The Mathematics of Mechanobiology: Cetraro, Italy 2018

Preface
Abstract
Contents
1 Cell Motility and Locomotion by Shape Control
	1.1 Introduction
	1.2 Swimming at Low Reynolds Numbers
	1.3 Locomotion Principles and Minimal Swimmers
		1.3.1 Looping in the Space of Shapes: No Looping? No Party!
		1.3.2 Minimal Swimmers With or Without Directional Control
		1.3.3 Steering by Modulation of the Actuation Speed
		1.3.4 Swimming by Lateral Undulations: Optimality of Traveling Waves of Bending
	1.4 Biological Swimmers
		1.4.1 Chlamydomonas' Breaststroke
		1.4.2 Sperm Cells and Flagellar Beat
	1.5  Euglena Gracilis: A Case Study in Biophysics and a Journey from Biology to Technology
		1.5.1 Metaboly and Mechanisms for Shape Change, Embodied Intelligence
		1.5.2 Flagellar Swimming, Helical Trajectories and a Principle for Self-Assembly
	1.6 Shape Control and Gaussian Morphing
		1.6.1 Controlling the Shape of Surfaces by Prescribing Their Metric
		1.6.2 Axisymmetric Surfaces
		1.6.3 Cylinders from Cylinders
		1.6.4 Axisymmetric Surfaces with Non-constant Metric
		1.6.5 Protruding Necks and Localized Bulges
	1.7 Discussion and Outlook
	References
2 Models of Cell Motion and Tissue Growth
	2.1 Introduction
	2.2 Bacterial Movement by Run and Tumble
		2.2.1 Modeling Run and Tumble
		2.2.2 Existence of Solutions
		2.2.3 Derivation of the Patlak/Keller–Segel System
		2.2.4 Modulation Along the Path
	2.3 Macroscopic Models of Chemotactic Movement
		2.3.1 Elementary Properties
		2.3.2 Blow-Up in the Keller–Segel System
		2.3.3 Keller–Segel System with Prevention of Overcrowding
		2.3.4 The Flux Limited Keller–Segel System
		2.3.5 Traveling Bands
		2.3.6 Instabilities
	2.4 Compressible Models of Tissue Growth
		2.4.1 A Simple Model with a Single Type of Cells
			2.4.1.1 Supersolution with Bounded Support
			2.4.1.2 Existence of Solutions and a Priori Bounds
			2.4.1.3 A Variant of Aronson-Bénilan Estimate
		2.4.2 Single Cell Type Population Model with Nutrient
		2.4.3 Models with Two Cell Types
		2.4.4 Two Cell Type Model with Different Mobilities
		2.4.5 Surface Tension and the Degenerate Cahn–Hilliard Model
	2.5 Incompressible Models of Tissue Growth
		2.5.1 Single Cell Type Free Boundary Problem
		2.5.2 Single Cell Type Model with Nutrient and Free Boundary
		2.5.3 Two Cell Types Incompressible Model
		2.5.4 Multiphase Models
			2.5.4.1 The One Phase Closure
			2.5.4.2 The Darcy/Stokes Closure
			2.5.4.3 The Single Pressure Closure
	2.6 The Incompressible Limit and Stiff Pressure Law
		2.6.1 Single Cell Type Model, Incompressible Limit
			2.6.1.1 Weak Formulation of the Hele-Shaw Problem
			2.6.1.2 The Complementary Relation
			2.6.1.3 From the Weak Formulation to the Free Boundary Statement
		2.6.2 Single Cell Type with Nutrient, Incompressible Limit
		2.6.3 Open Problems
	References
3 Segregated Algorithms for the Numerical Simulation of Cardiac Electromechanics in the Left Human Ventricle
	3.1 Introduction
	3.2 Mathematical Models
		3.2.1 Ionic Model and Monodomain Equation
		3.2.2 Mechanical Activation
		3.2.3 Passive and Active Mechanics
			3.2.3.1 Prestress
		3.2.4 Cardiac Cycle
	3.3 Space and Time Discretizations
		3.3.1 Space Discretization
		3.3.2 Time Discretization
			3.3.2.1 Discretization of the 0D Fluid Model
	3.4 Numerical Coupling: Segregated Strategies
		3.4.1 Fully Monolithic Strategy (IIEIAIMI)
		3.4.2 Partially Segregated Strategy (IIEIAI)–(MI)
		3.4.3 Partially Segregated Strategy (ISIESIASI)–(MI)
		3.4.4 Fully Segregated Strategy (ISI)–(ESI)–(ASI)–(MI)
	3.5 Numerical Results
		3.5.1 Preprocessing
		3.5.2 Benchmark Problem with Idealized Geometry
		3.5.3 Subject-Specific LV: The Full Heartbeat
	3.6 Conclusions
	References
4 Power-Stroke-Driven Muscle Contraction
	4.1 Introduction
	4.2 General Ratchet Model
	4.3 X-Tilted Ratchet
		4.3.1 Typical Cycles
		4.3.2 Force-Velocity Relations and Stochastic Energetics
	4.4 Y-Tilted Ratchet
		4.4.1 Typical Cycles
		4.4.2 Force-Velocity Relations and Stochastic Energetics
	4.5 XY-Tilted Ratchet
		4.5.1 Motor Cycles
		4.5.2 Force-Velocity Relations and Stochastic Energetics
	4.6 Comparison of the Three Models
		4.6.1 Soft Device
		4.6.2 Hard Device
		4.6.3 Stochastic Energetics
	4.7 XY-Tilted Ratchet with a Steric Feedback
		4.7.1 The Model
		4.7.2 Hysteretic Coupling
		4.7.3 Non-potential Models
	4.8 Active Rigidity
		4.8.1 Macroscopic Problem
		4.8.2 Mean Field Model
		4.8.3 Non-dimensionalization
		4.8.4 Phase Diagrams
		4.8.5 Zero Temperature Limit
	4.9 Conclusions
	References