ross-bridge y le. Before the uni ation we rst

modify the existing models to ast them into afully me hani alframework. Both

Huxley 1957 model [4℄ and Huxley and Simmons 1971 model [10℄, present ad ho

assumptionsregardingthe hemi alrate onstantsthatdrivethepro esses. Similar

assumptions were made in all re ent models to t the experimental data at the

expense of maintainingthe linkwith me hani s.

In Chapter 1 we des ribe the physiology of mus les and their me hani al

be-havior, as well as the orresponding experimental pro edures. There we alsogive

the detailsof the Huxley 1957 model and Huxleyand Simmons1971 model whi h

are importantfor the originaldevelopment in the subsequent pages.

en ounters problems in mat hing the observed time s ale of tension relaxation

when a realisti value of the stiness of the myosin head is taken. After a review

of how the more re ent models, whi h in orporate one or more aspe ts of the

original Huxley and Simmons 1971 model, deal with these problems, we present

our modi ation of the theory whi h pla es the power stroke me hanism entirely

in a me hani al framework. The novelty of our approa h from the perspe tive of

me hani s is that we deal with the me hani al behavior of a multi-stable system

in a Brownian domain, where the ee ts of thermal u tuations are important.

We obtain an analyti al des ription of the behavior of our model at equilibrium

and during the transients and show how the resulting modi ation of the Huxley

and Simmons 1971 model helps one to avoid the intrinsi problems of this model

indi ated above. Finally we show that our model gives a new meaning to the

powerstroke step,whi hisin quantitativeagreement withallre entexperimental

observations.

In Chapter 3, we turn to the atta hment-deta hment pro ess and review from

the new,fullyme hani alpointofviewtheHuxley1957 model. Weshowthat this

model anbeviewedasbelongingtoa lassofmodels ofBrownianrat hets. These

models,rst developed inthe early

′

_90s

,haveanimportantroleinthe des ription

of mole ularmotorsofwhi hthemyosin II isanexample. Weareinterestedinthe

Brownian rat hets theory be ause it allows one to have a ompletely me hani al

interpretation of the mus le ontra tion pro ess. We present dierent types of

rat hetsrepresentingthepro essofATPhydrolysis. Wemodifyoneofthesemodels

by in luding ooperative ee ts and adapt it to the des ription of the slow time

phase of the ontra tionphenomenon. Wealsodevelop and test inthis Chapter a

numeri alalgorithmtosolvethe oupledsystemofsto hasti dierentialequations

whi h islater used for our numeri al experiments.

In Chapter 4, we ombine the power stroke model from Chapter 2 with the

model of a ooperative Brownian rat het developed for the simulation of the

atta hment-deta hmentpro essinChapter 3. Wepresentdierentways oflinking

together thetwomodelsandstudybothadvantagesandlimitationsofea hversion

of the unied model. We nally ome up with a model apableof providing fully

me hani al des ription of all four stages of the bio hemi al Lymn-Taylor y le of

mus le ontra tion. The resultingmodel still has drawba ks and we present some

and present a general dis ussion and on lusions. In Appendix we review some

mathemati al results regarding sto hasti dierential equations whi h we used in

Mus le physiology and early

modeling

1.1 Mus le physiology

The movements of a mus le on a ma ros opi s ale appear as the result of the

on erted a tion of millions of elemental units that work in unison. The most

studied mus les are alled skeletal mus les be ause these mus les are atta hed

to the skeleton. The ontra tion of skeletal mus les is under voluntary ontrol.

They belong to the lass of striated mus les whi hare omposed of long, parallel,

ylindri al bers. Ea h of these bers is a multinu leate ell, of

1

− 100 mm

in

lengthand

10

− 100 µm

indiameter. Fibers ontain myobrils,also ylindri alin

shape with a diameter of

0.5

− 2 µm

. Myobrils are made of repeated segments

ea habout

2.5 µm

inlength, that are alled sar omeres (Fig. 1.1) [95℄.

Sar omere is the smallestelementof a mus le that an ontra t. Being pla ed

inseries,sar omeresgenerate the ontra tionofthewholemus le. Ea hsar omere

isformed by anarray of lamentsof twodierent types, whi hintera t with ea h

other: a thinner lament, ontaining the protein A tin, and a thi ker lament,

ontainingthe proteinMyosin. Thesar omere anbedividedinzones: inFig. 1.2

wesee alongitudinaland atransverse viewofit. In theregionwherethe laments

overlap, six thin laments are lo ated around ea h thi k lament (Fig. 1.2b).

Thinlamentsare an horedtothe Z-disk(fromgermanzwis hen, between) whi h

onne ts adja ent sar omeres. Thi k laments are an hored to M-line (Mittel,

middleofthesar omere,notshowedinFig. 1.2)andalsototheZ-diskviaanelasti

elementthe giant proteintitin. Theserepeating stru tures, (A=anisotropi atthe

Figure 1.1: Mus le'sanatomi almi rostru ture

generate the typi alstriatedstru ture that givesthe name tothis type of mus les

[19℄. A longitudinal view as it appears on a ele tron mo rograph is given in Fig.

1.3.

The sliding-lament hypothesis was proposed fty years ago. It assumes that

during ontra tionthethinlamentmovespastthethi kone,sothatboththe

sar- omere, andthemus le,shortenwithout hangingthelengthofthetwostru tures.

ThehypothesiswasbasedonthepapersofHughHuxleyandJeanHanson[2℄(using

aphase ontrast lightmi ros opy)and ofAndrewHuxley andRolfNiedergerke [3℄

(using aspe iallydeveloped interferen elightmi ros ope)both publishedin1954.

Both works showed that when the mus le ontra ts the laments keep a onstant

length, andthe on lusion wasmade thatthey must slide duringshortening. This

hypothesis has not been immediately a epted: the then urrent view was that

myosin was a long negatively harged polypeptide without mu h stru ture that

shorten down due to the addition of

Ca

2+

Figure 1.2: Longitudinal and transverseviewof asar omere: (a)Longitudinal viewof 3

sar- omeres(sket h). (b)Transverse viewat 3dierentse tion (sket h). ( ) Transverse viewof 3

sar omeres(mi ros ope). From[97℄

Figure 1.3: Longitudinalviewofasar omereas seeninele tronmi rgraph. From[98℄

the two laments intera t through the ross-bridges (later we use notation Xb);

these are the globular portions, or heads, that emergein regularly repeating

ou-ples from the thi k lament formed by the polymerisation of the dimeri protein

myosin II (Fig. 1.4). Ea h head has a site with an anity for a tin, and a site

with an anity for a high energy mole ule, alled ATP (adenosintriphosphate).

The rst site bounds an a tin monomer while the se ond site an ATP mole ule

whi h a ts as the fuel for the mus le motor. ATP is hydrolyzed by myosin in

ADP (adenosindiphosphate) and orthophosphate whi h subsequently are

disso i-ated with release of hemi alenergy [9℄.

