Sliding Filament Model Myosin filament Myosin head Actin binding site Actin filament x k +Rate constants k - x As the actin filament moves past the (fixed) myosin filament, the myosin head can bind to it at the red triangle. When it does, the springs are either stretched or compressed and a force x acts at the binding site. Equations governing the probability n(x,t) that a cross-bridge is attached dn( x,t) = n( x,t) ? v n(x,t) = [ 1? n( x,t) ] k + ( x) ? n(x,t)k ? ( x) dt t x Formation of new Detachment of existing bonds bonds At steady state [n = n(x)] ? v dn(x) = [ 1? n(x) ] k + ( x) ? n(x)k ? (x) dx k + k + = attachment rate; k - = k - detachment rate; n = probability of attachment x h 1 The sliding filament model x > h: In this region the actin binding site is approaching the free myosin head, unoccupied. Since both k + and k - are zero, no binding occurs: n(x) = n(h) = 0 h-x 0 < x <h: If binding is to occur, it has to do so (according to this simple model) within this narrow region where the binding rate constant is large, described by the equation: ?v dn = (1? n)k + 0 dx k + k - 0 ? k + x 0 ? n(h ? x 0 ) = 1? exp ? ? ? ? v ? x h 0 < x < h-x 0 Both the attachment and detachment rate constants are zero, so the myosin head can neither bind to nor detach from an actin filament, and the probability of attachment remains constant: n(x) = n(h-x 0 ) = constant x < 0 As the complex moves into the region x < 0, the force of interaction sustained at the actin-myosin bond changes sign and its probability of attachment begins to fall, as described by the equation: ?v dn =? k ? 0 n k + k - ? k ? x ? ? ? k + 0 x 0 ?? ? k ? x ? dx 0 0 n(x) = n(0)exp ? ? = ?1? exp ? ? ?? exp ? ? ? v ? ? ? v ?? ? v ? x h 2 W = xdx = 2 a 2 ? b 2 ( ) ?b a ∫ Work done by a single cross-bridge that attaches at x=a and detaches at x=-b: n(x) s As /2 [ ?∞ ∞ ∫ xdx ] lA = n(x)xdx = 2lA n(0)x exp k ? 0 x v ? ? ? ? ? ? ?∞ 0 ∫ dx + n(0)xdx 0 h ∫ ? ? ? ? ? ? 2lA ?∞ ∞ ∫ s As s As = s s h 2 4l 1? 2 v hk ? 0 ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? 1? exp ? k + 0 x 0 v ? ? ? ? ? ? ? ? ? ? ? ?= max = s s h 2 4l max = 1? v v max ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? 1? exp ? k + 0 x 0 v ? ? ? ? ? ? ? ? ? ? ? ? v max = hk ? 0 2 Predicted force-velocity curve from cross- bridge model F F max = 1 ? v v max ? ? ? ? ? ? 2 ? ? ? ? ? ? ? ? 1 ? exp ? k + 0 x 0 v ? ? ? ? ? ? ? ? ? ? ? ? -0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 V/Vmax F/Fmax 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 v/vmax F/Fmax or P/Pmax v v max = 1? F F max ( ) 1+ C F F max ( ) Hill’s equation 3 Introduction to Cellular Biomechanics References: R.D. Kamm Chapters 2.1, 2.2 (handed out) Molecular Cell Biology, Lodish et al. Goals for today: ? Why is cell mechanics important ? ? Important structural components of the cell. ? Plasma Membrane. Models Length scales and details Lumped parameters (Kelvin, Voight, Maxwell..) Coarse Grained Continuum Mechanics Statistical Mechanical Models Single Molecule 4 Why is cell mechanics important ? Critical to function: red blood cells Migration Cell-Cell/Cell-Matrix Adhesion Division Mechanotransduction- respond to mechanical stimuli - cell differentiation - gene expression -diseases (arthritis) Single Cell Mechanics: Aspiration by Micropipette What mechanical properties can we measure ? 5 During blood clotting, platelets change shape due to changes in the actin cytoskeleton 6 Images removed due to copyright considerations. 7 Important Structural Components in Cells 1. Membrane 2. Cytoskeleton 3. Nucleus and other organelles 4. Cytosol (excluding the cytoskeleton) 5. Adhesion sites Plasma Membrane Cytoskeleton 1. Actin 2. Microtubules 3. Intermediate Filaments ? Actin filaments (or microfilaments) are one of the three protein filament systems that comprise the cytoskeleton ? Eukaryotic cells contain abundant amounts of highly conserved actin Figure 18-1 Motility and the Cytoskeleton 8 Image courtesy of J. Hartwig. Used with permission. Images removed due to copyright considerations. See Figure 18-1 in [Lodish]. Organelles of the eukaryotic cell ? Lysosomes ? Peroxisomes ? Mitochondria ? Chloroplasts ? ? the Golgi complex ? the Nucleus ? the Cytosol the Endoplasmic Reticulum Viscosity=50-10 4 cp (water : 1 cp) Valentine and Weitz 2003 microrheology Cytosol (excluding the cytoskeleton) ? Inclusion bodies ? Proteins (actin monomers) ? Ions ? Water Xenopus egg extracts: 9 Image removed due to copyright considerations Image removed due to copyright considerations. Image removed due to copyright considerations. Case Study: Bacteria- 2 Primary Roles: 2) adhering to and colonizing environmental surfaces (rocks, hair, teeth…) In endothelial cells- compressible barrier from blood cells Adhesion sites Glycocalyx: ‘Cell Coat’, ‘Furry Coat’ 1) resisting phagocytosis ?Coupling to tissue ?Sensing ?Migration ?Communication 10 Image removed due to copyright considerations. See Holland, N.B., et al. Biomimetic engineering of non-adhesive glycocalyx-like surfaces using oligosaccharide surfactant polymers. Nature 392(6678):799-801 (1998 Apr 23). 11 Plasma Membranes Plasma Protein Classes: Integral membrane proteins Peripheral proteins Associated Skeletal (spectrin) 1) Lipid bilayer 2) Plasma proteins 3) Carbohydrates Functions of the Plasma Membrane ? Regulate transport of nutrients into the cell ? Regulate transport of waste out of the cell ? Maintain “proper” chemical conditions in the cell ? Provide a site for chemical reactions not likely to occur in an aqueous environment ? Detect signals in the extracellular environment ? Interact with other cells or the extracellular matrix (in multicellular organisms) ~5nm Main components: Lipid Membrane Permeability ion movements through channels Molecular Cell Biology Patch Clamp permit measurement of E. Neher and B. Sakmann 1976 (Nobel Prize 1991) Lodish et al., Chapter 21 12 Image removed due to copyright considerations. Image removed due to copyright considerations. See Figure 21-19 in [Lodish]. Fig. 21-19a is available online via PubMed Bookshelf at http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=mcb.figgrp.6162. Cortical networks in erythrocytes Due to the amphiphilic nature and structure (2 tails) of phospholipids, these molecules spontaneously assemble to form closed bilayers. Liposomes: Drug delivery systems Lipid Bilayers Phospholipids are the main (lipid) constituents of most biomembranes. Some common head groups tails heads Phospholipid structure Source: Molecular cell biology, Lodish et al. 14 Image removed due to copyright considerations. Images removed due to copyright considerations.