The Lorenz attractor arises in the study of the Lorenz Oscillator, a dynamical system. Overview Before the advent of computers, finding an orbit required sophisticated mathematical techniques and could be accomplished only for a small class of dynamical systems. Numerical methods implemented on electronic computing machines have simplified the task of determining the orbits of a dynamical system. For simple dynamical systems, knowing the trajectory is often sufficient, but most dynamical systems are too complicated to be understood in terms of individual trajectories. The systems studied may only be known approximately—the parameters of the system may not be known precisely or terms may be missing from the equations. History Many people regard Henri Poincaré as the founder of dynamical systems. Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). Basic definitions Flows
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Feedback"...'feedback' exists between two parts when each affects the other."(p53, §4/11) A feedback loop where all outputs of a process are available as causal inputs to that process "Simple causal reasoning about a feedback system is difficult because the first system influences the second and second system influences the first, leading to a circular argument. In this context, the term "feedback" has also been used as an abbreviation for: Feedback signal – the conveyance of information fed back from an output, or measurement, to an input, or effector, that affects the system.Feedback loop – the closed path made up of the system itself and the path that transmits the feedback about the system from its origin (for example, a sensor) to its destination (for example, an actuator).Negative feedback – the case where the fed-back information acts to control or regulate a system by opposing changes in the output or measurement. History Types Positive and negative feedback Biology
Cell signalingTraditional work in biology has focused on studying individual parts of cell signaling pathways. Systems biology research helps us to understand the underlying structure of cell signaling networks and how changes in these networks may affect the transmission and flow of information. Such networks are complex systems in their organization and may exhibit a number of emergent properties including bistability and ultrasensitivity. Analysis of cell signaling networks requires a combination of experimental and theoretical approaches including the development and analysis of simulations and modeling. Unicellular and multicellular organism cell signaling Classification Signaling within, between, and among cells is subdivided into the following classifications: Intracrine signals are produced by the target cell that stay within the target cell.Autocrine signals are produced by the target cell, are secreted, and effect the target cell itself via receptors. Figure 2. Figure 3.
CalculusHistory Modern calculus was developed in 17th century Europe by Isaac Newton and Gottfried Wilhelm Leibniz (see the Leibniz–Newton calculus controversy), but elements of it have appeared in ancient Greece, China, medieval Europe, India, and the Middle East. Ancient The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Medieval Modern In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. These ideas were arranged into a true calculus of infinitesimals by Gottfried Wilhelm Leibniz, who was originally accused of plagiarism by Newton. He is now regarded as an independent inventor of and contributor to calculus. Leibniz and Newton are usually both credited with the invention of calculus. Foundations
Cell migrationCell migration is a central process in the development and maintenance of multicellular organisms. Tissue formation during embryonic development, wound healing and immune responses all require the orchestrated movement of cells in particular directions to specific locations. Errors during this process have serious consequences, including intellectual disability, vascular disease, tumor formation and metastasis. An understanding of the mechanism by which cells migrate may lead to the development of novel therapeutic strategies for controlling, for example, invasive tumour cells. Cells often migrate in response to specific external signals, including chemical signals and mechanical signals. Due to the highly viscous environment (low Reynolds number), cells need to permanently produce forces in order to move. Cell migration studies The migration of cultured cells attached to a surface is commonly studied using microscopy. Common features Molecular processes of migration
State-space representationIn control engineering, a state-space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations. "State space" refers to the space whose axes are the state variables. The state of the system can be represented as a vector within that space. To abstract from the number of inputs, outputs and states, these variables are expressed as vectors. inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the information about a system. State variables The internal state variables are the smallest possible subset of system variables that can represent the entire state of the system at any given time. The minimum number of state variables required to represent a given system, , is usually equal to the order of the system's defining differential equation. Linear systems Block diagram representation of the linear state-space equations inputs, outputs and where: ). .
Biochemical systems theoryBiochemical systems theory is a mathematical modelling framework for biochemical systems, based on ordinary differential equations (ODE), in which biochemical processes are represented using power-law expansions in the variables of the system. This framework, which became known as Biochemical Systems Theory, has been developed since the 1960s by Michael Savageau and others for the systems analysis of biochemical processes. According to Cornish-Bowden (2007) they "regarded this as a general theory of metabolic control, which includes both metabolic control analysis and flux-oriented theory as special cases". Representation The dynamics of a species is represented by a differential equation with the structure: where Xi represents one of the nd variables of the model (metabolite concentrations, protein concentrations or levels of gene expression). j represents the nf biochemical processes affecting the dynamics of the species. ij (stoichiometric coefficient), See also Books:
Conceptual modelA conceptual model is a model made of the composition of concepts, which are used to help people know, understand, or simulate a subject the model represents. Some models are physical objects; for example, a toy model which may be assembled, and may be made to work like the object it represents. The term conceptual model may be used to refer to models which are formed after a conceptualization (generalization) process in the mind. Conceptual models represent human intentions or semantics[dubious ]. Conceptualization from observation of physical existence and conceptual modeling are the necessary means that humans employ to think and solve problems. Models of concepts and models that are conceptual The term conceptual model is ambiguous. Type and scope of conceptual models Overview A conceptual model's primary objective is to convey the fundamental principles and basic functionality of the system in which it represents. Techniques
Metabolic control analysisMCA was originally developed to describe the control in metabolic pathways but was subsequently extended to describe signaling and genetic networks. MCA has sometimes also been referred to as Metabolic Control Theory but this terminology was rather strongly opposed by Henrik Kacser, one of the founders. More recent work has shown that MCA can be mapped directly on to classical control theory and are as such equivalent. Control Coefficients A control coefficient measures the relative steady state change in a system variable, e.g. pathway flux (J) or metabolite concentration (S), in response to a relative change in a parameter, e.g. enzyme activity or the steady-state rate ( ) of step i. and concentration control coefficients by: Summation Theorems The flux control summation theorem was discovered independently by the Kacser/Burns group and the Heinrich/Rapoport group in the early 1970s and late 1960s. Elasticity Coefficients We assume that and . .