AsimpliedmodelofXb y leisshown inFig. 1.5,whereone anseefourmost

important states in whi h Xb an exist. When atta hed to a tin (state 2 in the

gure),ea hXbusesitspotentialenergytopullthea tinlamentthroughapower

stroke (state3)whi h,a ordingto rystallographi studies, onsistsinatiltingof

Figure 1.4: Myosin lament stru ture. (a) Myosinmole ule ( ouple). (b) Bundle of oupled

myosinmole uleswhi hgeneratethethi klament. From[100℄

pla e in one dire tion (plus dire tion), but due to the antiparallel arrangement

of the two halves of the sar omere, the two Z-disks are pulled towards the enter

of the sar omere, redu ing its length. In this sense, the half-sar omere, the zone

between one Z-disk and the next M-line, an be seen asthe smallest element that

an ontra t. Togoba ktoitsoriginal onguration(state1)theXbneedsanother

ATP to deta h from a tin and start another y le (state 4). It then binds to a

new a tive site onthe a tin lament (state 2) and the whole pro ess starts again

[64℄. This inner working is des ribed in the bio- hemi al Lymn-Taylor model of

a ross-bridge y le [9℄. The y le in Fig. 1.5 is a simplied four-states model

that omits anumber of intermediate states, nevertheless it des ribes the essential

steps of the pro ess. An important general observation is that mus le needs ATP

for both the ontra tionand the relaxation;the unphysiologi al depletion of ATP

below a ertain on entration will prevent the deta hment of the heads from the

a tin lament, whi h auses rigor mortis [19℄.

Thestru tureofthehead anberesolvedwithapre isionofonenanometer[33℄,

[87℄. It has been proved that the relative displa ement of the laments is mainly

taking pla e during the power stroke (state 3). It is a hieved by a rotation of the

distal part (C-terminal) of the head that a ts like a lever arm. This me hanism

gives tothe whole approa hthe nameof swinging lever armtheory.

Regulationofthe ontra tionisdue tothe fa tthat Xbs an bound a tinonly

Figure 1.5: SimpliedCrossBridge(Xb) y le(seetext). From[44℄

thatisatta hedtotheproteintropomyosin andlieswithinthegroovebetween the

twostrandofthe a tinlamentinmus letissue. Inarelaxedmus le,tropomyosin

preventtheintera tionofmyosinwiththeatta hmentsiteona tin,thuspreventing

ontra tion. Whenthe mus le ellisstimulated, al ium hannelsopen inthe

sar- oplasmi reti ulumandrelease al iumintothesar oplasm. Someofthis al ium

ions atta h totroponin, ausinga onformational hange that moves tropomyosin

out of the way so that the Xbs an atta h to a tin and produ e mus le

ontra -tion. The ions

Ca

2+

are stored inthe sar oplasmati reti ulum (SR) surrounding

the myolaments. The a tion potential originated at the neuromus olar jun tion

triggers the release of al ium from the SR almost syn hronously everywhere by

indu inganin rease inCa2+ permeability of the SR membrane. The ontra tion

ismaintained untilthe nerve ontinues to re; when the train of a tionpotentials

stops,the Ca2+ permeabilityfalls, whilethe Ca2+ pumpbringsba k the al ium

intotheSR.The de reaseof al ium on entrationbelowthe thresholdina tivates

Figure 1.6: Ex itation ontra tion oupling: s hemati des ription. From[97℄

Wehavegiven hereabriefoverviewofthe omplexeventsthat leadtothe

on-tra tionofskeletalmus les. A omplete des riptionshouldtake into onsideration

how the ele tri al signal generates the bio hemi al pro esses, with all their

om-plexity, whi h nallyleads to the me hani al for eor displa ement insar omeres.

Wewouldliketoemphasizethatinthisworkweshall onsideronlytheme hani al

aspe ts ofthe ontra tion,avoidingthe des riptionofproteinsintera tion through

hemi al rate onstants. Despite some limitations in the physi al interpretation

of the nal model, dis ussed at the end of the thesis, this approa h allows one

to produ e a fully me hani almodel of the ontra tion inthe sar omere, opening

the way to the onstru tion of arti ial mus le type ma hines. Moreover, as we

shall show the new approa h improves, in some aspe ts, the predi tive power of

1.2 Me hani al experiments

There exist dierent experimental approa hes to the study of the me hani s of

mus le ontra tion(see[76℄,[75℄,[78℄andreferen estherein). Thete hnologyused

in these experiments has been often highly innovative, leading to te hnologi al

spin-o. We have already mentioned dierent mi ros opy te hniques. Another

te hniqueistheinvitromotilityessay,wheresinglemyosinmole ulesatta hedtoa

beadtrappedby alaser beamareused tomeasure thegeneratedfor e. Dierently,

the IVMA measures the speed of sliding of a tin laments, atta hed to a bead,

gliding on a bed of myosins. Then the syn hrotron radiation (an intense X-ray

sour e) was developed to study the Xb movements in situ in whole mus le or

single bres. Finally protein rystallography was applied toinvestigate the power

stroke inthe myosin mole ule atatomi resolution (see Fig. 1.7) [76℄.

Figure1.7: Stru tureofmyosinS1from hi kenskeletalmus le. From[33℄

In thisThesis weapproa hthe modelingof skeletalmus les ontra tionfroma

me hani alpoint of view. Therefore, we shall be mainlyinterested in aparti ular

setofexperimentsperformedonthemus lebers ormyobrils. Theseexperiments

have ommon aspe ts with the usual me hani al measurements aimed at testing

the behavior of passivematerials [19℄.

A mus le responds to a single stimulus with a single transient rise in tension,

alled twit h. Two stimuli, generated after a suitable interval of time, produ e

identi al for e transients. When the se ond twit h starts before the rst one is

over, the se ond one develops a larger peak tension. Witha train of stimulations

the for e rea hes a steady state value, alled unfused tetanus, and hara terized

frequen y themean for erises toanalmost onstantvalue: this situationis alled

tetanus. The required frequen y depends onthe type of mus les and onthe

tem-perature (50-60 Hz in mammalian mus les at body temperature, not used in the

gure) [19℄. An experiments in whi h we are interested have been made in the

state oftetanus, that an beviewed as asteady state ondition.

Figure 1.8: For egenerated atdierentstimulationfrequen y. 1pps orrespond tothesingle

twit h,at 80pps isrea hedthetetanus. From[97℄

The me hani al experiments, either on a ber or on a myobril, are usually

performedwith one end ofthe spe imen xed and the otherlinked toaleverwith

a at h me hanism and a transdu er of for e (Fig. 1.9). In this Se tion we shall

explain indetail the three major proto ols used in these type of experiments and

present theirmain results.

For e-length urves

Whenthe at hme hanismisxed,themus leundergoesanisometri ontra tion.

Byimposingtetanizationwiththeendsxedone anregisterthetensiongenerated

by the mus le. Byvarying the initiallengthof the mus le beforethe tetanization,

a for e-length urve an be onstru ted[95℄.

