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Porto Ercole, M&MKT 2016
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
Skeletal Muscle
 Striated and voluntary
 Attaches to skeleton via tendons
 Most abundant tissue in the body (45-75% of body weight)
Structure of a muscle cell
A. Fascicles
– fiber bundles
B. Fibers
– muscle cell
– bundles of myofibrils
C. Myofibrils
D. Sarcomeres (series)
E. Actin & Myosin Filaments
Fascicles
A muscle is composed of multiple fascicles in parallel
– A sheath of connective tissue surrounds the muscle
(epimysium)
– Each fascicle is surrounded by connective tissue
(perimysium)
– Fascicles composed of bundles of muscle fibers
Muscle Fiber
• Long, cylindrical,
multinucleated cells
• Between fibers are blood
vessels
• Surrounded by endomysium
• Composed of 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
sarcomere
A-band
filament
Z-line
Myofibrils
I-band
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.
Myosin
•
•
•
•
Composed of two heavy chains twisted together. Has a ‘head’ and a ‘tail’.
The myosin head has 2 binding sites for actin and ATP.
The head can swivel about its hinge.
They bind to actin filaments to form cross- bridges to span the space
between filaments.
• In skeletal muscle, about 250 myosin molecules create to make a thick
filament.
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
Sir Andrew Huxley
• The I-band and H-zone shorten but the A-band remains constant.
• Sliding-filament theory of contraction. Overlapping filaments slide past each other.
Muscle Force Depends (mainly) on Four
Factors
•
•
•
•
Sarcomere (muscle) length
Velocity of muscle contraction
Activation level
Previous contraction history
Force-Length Relationship
Fact:
Muscles at very long and very short
lengths can not produce high forces
Fact:
Maximal force production of a muscle
depends on its length
Force-Length Relationship
Sliding filament model, 1954, hypothesis
• No deformation of filaments
during the contraction
• Only the distance between the Z
disc changes
Sliding filament model (cross bridges)
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 (crossbridge 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.
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).
Contractile Element: Half -Sarcomere
Let u(x,t) denote the density of cross-bridges which. at time t , are halfattached
between myosin and actin at the distance x from the equilibrium position in the element
CE.
x
actin
Cross bridge
myosin
The dynamics of the cross-bridges population is given by
𝑑𝑢
(x,t) =  𝑥, 𝑡, 𝑢
𝑑𝑡
where  gives the balance of the formation and the breakdown of bridges. By
considering a fixed frame, the material derivative becomes
𝑑𝑢
𝜕𝑢 𝑑𝑥 𝑢
(x,t) =
+
=  𝑥, 𝑡, 𝑢
𝜕𝑡 𝑑𝑡 𝑥
𝑑𝑡
where v(t)= dx/dt is the speed of the contraction of the half sarcomere.
Assuming, for simplicity, a linear elastic behavior with stiffness k, we
have that the force developed at time t by all interacting cross-bridges in
the half-sarcomere is given by:
The PE and SE elastic forces are
𝐹𝑃𝐸 t = 𝐺1 𝐿𝑃𝐸 𝑡 − 𝐿𝑃𝐸0
𝐹𝑆𝐸 t = 𝐺2 𝐿𝑆𝐸 𝑡 − 𝐿𝑆𝐸0
where 𝐺1 and 𝐺2 are given nonlinear
one-to-one real maps, and 𝐿𝑃𝐸0 , 𝐿𝑆𝐸0
are the initial lengths (𝐿𝑃𝐸0 (t)=L(t))
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,
Note. Here we write v(t)=z(t)
Theorem. The ISOMETRIC PROBLEM has one and only one
solution.
Outline of the proof.
Step 1 Consider the hyperbolic problems with z(t) given (for the
moment): the characteristics technique allows us to find
a unique solution Uz (x, t) (depending obviously on z ) such that
Step 2 By means of the data assumptions one can state the a priori
boundedness of the solutions z(t), and Uz
Step 3 Given function z(t), z(0), we can compute Uz(x,t), then (by using
the definition of z(t) and integrating the hyperbolic equation),
Step 4 Solve the problem: find z(t) in C0,1([0,T]) such that
z(t) = Gz(t)  t  [0,T]
By using a suitable fixed point Theorem.
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
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.
Series
Parallel
Example, from P. Colli, V. Comincioli, G.N., C. Reggiani, 1988.
Open problems:
-
Analysis of the networks of the sarcomere
Efficient numerical methods
Parameter identification and experiments
….