Understanding Muscle Contraction: A Multiscale Approach to Muscle Physiology, Slides of Human Physiology

Download Understanding Muscle Contraction: A Multiscale Approach to Muscle Physiology and more Slides Human Physiology in PDF only on Docsity! Multiscale Systems from Particles to Continuum: Modelling and Computation Modelling Muscle Contraction … a multiscale approach Giovanni Naldi Dipartimento di Matematica ``F. Enriques’’ Università di Milano Porto Ercole, M&MKT 2016 Skeletal Muscle  Striated and voluntary  Attaches to skeleton via tendons  Most abundant tissue in the body (45-75% of body weight) Muscle Fiber • Long, cylindrical, multinucleated cells • Between fibers are blood vessels • Surrounded by endomysium • Composed of myofibrils filament Z-line sarcomere A-band I-band Myofibrils • Literally (muscle thread) • Contractile element of muscle • Made up of filaments aligned in parallel • filaments make striations - Banding pattern • One repeating unit is called a sarcomere • string of sarcomeres in series Sarcomeres • Functional unit of muscle contraction • Literally ‘muscle segment’ • Number of sarcomeres in a fiber is very important to muscle function • When each sarcomere shortens the same amount, the fiber with more sarcomeres will shorten more. • Made up of myofilaments – Thick and thin filaments A sarcomere is about 1.5-3.5 m long. Actin • Multiple actin molecules (G-actin) polymerize to form a long chain (F-actin). • A pair of F-actin chains twist together to form the thin filament. • G-actin has a binding site for a myosin head. Muscle contraction Sliding filament theory  AF Huxley and HE Huxley (1954, Nature)  Light and Electron microscopy  Both published results same time in Nature  Does not explain lengthening contractions  The exertion of force by muscle is accompanied by the sliding of thick and thin filaments past one another  Commonly explained by cross-bridges (AF Huxley, 1957) Video • Sarcomere Length Changes During Contraction • The I-band and H-zone shorten but the A-band remains constant. • Sliding-filament theory of contraction. Overlapping filaments slide past each other. Sir Andrew Huxley Sliding filament model, 1954, hypothesis • No deformation of filaments during the contraction • Only the distance between the Z disc changes According to the sliding filament theory, the contractile component is able to develop force or to shorten because of the interaction between the proteins belonging to the two sets of filaments, actin and myosin (cross- bridge cycle) The cross-bridge cycle is dependent on the level of activation of the contractile proteins. The activation is considered to be proportional to the amount of Ca ++ bound to troponin belonging to the thin filament. The amount of bound Ca ++ can be obtained as output of a compartmental system that describes the Ca ++ movements inside the muscle cell. The input of the model is the action potential, which affects some parameters of the Ca ++ compartmental model. Sliding filament model (cross bridges) Here, for simplicity, we consider a two state model for the cross-bridges Rheological model (Eisenberg- Hill) According to the classic view of Hill the muscle’s mechanical properties can be separated into three elements. Two elements are arranged in series: (CE) an active force generating contractile freely extensible at rest, but capable of shortening when activated; (SE) a series elastic (i.e. depending on the amount of strain) or viscoelastic (i.e. depending also on strain rate), which represents the structures on which the CE exerts its force during contraction (tendons, Z-band, connective tissue). To account for the mechanical behavior of muscle at rest, a parallel (visco-)elastic element (PE) is added (sarcolemma, collagen or elastic fibers). An example for G1 and G2 has been experimentally obtained by Capelo, Comincioli, Minelli, Poggesi, Reggiani, and Ricciardi (J. Biomech. 1981) Let us remark that we are also now assuming that the actin and myosin filaments do not deform. For the forces, we have 𝑃(𝑡) = FPE(t) + FSE(t), FCE(t)=FSE (t) where P(t) denotes the muscle tension. We can consider different experimental situations:  isometric, L(t) as input and P(t) as output;  isotonic, P(t) as input and L(t) as output;  and isometric-isotonic, sequence of the previous situations. Denoting by v(t) the rate of contraction we have (by using the parameter identification of CCMPRR) Limiting ourselves to the isometric case with L(t) = const., we obtain, Assumptions about the data. Given T > 0, and positive parameters k, bsg, agg, and function a, with wo: Rx [0,7] x R> R continous, and YR > 0, 3M(R) such that for any t € [0,T], 2, Z, u, 7 € R with mar{|ul, |z|, |u|, |@|} < R one has hit, tu) — Wz, t,a)| < M(B) (le —2| + lua), there exist two functions f(,¢), g(a,t) which are Lipschitz continuous in R, ¥¢ € [0,T], 0 < f(a,t) < g(2,t) Y(a,t), f(-,t) has a compact support Vt € [0,T]), such that —g(x, thu s ola, t,u) < f(x, t) — g(x, thu for any (x,t,u) € R x (0,T] x R. Moreover let 6 € C°:'(IR) with compact support and 0 < d(x) <1, ¥r € R. ISOMETRIC PROBLEM Find z € C®"((0,T]), and u € C'(R x {0,T]) such that Ou dz du at Hd = u(4,t,u(x,t)) (x,t) € Rx (0,7), u(x,0) = ¢(x), for any x € R, u(-,t) has compact support Vt € [0,T], 2(t) = ——— 1 og f + ua, t)eds| . bse ase JR Note. Here we write v(t)=z’(t) Note. The cross bridges could be in different states (and produce different forces), we have a system of uj(x,t) with j=1,…,M with a transition matrix. The whole picture: From the input Stimulus to the Mechanobiological transduction Series Parallel Heterogeneity has been shown in skeletal muscle, but, how the heterogeneity of functional properties inside each fibre determines the behaviour of the fibre taken as a whole, and how the heterogeneity among adjacent fibres in a muscle can influence the contractile performance of that muscle taken as a whole.