InFig. 1.10weshowthe s hemati for elength urveforthetotalfor eandfor

its two omponents: a tive for eand passive for e. Passive for eis the resistan e

generated by elasti omponentsin parallel tothe ontra tile element,it be omes

relevant when the sar omeres are overstret hed. The passive resistan e is almost

zero until a ertain riti al elongation of the sar omere, and then in reases fast

showing nonlinearelasti ity. Subtra tingthe passivefor efromthetotal isometri

Figure 1.9: Experimental devi es (a,d),experimental urves(b, ) andone versionof theHill's

model(e)(seeSe tion1.3.1). (a)Whenthe at hme hanismisa tingthemus le anbetetanized

ata onstantlength,leftpartof( ),rea hingthetetanusat

T

0

in(b). Whenthesystemisreleased

(d),theisotoni ontra tionagainsta onstantload,

T

in(b),generatesthelength-time urvein

therightpartof( ). From[19℄

Dierent types of mus les have dierent passive responses and so dierent total

for es, but the a tive for e-length urve, for most of them, shows the same

non-monotone behavior [7℄. This behavior (Fig. 1.11) is in agreement, with the fa t

thatthetwolamentsmustoverlaptogeneratefor e. Infa t,themaximuma tive

for eisgeneratedwhentheoverlapbetweenthe twolamentsisoptimal,i.e. when

allthe Xbssee ana tinsite where they an bound and,at the sametime, there is

nointerferen e between the two halfparts ofasar omere. When theinitiallength

in the passive state is su h that some Xbs, the ones near the M-line, donot have

any a tive site toatta h,the a tive for estarts tode rease linearlywith the total

number of Xbs available to intera t with a tin. For shorter initial lengths than

those orresponding to the plateau of the for e-length urve, two opposite a tin

lamentsstart tointerfere withea hother,that again ontributestoade rease of

for e[7℄.

Animportantthingtonoteisthat thefor e-length (T-l) urve is reated point

by point: rstwex alengthinthe passivestate,thenwetetanizethe mus leand

registerthefor ewhi hdevelopsinisometri ontra tion. TheT-l urverepresents

therefore a series of isometri a tivations at dierent initial passive lengths. In

parti ular this urve does not represent the response of the mus le to quasi-stati

Figure1.10: Total,passiveanda tivefor easafun tionofthelengthfortwodierenttypesof

mus les. Thepassivefor eisanalyzedstret hingthemus leinthepassivestate. Thetotalfor e

is analyzedtetanizingthemus lefrom a onstantpassivelength. Thea tivefor e isderivedby

subtra tion. From[19℄

Figure 1.11: Upper gure: A tivetension generated by isometri tetanization from dierent

passivelengthsofthesar omere. Lowergure: relativepositionsofthethinlament(bla kline)

For e-velo ity experiments

Thedynami albehaviorofskeletalmus les isusuallystudiedinadierent typeof

experiments[1℄, [67℄,[66℄,[93℄aimedat onstru tingthefor e-velo ity urve. This

urverelatestheload imposedtoa ontra tingmus letothe velo ityatwhi hthe

mus le shortens. It an be obtained, still point by point,within the experimental

setup dis ussed before [5℄: a mus le is tetanized at a xed passive length, then

the leveris released, while a onstant load is applied. The length of the mus le is

plottedagainst time (Fig. 1.12). As soon astension isredu ed, the mus le length

de reases: this typi ally fast response shows the presen e of an elasti element

whose shortening takes pla e before aslower time s ale dynami s of the Xb y le

gets a tivated. After this fast transient, the mus le starts to shorten at onstant

velo ity. Repeating the experiment with dierent loads, one an onstru t the

urve plottedin Fig. 1.13.

Figure 1.12: Shorteningvs. time urves, forone load(A)and fordierentloads (B). Length

hange axes refers to shortening. The os illating regime is due to the me hani al apparatus.

From[5℄

As we an see, thereis amaximum velo ity

v

0

that the mus le ontra tion an rea h under free (unloaded) shortening; this velo ity is independent of the length

of the mus le in the passive state. There is also a load against whi h the mus le

undergoesanisometri ontra tionat

v = 0

,thisvalueisprovidedbytheT-l urve.

Applying a onstant load greater than this value gives the for e velo ity urve in

lengthening (or e entri ) portion. This latter range ismu h less known than the

shortening range be ause of the higher dispersion of the experimental points. A

Figure1.13: For e-velo ity urve. Inshorteningthebehavior anbedes ribedbyanhyperbola

whi hinterse ts theabs issaatthemaximumunloadedspeed(normalized) andtheordinateat

the isometri tetani tension. The slope has adis ontinuity in the isometri point

v = 0

. A

plateau isobservedathigh lengtheningvelo ities. From[99℄

toinnity. Moreoverthereisadis ontinuity inthe derivativeofthe

F

− v

urveat

the isometri point

v = 0

: inthe e entri portionthederivativeofthe urveissix

timegreaterthanintheshorteningportion. Inorderto onstru tthefor e-velo ity

urvethesteadyshorteningstatemustberea hed,whi hhappensinatypi altime

s ale oftens ofmillise onds[93℄. Thetransientthatpre edes this statetakespla e

in a typi al time s ale of some millise onds and an be analyzed alsoin aslightly

dierent me hanism, whi h we introdu e inthe next Se tion.

Experiments on fast for e re overy

There is athird type of experiments with whi hwe shall mainly o upy ourselves

in this Thesis. Imposing on a tetanized mus le a small in rement, say negative,

of length

δ

generates a hange in tension as shown in Fig. 1.14(a) (see [10℄ and

referen es below). Thereis an instantaneous (hundreds of mi rose onds) de rease

in tension to a new value alled

T

1

, just as it would be if the thin and thi k lamentswereatta hedtoea hotherby elasti springs. Almostimmediatelyafter

this elasti stress drop,the tensionrisesand thenforsome time(millise ondstime

s ale) remains lose to a plateau level ( alled

T

2

) before nally re overing fully the value it had had before the length hange (tens of millise onds time s ale).

The hanges in length in these experiments are very small, about 4-10 nm per

su hthat itis realisti to assume that the number of atta hed Xbs remainsxed.

Imposing dierent length in rements, one obtains the relationship between the

imposed lengthin rements and the tensions

T

1

and

T

2

shown in Fig. 1.14(b).

(a) (b)

Figure 1.14: Fast re overyexperiments. (a)A rapidsmallshortening isappliedto themus le

(uppertra e)andtheresultingtensionhistoryismeasured(lowertra e). (b)The urves

T

1

and

T

2

vs. the imposed length in rement fortwodierentvalues ofinitial length, normalizedwith respe ttothehigherisometri tension

T

0

. Symbolsaredenedin thetext. From[19℄

Animportantunderstandingthatderivesfromthisexperimentisthatthevalues

of

T

1

at various shortenings lay pra ti ally on a straight line. Another important result is that the rate of re overy of tension hanges with the step imposed in a

highly non linear manner (see Fig. 1.15). It tends to in rease in an exponential

way from positive lengthsteps tohigher negative steps.

The behavior exhibited by a mus le in this set of experiments is an

impor-tantsour e of informationabout mus le me hani s, be ause atleast the fasttime

response produ ing the fun tions

T

1

(δ)

and

T

2

(δ)

appears to be independent of the atta hment-deta hment pro ess. Sin e the pioneering paper of Huxley and

Simmons[10℄ these experimentshave been repeated by many groups [14℄ [41℄ [43℄

[67℄.

