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Chromosome Oscillations in Mitosis Otger Campàs 1,2 and Pierre Sens 1,3 1 Institut Curie, UMR 168, Laboratoire Physico-Chimie «Curie» 2 Dept. Estructura i Constituents de la Matèria, Universitat de Barcelona 3 ESPCI, UMR 7083, Laboratoire Physico-Chimie Théorique

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Phases of cell division Scholey et al., Nature 422, 746 (2003) Prometaphase Prophase Metaphase Anaphase A Anaphase B Telophase

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Schematic representation of "typical" chromosome motions. M - monooriented, characterized by oscillatory motion; C - congression, characterized by movement away from the attached spindle pole by the trailing kinetochore and movement toward the attached spindle pole by the leading kinetochore; B - bioriented and congressed, characterized by oscillatory motion; A - anaphase, characterized by poleward movement of all kinetochores. G refers to a short rapid poleward glide that often occurs during initial monooriented attachment. R. V. Skibbens, Victoria P. Skeen, and E. D. Salmon J. Cell Biol., 122 (1993) 859-875 Chromosome mouvement in mitosis Rieder et al., Science 300, 91 (2003)

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Poleward force (kinetochore) chromokinesin on microtubule Away from pole force (Chromokinesines) Poleward force Away from pole force Balance of forces on the chromosome Levesque & Compton, J. Cell Biol. 154, 1135 (2001) kinetochore Kid DNA kinetochore centrosome Kid No Kid

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Balance of forces on the chromosome Equation for chromosome motion Phenomenological friction Kinetochore Force (poleward) Chromokinesins Force (Away-from-the-Pole) chromokinesin on microtubule Maximal velocity (at zero force) Stall force Force - Velocity Svoboda & block Force - Unbinding Force unbinding Unbinding rate Microscopic length

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Equation for the number of bound motors (that can exert a force on the chromosome) Binding rate (depends on MT concentration) Unbinding rate (depends on motor load) Available motors Chromokinesin Kinetics Velocity (depends on load) r for an isotropic aster Spatial information The total force is equally shared by all motors Collective effects

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Motor speed = chromosome velocity Dynamical equations chromosome motion & motor binding Stall number (number of motors needed to balance the kinetochore force) Friction number (kinetochore friction compared to Effective chromokinesin friction ) Detachment force (influence of applied force on detachment rate) Main control parameters

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Dynamical system Linear Dynamics Stability of the fixed point gives fixed point & Stability of the fixed point Linear stability analysis Eigenvalues

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Dynamical behavior unstable stable Independent of

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Linear Dynamics Stability of the fixed point Stable fixed point Unstable fixed point Unstable Chromosome friction Kinetochore force Unbinding rate High friction each motor feels its stall force (small velocity) and is independent from the other ones Low friction, the motors share the kinetochore force (no friction force) Very cooperative:the unbinding rate depends strongly on ‘n’ Unstable at low friction

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Stable fixed point distance to centrosome number of bound motors distance to centrosome number of bound motors The motor kinetics is much faster than the motion of the chromosome Non-Linear Dynamics Nullclines and Phase flows NullCline

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distance to centrosome number of bound motors distance to centrosome number of bound motors Unstable fixed point Non-Linear Dynamics Nullclines and Phase flows NullCline The n-Nullcline is non-monotonous The system reaches a LIMIT CYCLE

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Chromosome position Number of bound motors Experimental oscillations Skibbens et al., J. Cell Biol. 122, 859 (1993) parameters Numerical Integration and comparizon with experiments (slow) linear motion at constant speed Lot of bound CK motion at maximum CK speed (slow) linear motion at constant speed No of bound CK motion at natural Kinetochore speed Sharp change of direction Collective CK binding or unbinding Asymmetry of the oscillations:

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Experimental predictions Easiest parameter to modify: Total number of motors: N There is a critical number of motors below which oscillations stop

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Chromosome instability occurs because of collective motor dynamics Chromosome oscillations occur because the astral MTs provide a ‘position-dependent’ substrate for CK binding - The kinetochore is treated like one big (infinitely processive) motor Maximum velocityStall force Main outcome of the model Refinement: Force-sensitive kinetochore ‘Concept: ‘smart kinetochore’ Can sense both position and force Our model so far: ‘very dumb kinetochore’ Constant force and constant firction

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Conclusions Oscillations arise from the interplay between the cooperative dynamics of chromokinesins and the morphological properties of the MT aster. Highly non-linear oscillations, similar to those observed in-vivo, appear in a considerable region of the parameter space. Sawtooth shaped oscillations come from different kinetics of chromosome motion and motor binding dynamics Testable prediction - critical number of CK for oscillations

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