Thefor e-length,thefor e-velo ityandthe

T

1

and

T

2

vs. step-length urvesare themostimportantexperimentalresultsthatdeal dire tlywiththeme hanismsof

mus le ontra tion. Wehavegivenreferen es tosomere entexperimentsrevealing

for instan e the history dependen e in the me hani al response when mus le is

stret hed after tetanization [52℄. Nevertheless, in what follows, we shall fo us on

the explanationof onlythe mainexperimentalfa tsthat are onsideredtobe well

Figure1.15:Rate onstant

r

ofqui ktensionre overyfollowingalengthin rementofmagnitude

y

. Estimatedas

ln(3)/t

1

/3

where

t

1

/3

isthetimeforre overyfrom

T

1

to

(2T

2

+ T

1

)/3

. From[10℄

1.3 Me hani al modeling

In this Se tionwe introdu eseveralbasi modelsaimed atexplainingthe

me han-i al behavior of mus les. They are: the Hill 1938 model, the Huxley 1957 model

and the Huxley and Simmons 1971 model. These models represent the basis on

whi h the majority of more re ent models are based. Some of this more re ent

models are reviewed later inthe Thesis.

1.3.1 Hill 1938 model

An analyti alexpression for the on entri part ofthe for e-velo ity urve was

ob-tainedbyHillin1938[1℄. Heusedhisownexperimentsfo usedontheenergeti sof

mus le ontra tionagainsta onstantfor e. Firsthe observed thatwhen the

mus- le is allowed to shorten, it liberates more energy (thermal and me hani al) than

during isometri ontra tion. He divided the totalenergy rate

E

intothreeterms:

the maintenan e heat rate (

A

) liberatedby amus le inisometri ontra tion, the

shortening heat rate (

H

), that is the total heat liberated during the ontra tion

minus

A

, and the rate of work done (

W

) equal to

F

· v

where

F

is the onstant

dis ussion of the for e velo ity urve. Hill wrote the energy balan ein the form:

E = A + H + W

⇒ E − A = H + W.

(1.1)

Bya very pre isemeasurementof the rst term

A

and of the totalenergy rate

E

, Hillobserved empiri ally the relation:

H + W = b(F

0

− F ).

(1.2)

In the right hand side of (1.2) we see the dieren e between the for e

F

applied

to the mus le and the maximal for e

F

0

exerted by it in an isometri ontra tion when

v = 0

. Independently Hill observed that

H

depends linearly onthe velo ity

of ontra tion,

H = av

. In this way we have:

H + W = av + F v = b(F

0

− F ).

(1.3) Byrearranging terms in(1.3), Hill obtained:

(a + F )(v + b) = b(a + F

0

).

(1.4)

Figure 1.16: For evelo ityrelation. The ir les representtheexperimental observation(frog

mus le),theline orrespondsto the urve(1.4). From[52℄

Inthe

F

−v

spa eequation(1.4)des ribesahyperbolawithasymptotes

−a

and

−b

(seeFig. 1.16)[46℄. Ittstheexperimentalpointsverywell(usingappropriates values of

a

and

b

) for a large variety of mus les. In the free ontra tion

F = 0

,

the velo ity be omes maximal

v

max

and it has been observed that for many type of mus les a ross spe ies and temperatures [52℄:

a

F

0

=

b

v

max

Figure 1.17: Hill1938 model withwith aseriesof apassivespringand a ontra tileelement,

bothinparallelwithase ondpassivespring. From[19℄

With these results, Hill proposed a model (see Fig. 1.17) where the a tive

mus leis represented by anelasti elementSE inseries witha ontra tileelement

CEwhosefun tionistolinktheappliedfor etothevelo ity,inabla kbox manner.

Su essively, to a ount for passive elasti ity, anelasti element PE was addedin

parallel with the CE and the SE(Fig. 1.17).

In the passive state the CE an be stret hed without any resistan e. During

the ontra tion, the total for e generated by the system is

F = k

p

u + k

s

(u

− w)

. Here

k

s

(u

− w) = f

CE

[w

′

, l

0

]

,

k

s

isthe stiness of the SE and

k

p

of the PE,

u

the totaldispla ement,

w

thedispla ementoftheCE,

f

CE

isthefor einthe ontra tile elementwhi hdependsontherateof hangeofthedispla ement

w

′

= v

. A ording to observations made by Hill, the CE exerts afor e of the type

f

CE

=







































0

w 6

˙

−F (l

0

)

b

a

F (l

0

)b + a ˙

w

− ˙

w + b

−F (l

0

)

b

a

6

w 6 0

˙

1.5F (l

0

)

− 0.5

F (l

0

)b

′

− a

′

˙

w

˙

w + b

0 6 ˙

w 6 F (l

0

)

b

′

a

′

1.5F (l

0

)

F (l

0

)

b

′

a

′

6

w

˙

that a ounts also for the e entri ontra tion. The isometri for e

F

0

has a dependen e on the initial length of the mus le

l

0

, as shown by the for e length urve. Thefor eis equal tozero forlarge negativevaluesof

w

′

(shortening),while

it an be dire tly obtained from (1.4) for smaller ontra tion velo ities. In the

e entri region,

w

′

> 0

, the values are taken to mat h the behavior observed experimentally(see Fig. 1.13).

time dependen e of the for e for dierent given proto ols of stret hes. Thus if we

introdu eaparameter

β

toa ountforthe on entrationof al ium,asintherst

and third part of Fig. 1.18, weobtain:

F (l

0

, t)

− k

p

u = βf

CE

"

1 +

k

p

k

s



˙u

−

F (l

˙

0

, t)

k

s

, l

0

#

(1.6)

where

0 < β < 1

modulatesthefor einthe ontra tileelement

f

CE

. Twoexamples of loading programs are presented in Fig. 1.18. The elongation, equal for both

experiments,is given by arampthat in reases the lengthofthe mus le, maintains

it onstant and then shortens it to the initial state. The a tivation parameters

were dierent. The experimental observations obtained for the given elongation

history is shown in Fig. 1.19. The predi tions of the model are in Fig. 1.20: the

two responses are rather similar.

For 50 years Hill 1938 model dominated the eld. In this period many ideas

havebeenaddedtothemodelinordertoa ommodatenewlydis overedfa ts[52℄.

Originallyquitesimplethe model be amemore and more ompli atedand lostits

appeal;howeverthe simplestversion isstilltoday usedtosimulatethe me hani al

behaviorof mus les.

Figure 1.18: Two dierent experimental pro edures for Hill's 1938 model. The response is

Figure1.19: ExperimentalresultsfortheelongationhistoryshowninFig. 1.18. From[13℄

Figure 1.20: Response of the Hill's 1938 model for twodierentpro edures I (upper partof

The main reason for the sear h of dierent on epts in mus le modeling was

the following: Hill's model does not provide insights into the me hanism of the

produ tion of for e. Its bla k box nature is su ient to give a good t to the

experimental urves, but it does not provide a tool for the understanding of the

me hanismsthatoperatesatthe mi ro-s aleswhi harenot visibleinthestandard

me hani alexperiments.

1.3.2 Huxley 1957 model

Before1954,mosttheoriesofmus le ontra tionwere basedontheideathat

short-eningandfor eprodu tionweretheresultofsomekindoffoldingor oilingoflarge

proteinmole ules. In1954, HEHuxleyand JHansen[2℄aswellasAFHuxleyand

RM Niedergerke [3℄ demonstrated that mus le ontra tion is not asso iated with

any hangeoflength insidethe mi rostru ture. These authorspostulated thatthe

for eis generated through the intera tion of a tin and myosin laments.

Based on this understanding, AF Huxley developed in 1957 a new theory of

mus le ontra tion [4℄. The thi k myosin lament is assumed to be xed inspa e

whilethethinlamentisassumed toslideparalleltomyosinwith onstantvelo ity

v

. The movementisgeneratedby ame hani alstru ture(thatisnowknown tobe theXb)that an o upy dierentpositionsalongtheba kbone ofa tin,and whose

movement is limited by an elasti element (Fig. 1.21(a)). The model postulates

that thenumberof a tiveXbs is onstant and onsiders onlythe fulla tivationof

the mus le (tetani response).

Figure1.21: Huxley1957model. (a)ThemyosinheadMiselasti ally oupledtotheba kbone.

Theintera tion between thelaments an beestablished whenM rea h theatta hmentsite A

The stru ture in question an atta h itself only to spe i sites on the a tin

lament. When it is atta hed, then there is a for e between a tin and myosin,

whi h depends on the position of Xb. To al ulate the total for e generated by

the mus le one needs to know the total number of atta hed Xbs at ea h position

x

relative tothe referen e position of the stru ture, at every time

t

.

As a result of thermal u tuations Xbs atta h to the a tin in a range of axial

position. They exert a for eif they rea h the atta hed position where the elasti

elementisstret hed;noti ethatasour eofasymmetryisneeded togenerateanet

for e in one parti ular dire tion [4℄. It is assumed that the probability

f

that a

deta hedXb an atta hand theprobability

g

thatanatta hed Xb andeta h,are

fun tions ofthe variable

x

, asshowed inFig. 1.21(b). The atta hmentprobability

f (x)

is assumed to belinear in

x

and is zeroboth beyond amaximum distan e

h

, and for

x < 0

(the Xb an not atta h toan a tivesite whenthe elasti elementis

ompressed). The deta hment probability fun tion

g(x)

is alsolinear for positive

x

, the probability in reases even beyond

h

, and islarge and onstant for negative

x

. If

n(x, t)

is the fra tion of the total populationof atta hed Xbs whose distan e fromthe a tivesite is

x

attime

t

,then itstimeevolution anbefound fromarst

order kineti equation [4℄:

∂n(x, t)

∂t

− v

∂n(x, t)

∂x

= (1

− n(x, t))f(x) − n(x, t)g(x).

(1.7) Huxleylimitedtheanalysistosteadystate ase,whenthesolutionis onstantin

time,sothersttermintheleftiszero. Theequation(1.7)allowsthe omputation

of

n(x)

at dierent

v

: at zero velo ity

n(x)

rea hes the onstant value

f /(f + g)

.

Athighervaluesofvelo itytherearetwofa torsthatredu eitsvalue: rst thereis

less timefortheXbstoatta h,se ondthe Xbsare broughtfastertowards negative

values of

x

. The predi tions of the model (1.7) are illustrated in Fig. 1.22. The

analyti al solutionis given by:

n(x, v) =























f

1

f

1

+ g

1

[1

− exp(−φ/v)] exp(

xg

2

v

)

x 6 0

f

1

f

1

+ g

1



1

− exp



x

2

h

2

− 1



φ

v



0 6 x 6 h

0

x > h

(1.8) where

φ =

f

1

+ g

1

2

h.

Usingthissolutionone anwriteanexpli itexpressionforthefor evelo ity

Figure1.22: RelativedistributionofXbsatvariousvelo itiesa ordingtoHuxley1957model.

From[4℄

k

, generating a for e proportional to its displa ement

kx

. Then the total tension an be writtenas:

T (v) = ρ

R

_−∞

∞

kxn(x, v)dx =

ρk

f

1

+ g

1

2

h

2

2



1

−

v

φ

1

− e

−φ/v



1 +

1

2



f

1

+ g

1

g

2



v

φ



(1.9)

where

ρ

standsfor thedensity ofXbsperunit volume. Optimizingthe parameters

tottheHill'sdata,Huxleyobtainedanex ellenttasinFig. 1.23. Theisometri

tension be omes

C(f /(f + g))

, where C depends on the number of Xbs present

in the segment of mus le under onsiderationand on the other parameters of the

model, for instan e the elasti onstant.

Inadditiontothe on entri part ofthefor evelo ity urvethe modelpredi ts

alsootherfeatures of the mus le response, even if onlyqualitatively. Forinstan e,

the model predi ts the e entri part of the for e velo ity urve, showing both a

dierent slope of the urve at the isometri point and an asymptoti behavior of

the for e at high velo ities. Both values however are highly overestimated. The

model alsooverestimates the rate of heat releaseduring lengthening,however this

Figure 1.23: Huxley's predi tion for the for e velo ity urve (line) and experimental data

(points). From1.23

during lengthening there is a me hani al breakdown of the Xbs, whi h deta h

withoutATP release. Manymorere entdevelopmentshavebeen donealongthese

lines, see for instan e [21℄, [22℄, [27℄, [58℄, [64℄. Overall, the Huxley 1957 model

represents an improvement over the Hill 1938 model be ause it gives a pre ise

mathemati aldes ription of the mi ros opi events behind the bla k box behavior

postulated by Hill.

1.3.3 Huxley and Simmons 1971 model

The experimental response of mus les to rapid length in rements, des ribed in

Se tion1.2, annotbeeasilyexplainedbythe1957Huxley'smodel. Thepioneering

experiments of this type, made in [10℄, have lead to the development of another

importantmathemati almodel: Huxleyand Simmons'modelof 1971. This model

isnot anexpansionof theHuxley 1957model,but isaquitedierentmodel whi h

deals only with for e generated by the atta hed Xbs. In parti ular it does not

take into a ount the deta hment pro ess. What brings the ne essity of a new

model is the fa t that the rapid re overy of for e takes pla e in the millise onds

time s ale,whi hisdi ulttoexplaininthe frameworkofthe slower

atta hment-deta hment pro ess, relatedto the time s ale of tenthof a se ond. The approa h

used by Huxley and Simmons, whi h we shall des ribe below, is the predominant

onrmation from the measurements of the axial motions of the myosin heads at

angstrom resolution by X-ray interferen e te hnique [66℄.

The parti ularme hanismsuggested byHuxley andSimmonsforthe stru ture

ofthe Xbsisshown inFig. 1.24. First ofall,they assumed thatthe Xb ontains a

linearelasti spring linked to the head ofthe myosin. When atta hed tothe a tin

lament, the head of the myosin an be in two states, and an swit h from one

state to another in a jumpfashion. The ratio of the rates of jumps are ontrolled

by the relative energy of the two states. The energy

U

h

of the head is a double wellfun tionof onguration oordinate

x

,it isplottedinFig. 1.25 togetherwith

the paraboli energy of an elasti element. The swit hing an stret h or relaxthe

elasti element, sowe an referto the states as alow for e generating state and

a high for e generating state. The total potential energy

U

tot

, given by the sum of the potential energy of the elasti element

U

s

and the potential energy of the hemi alstate

U

h

, is plottedin Fig. 1.26.

Figure 1.24: Huxley and Simmons 1971 model. The myosin head S-1 is linked to the thi k

lamentthroughanelasti elementS-2and hastwostable positions. From[19℄

The model analyzes the distribution of Xbs inea hof the energy wellin order

toobtain the total for e generated by the mus le. Be ause inea h halfsar omere

the Xbsarearranged inparallelbetween ana tinlamentand therelativemyosin

lament, the total generated for e is the sum of the for es generated by ea h Xb.

It is assumed that when the mus le is isometri allytetanized, the two states have

the same total energy.

Due tothe linearity of the elasti element:

U

s

=

1

2

Kl

2

₌

K

2

(y

0

±

h

2

+ y)

2

(1.10)

where

l

is the stret h whi h an be written as the sum of

y

0

± h/2

(where

y

0

is a point lo ated at equal distan e from both wells,

h

is the distan e between the

Figure 1.25: Thepotentialenergiesin HuxleyandSimmons1971model. (a) Elasti energyin

isometri ontra tion(

y = 0

), after stret h(

y > 0

)and afterreleaseofthe mus le(

y < 0

). (b)

Congurationalenergyofthehead,twostablestates1and2arepresent. From[19℄

Figure1.26: Thetotalenergy

U

tot

= U

h

+ U

s

inHuxleyandSimmons1971modelasafun tion

twopotentialenergy wells, orrespondingtothetwostates1and 2. Theheightsof

the potential energy barriers

E

1

and

E

2

in Fig. 1.25, are assumed to be the same for both wells. Sin e in the state of isometri ontra tion the two minima of the

total potentialenergy havethe same levelwhen

y = 0

,the totalnumber ofXbs in

the two ongurations is the same. When a length in rement is imposed,

y

6= 0

,

there isa hange in

U

s

(upperpart of Fig. 1.25) and therefore inthe total energy

U

tot

= U

s

+ U

h

,as shown in Fig. 1.27.

Before giving the mathemati al details, we des ribe briey how the model

works. The hange in the total length

y

rst ae ts the tension in the linear

spring, and is therefore responsible for the

T

1

for e observed in the experiments. After the step, the levels of the energy in the two minima be ome dierent, and

a hange in the total number of Xbs in ea h state is generated. This adjustment

pro essfollowskineti spostulatedforthejumppro ess,andtakespla einaslower

time s ale than the time s ale responsible for the

T

1

response. The nal steady state isresponsible for the value of for e

T

2

.

To ompute the for es

T

1

and

T

2

, we needto know the relative number of Xb,

n

1

and

n

2

inea hwell,

n

1

+n

2

= 1

. Undertheassumptionthatthestateofdetailed balan eisrea hed,the rate onstants

k

+

,des ribing transitionsfromposition 1to position 2,and

k

−

, des ribing transitionfrom2 to1, are related through:

k

+

k

−

= C exp



(B

12

− B

21

)

k

B

T



,

(1.11)

where

T

isthe absolutetemperature,

k

B

the Boltzmann onstant,

C

a onstantto be determinedand

B

12

and

B

21

the a tivationenergies for passingfrom state1 to state 2 and vi e versa. In Fig. 1.27, we also see that

k

−

is onstant sin e

B

21

is a xed quantity independentof the tensioninthe elasti element. Therefore we an

write

B

21

= E

1

and

B

12

= E

1

+ ∆U

tot

, where

∆U

tot

is given by

∆U

tot

= (E

2

− E

1

) +

1

2

K

"

y

0

+ y +

h

2



2

−



y

0

+ y

−

h

2



2

#

=

(1.12)

kh(y

0

+ y) + (E

2

− E

1

)

Sin e inisometri ontra tionthe two states have the same energy:

∆U

tot

|

y=0

= 0

⇒ −Khy

0

= (E

2

− E

1

),

(1.13) the relation(1.11) be omes:

k

+

k

−

= exp



−Khy

k

B

T



.

(1.14)

Figure1.27:ThedierentbehaviorofHuxleyandSimmons1971model: totalenergyinstret h

(

y > 0

, where the onguration 1 is energeti ally preferred) and in release (y<0, where the

onguration2isenergeti allypreferred)modes. From[10℄

The dierential equation des ribing the number of Xbs in the state 2 during

transientsis:

dn

2

(t)

dt

= k

+

n

1

(t)

− k

−

n

2

(t) =

−(k

+

+ k

−

)n

2

(t) + k

+

.

(1.15) Due to the hypothesis of equal energies of the states duringisometri ontra tion

weobtain that

n

2

(0) = 1/2

. We an now solve (1.15)and write:

n

2

(t) = n

∞

2

+ (0.5

− n

∞

2

) exp[

−t(k

+

+ k

−

)]

(1.16)

where:

n

∞

₂

=

k

+

k

+

+ k

−

.

One an see that the fra tion of Xbs instate 2, starts at one-half and rises to the

value

n

∞

2

exponentially with rate

k

+

+ k

−

. To ompute the steady state tension, only the ratioof the rates onstants (1.14)is needed. The transientof tension an

be writtenas:

T (y, t) = n

1

(t)K



y

0

+ y

−

h

2



+ n

2

(t)K



y

0

+ y +

h

2



= K[y

0

+ y + (n

2

(t)

− 0.5)h].

The hara teristi values of tensionpredi ted by this modelare

T

1

(y) = T (y, 0) =

K(y

0

+ y)

and

T

2

(y) = T (y,

∞) = K[y − 0 + y + (n

∞

2

− 0.5)h]

.

In order to ompute these fun tions, one needs the elasti onstant

K

. As it

was not known at that time, Huxley and Simmons used the data on the rate of

re overy

r(y)

(Fig. 1.15). They obtained a t:

r(y) = r

0

(1 + e

−αy

)

with

r

0

= 0.2 ms

−1

and

α = 0.5 nm

−1

. As we have seen, the same urve an be

predi ted fromthe model:

r(y) = k

+

+ k

−

= k

−

(1 + e

−Kh/k

B

T y

).

(1.17)

Huxleyand Simmonsused this formulatoobtainthe values ofboth

k

−

and

Kh =

α k

B

T

. Then they ould ompute the equilibriumfor e:

T

2

(y) =

αk

B

T

h



y

0

+ y

−

h

2

tanh

αy

2



(1.18)

whi hisshowninFig. 1.28. Theresulting urves

T

1

(y)

and

T

2

(y)

exhibitthesame

Figure1.28: Predi tion oftheHuxleyandSimmons1971model. From[10℄

value for the stiness be omes

K

≃ 0.2 pN/nm

, however a value of one order of

magnitude greater [35℄ [38℄ [36℄ [80℄ [92℄ was proven later. Weshall ome ba k on

the importan e of this value extensively in Chapter 2, however we mention here

that the more realisti value

K = 2 pN/nm

ae ts dramati allythe predi tion of

the model. First,we see immediatelyfrom(1.17)that the

r(δ)

dependen e willbe

highly overestimated, be ause it depends on

K

exponentially. Se ond, the

T

2

(δ)

urve is more ine ted with the higher value of the parameter

K

, thus it shows a

negative slopeat

δ = 0

whi h isin ontrast withthe experimental result.

There were several re ent attempts toimprove the quantitative predi tions of

the theory[18℄[43℄[48℄[59℄[73℄[96℄,howevernothingfundamentallynewhas been

added to the model. The most attra tive feature of the Huxley and Simmons

1971 model is thatit attempts tolink bio hemistrytome hani s. Notonlyin the

more re entmodelsthis linkwasnotimproved, but,onthe ontrary, itwasalmost

lost. In the next Chapter we shall give abriefreview of some of these models and

Power Stroke

2.1 Introdu tion

The spe ial hara ter of the me hani al response of skeletal mus les des ribed

in Chapter 1, allows one to lassify them as a tive materials, be ause they an

adapt to external stimuli. The advan es in te hnology are often linked to the

developmentofsu hmaterialsthat anprovidea tivefun tioning,likesensingand

a tuation. In the past, a tiveresponse was a hieved through organizing elements,

with passive response at the mi ro-level, into omplex stru tures with multiple

equilibrium states. However modern te hnologies require that su h me hanisms

fun tionatmi rometerandevennanometers ales,sotraditionalsolutionsbe ome

una eptable, and there is a demand for materialswhere the omplex behavior is

realized already at the mole ularlevel. An example of su h materialsis given by

shapememoryalloys,wherethemulti-stabilityofthesystematthemole ularlevel

isduetophasetransformationwhi hdoesnotrequirediusion,and anbeindu ed

bystress,temperatureorele tro-magneti eld. Theanalysisandmodelingofsu h

a tive materials has rea hed a level of pre ision that one would want to a hieve

in the des ription of skeletal mus les,given some similarity of the behavior of the

two types of systems. The similarity is based on the idea of multi-stability of the

mi ros opi elementsof the system.

As we have seen in Chapter 1 the model of Huxley and Simmons (HS71) an

des ribe fast response of skeletal mus les assuming the presen e of bi-stable

ele-ments with double well energy. In this model the energy lands ape is degenerate

be ause the wells are innitely narrow. This leads to a des ription in terms of

a jump pro ess, whi h requires the knowledge of hemi al rate onstants. In the

not needed: the Xb swit hes between the states instead of ontinuously moving

between them. This makes the pre ise analyti al omparison of this model with

me hani al models of shape memory alloys di ult. Despite these apparent

dis-tin tions, the main ingredients in both types of models are similar whi h leaves

a possibility to link the Huxley and Simmons model to the ontinuum theory of

martensiti transformations ina tive materials.

We re all that, the main dieren e between the multi-stableand onventional

linear elasti elements is that the energy of the former is non- onvex. As it was

shown inthe pioneering work of Eri ksen [11℄, this non onvexity isof

fundamen-tal importan e for the interpretation of the behavior experimentally exhibited by

shape memory alloys, whi h is related to the presen e of multiple stable

mi ro- ongurations. Eri ksen onsidered the behavior of a ontinuum 1-D problemfor

a material with a non- onvex energy under slowly varying load showing that a

mathemati al model based on bi-stability an explain hysteresis. After that, a

thorough study of the problem was performed, in parti ular a pre ise des ription

of themi ros opi eventswasobtained bydis retizing the1-D ontinuum problem

and viewingit asa hain of bi-stable elements [11℄, [16℄, [29℄, [62℄.

In this Chapter we reformulate the original Huxley and Simmons 1971 model

in this pre ise mathemati al framework. We show that this reformulation an

produ e a pi ture whi h avoids some drawba ks of the original HS71 model. We

start by briey des ribing the way in whi h other re ent models have dealt with

these drawba ks. Then we introdu e our new me hani al model aimed rst at

modelingthe power stroke only and present a quantitative analysis of this model

in ludingthe omparisonwith experimental urves.

2.2 Re ent Models

Already in 1978 [17℄ it was realized that the Huxley and Simmons 1971 model

an not predi t orre t time s ale of tension relaxation, if a realisti value of the

stinessoftheelasti elementisused. Thequantitativeresolutionofthisandother

problemsoftheHS71model, alreadymentionedinSe tion1.3.3,willbegivenlater

in this Chapter, while now we would like to briey review the main approa hes

used to ir umventthese problems. In parti ular we show that the way hosen by

the authors of the re ent models to deal with the drawba ks of the HS71 model,

leads toalmost omplete lossof ouplingbetween the two aspe ts of the problem:

un orre t time s ale predi ted by the HS71, however here we shall fo us only on

this aspe t of the problem.

Eisenberg and Hill model

An earlymodi ation of the originalHS71 model was proposed in 1978 by

Eisen-berg and Hill [17℄. The model was extended in 1980 on amore quantitativebasis

[18℄. Itisbasedontheobservationthatby assumingtwoverynarrowenergywells,

Huxley and Simmons made impli itly the hypothesis that the transition between

the states takes pla e only after a Xb had olle ted the total amount of energy

needed to over omethe barrier.

Eisenberg and Hill proposed to make the wells wider in order to allow the

transitiontostart atlowerenergy. They alsolinked the for egeneratedby the Xb

in ea h state with the rst derivative of the free energy, instead of the stret h of

the elasti element, that has now been formally eliminated. Without the elasti

element, the oordinate

x

of the Xb is ontrolled by the imposed length of the

sar omere,a tuallythe oordinate

x

has be omeameasure ofaxialpositionofthe

parti ular a tin site at whi h the Xb is atta hed. The origin

x = 0

was hosen

in su h a way that the Xb in the pre-power stroke state is in itsresting position.

At every value of

x

the Xbs an be in four dierent states whose free energies

were postulated to have an

x

-dependen e shown in Fig. 2.1. Here

AM

‡

_DD

is

the a tin-myosin omplex in the pre-power stroke state and

AMD

is the

a tin-myosin omplex in the post-power stroke state, (phases 2 and 3 in Fig. 1.5).

Similarly

M

∗∗

_D

and

M

‡

_D

are two deta hed states, the refra tory state and the

non-refra torystate,respe tively(phases4and 1inFig. 1.5). Thedierentangles

of the lever arm in the two atta hed states are ree ted by the dierentpositions

oftheminimainthe relativeenergies. The hangeofstateisassumedtobeajump

pro ess, allowing the Xb to follow the entire Xb- y le as shown by the arrows in

Fig. 2.1.

In dening the rate onstants of the individualjump pro esses, Eisenberg and

Hill used the hypothesis of detailed balan e whi h imposes one ondition for the

ratioofea hpair ofrate onstants. Inthe mostgeneralformthis ondition anbe

writtenas:

k

jm

(x)/k

mj

(x) = exp



G

j

(x)

− G

m

(x)

k

B

T



(2.1)

Theelasti elementispresentinthis model throughthe

x

-dependen e of the

Figure 2.1: Freeenergiesfortwoatta hedstatesandtwodeta hedstatesintheEisenbergand

Hillmodel. Thestate

M

∗∗

_{D + D}

isshiftedwithrespe tto

M

∗∗

_{D + T}

byanamountofenergy

givenbytheATPhydrolyzation. ThearrowsshowapossibleXb y le. From[17℄

the shape of the energy barrier between the states at a given

x

. That leaves for

ea h transition one of the rate onstants

k

ij

as a free parameter. In the Huxley and Simmons 1971 model the dependen ies of both rate onstants on the step

x

were ompletely dened by the shape of the total energy. Instead in Eisenberg

and Hill model these onstants retain some freedom (used in Fig. 2.2), sin e only

one ondition, equations (2.1), is imposed. To our knowledge, the EH78 model

wasrst toabandon the me hani altransparen y of the HS71 model even though

some features of the HS71 model were preserved. In a sense the bio hemi al

in-terpretation of dynami s has over ome the me hani al basis of the HS71 model.

The freedom left by the EH78 model was used to hoose the

k(x)

dependen ies

phenomenologi ally inorder to tthe experimentalobservations almost perfe tly.

Piazzesi and Lombardi model

The model of Piazzesi and Lombardi developed in 1995 [43℄, deals with the entire

y leoftheXb,andisabletopredi tboththefor evelo ity urveandthebehavior

of the mus le subje ted to rapid in rementsin the total length as well asthe ux

of energy and the e ien y of the ontra tion. There are two deta hed states,

D1 and D2, and three atta hed states, A1, A2 and A3 in Fig. 2.3 A. Moreover

there are two distin t paths in whi h Xbs an split ATP to generate for e. Two

a tive states, A1 and A2, are ommonfor the two paths, as shown in Fig. 2.3(a).

From A2 there is a long path, whi h ontains a se ond a tive state A3 beforethe

deta hmentof thehead D1. Thispath an ompete with the short one, wherethe

head deta hes immediatelyafterA2into D2. The long path generateslarger for e

but it has a lower rea tion rate, about 20/se ond. The rea tion rate for the short

path is about 100/se ond. The orresponding rate onstants satisfy the detailed

balan e equation (2.1). As in the Eisenberg and Hill model, the fun tions

k

jm

(x)

and

k

mj

(x)

depend onthe imposed step, and this dependen e an be hosen tot the experimental data. The spe i hoi e of the authors is shown in Fig.

2.3(b-e). The system of dierentialequations governing the distribution of the number

of Xbs in various states,

a

1

,

a

2

,

a

3

and

d

1

,

d

2

, generalizes the equation proposed by Huxley in 1957. Without going in all details we just mention that the system

onsistsof the equationsof the type:

∂a

1

(x, t)

∂t

= k

1

(x)d

1

(x) + k

−2

(x)a

2

(x) + k

6

(x)d

2

(x)

−

(k

−1

(x) + k

2

(x) + k

−6

(x))a

1

(x, t)

− v

∂a

1

(x, t)

∂x

(2.2)

for ea h of the vepossible states and issolved numeri ally.

Small length in rements bring the Xb in the region where the short y le is

favored and, being rapid, it an explain the behavior observed experimentally at

moderate shortening velo ities. For higherlength step the long y le is preferred,

leading to the possibility to t the

11 nm

power stroke. In this way the model

is able to predi t both for e-velo ity urve ( omputed onsidering the onstant

urvature ofthe freeenergy

G

i

(x)

asinthe EH78modeland showninFig. 2.4(a)) and step inlength type experiments (Fig. 2.4(b)).

Huxley and Tideswell model

Huxleyand Tideswell proposed in1996 amodel whi h wasexpli itlydeveloped to

Figure2.3: Generals hemeandrate onstantsforthePiazzesiandLombardimodel. A:(Left)

S hemeof thePL95model, A=atta hed, D=deta hed. (Right)Basi freeenergyof ea hstate.

(a) (b)

Figure 2.4: Comparison between simulations(lines) and experimental results (points) of the

PL95model. (a) For e-velo ity urve. (b)ModelingofT1andT2. From[43℄

on the same idea as the HS71 model, however new features were added. Among

them, the most important for our approa h are that, rst, three atta hed states

A

1

,

A

2

, and

A

3

are onsidered, with two onstant power strokes between them of

5.4 nm

and

4.5 nm

. Se ond, the rate oe ients for ea h of the four possible

rea tions

A

1

→ A

2

,

A

2

→ A

1

,

A

2

→ A

3

,and

A

3

→ A

2

, aregivenbythe expression

A/(1 + exp[B(x

− C)])

where

x

isrelatedtothestret hoftheelasti elementand

A

,

B

and

C

aredierent

for ea h transition, so that 12 parameter have tobe spe ied. Observe that as in

the previous models, there is an

x

-dependen e of the rate onstants. In order to

respe t the ondition(2.1) some onstraintsare added,however, noexpli it shape

of the free energywaspres ribed, whi h meansweakerme hano hemi al oupling

than in HS71 model. Finally, to mimi the in ommensurability of the spa ing of

thea tivesitesonthea tinlamentwithrespe t tothespa ingofthe headsalong

the myosin lament, ve populations of Xbs were equally spa ed relative to the

a tivesites. The authorsobserved that withthe lastassumptionthe tension

T

2

(δ)

ends up averaged over the range of spa ing of the a tive sites (see also Se tion

2.7.2). Under the assumptionthatthis spa ingisequal to

5.5 nm

,the diameterof

the a tin monomer, the omputed

T

2

(δ)

urve ontains a at region around

δ = 0

even with

K = 2 pN/nm

. This solves the rst in oheren e of the HS71 model

Despite this su ess the single stroke size of

5.5 nm

was too small to justify

other experimentalobservations,and forthis reason theauthors were for etoadd

ase onda tivestateinthemodel. Thebehaviorofthemodelwasanalyzed

numer-i ally,and arathergoodtofthe experimentaldataregardingtherate ofre overy

was rea hed. The results are shown in Fig. 2.5, together with unsatisfa tory

pre-di tions of HS71 model. This solved even the se ond problemof the original1971

model.

Figure 2.5: Re ipro al of half-time of tension re overy vs. imposed length step. Solid line:

experimental values from [14℄. Points: Huxley and Tideswell model predi tion. Dotted line:

Huxleyand Simmonsmodelpredi tion

Weemphasize,however, thatthepri eofalargernumberofpossiblestateswas

a largernumberof freeparameters, and that the pre iserelationbetween the rate

onstants andthe shapeof thetotal energyused inHS71model hasbeenpartially

lost. We shouldmention though that,the hypothesis that the rate onstants vary

exponentiallywiththeworkdoneinstret hingoftheelasti element,waspreserved

(see [39℄). Asthe authorshaveobserved, thisrepresentsa dieren ebetween their

model and the Eisenberg and Hill model, where the authors used, as a typi al

rea tion rate, an arbitrary fun tion adjusted to obtain agreement.

Smith et al. model

Asalastexample,wedis ussaveryre entandthemost omplete modelofmus le