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| '''Biological applications of bifurcation theory''' provide a framework for understanding the behavior of biological networks modeled as [[dynamical system]]s. In the context of a biological system, [[bifurcation theory]] describes how small changes in an input parameter can cause a bifurcation or qualitative change in the behavior of the system. The ability to make dramatic change in system output is often essential to organism function, and bifurcations are therefore ubiquitous in biological networks such as the [[biochemical switches in the cell cycle|switches of the cell cycle]].
| | Howdy and welcome. My title is Korey Westhoff. Just one of my beloved hobbies is ice skating and I am attempting to make it a career. My spouse and I selected to reside in North Dakota and my spouse and children enjoys it. I am currently a manufacturing and distribution officer but I program on shifting it. Go to my web-site to find out far more: http://wiki.acceed.de/index.php/Benutzer:[http://Www.adobe.com/cfusion/search/index.cfm?term=&RachellTQEN&loc=en_us&siteSection=home RachellTQEN]<br><br> |
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| ==Biological networks and dynamical systems==
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| Biological networks originate from [[evolution]] and therefore have less standardized components and potentially more complex interactions than many networks intentionally created by humans such as [[electrical networks]]. At the cellular level, components of a network can include a large variety of proteins, many of which differ between organisms. Network interactions occur when one or more proteins affect the function of another through [[transcription (genetics)|transcription]], [[translation (biology)|translation]], [[Protein targeting|translocation]], or [[phosphorylation]]. All these interactions either activate or inhibit the action of the target protein in some way. While humans build networks with some concern for efficiency and simplicity, biological networks are often adapted from others and exhibit redundancy and great complexity. Therefore, it is impossible to predict quantitative behavior of a biological network from knowledge of its organization. Similarly, it is impossible to describe its organization purely from its behavior, though behavior can indicate the presence of certain [[network motif]]s.
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| [[File:Skotheimsystem.jpg| right | thumb | 300px | framed image | fig.1. Example of a biological network between genes and proteins that controls entry into [[S phase]]]]
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| However, with knowledge of network interactions and a set of [[parameter]]s for the proteins and protein interactions (usually obtained through [[empirical]] research), it is often possible to construct a model of the network as a [[dynamical system]]. In general, for n proteins, the dynamical system takes the following form<ref name = "Strogatz">Strogatz S.H. (1994), Nonlinear Dynamics and Chaos, Perseus Books Publishing</ref> where x is typically protein concentration:
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| : <math> \dot{x_1} = \frac{dx_1}{dt} = f_1(x_1, \ldots, x_n) </math>
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| : <math> \vdots </math>
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| : <math> \dot{x_i} = \frac{dx_i}{dt} = f_i(x_1, \ldots, x_n) </math>
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| : <math> \vdots </math> | |
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| : <math> \dot{x_n} = \frac{dx_n}{dt} = f_n(x_1, \ldots, x_n) </math> | |
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| These systems are often very difficult to solve, so modeling of networks as a [[linear dynamical system]]s is easier. [[Linear systems]] contain no products between ''x''s and are always solvable.
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| They have the following form for all i:
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| : <math> f_i = a_{i1}x_1 + a_{i2}x_2 + \cdots + a_{in}x_n \, </math> | |
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| Unfortunately, biological systems are often [[nonlinear]] and therefore need nonlinear models.
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| ==Input/output motifs==
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| Despite the great potential complexity and diversity of biological networks, all first-order network behavior generalizes to one of four possible I/O motifs: hyperbolic or [[Michaelis–Menten kinetics|Michaelis–Menten]], [[Ultrasensitivity|ultra-sensitive]], [[Bistability|bistable]], and bistable irreversible (a bistability where negative and therefore biologically impossible input is needed to return from a state of high output).
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| Ultrasensitive, bistable, and irreversibly bistable networks all show qualitative change in network behavior around certain parameter values – these are their bifurcation points.
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| ==Bifurcations==
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| [[File:Saddle node bifurcation - animation.gif|thumb|fig. 2. Saddle node bifurcation – The phase portrait changes with values of ε. As ε decreases, the fixed points come together and annihilate one another; As ε increases, the fixed points appear. dx/dt is denoted as v.]]
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| Nonlinear dynamical systems can be most easily understood with a one dimensional example system where the change in some measurement of protein x's abundance depends only on itself:
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| : <math> \dot{x} = \frac{dx}{dt} = f(x) \, </math>
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| Instead of solving the system analytically which can be difficult for many functions, it is often best to take a geometric approach and draw a [[phase portrait]]. A phase portrait is a qualitative sketch of the differential equation's behavior that shows equilibrium solutions or [[fixed point (mathematics)|fixed point]]s and the [[vector field]] on the real line.
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| Bifurcations describe changes in the stability or existence of fixed points as a control parameter in the system changes. As a very simple explanation of a bifurcation in a dynamical system, consider an object balanced on top of a vertical beam. The mass of the object can be thought of as the control parameter. As the mass of the object increases, the beam's deflection from vertical, which is x, the dynamic variable, remains relatively stable. But when the mass reaches a certain point – the bifurcation point – the beam will suddenly buckle. Changes in the control parameter eventually changed the qualitative behavior of the system.
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| For a more rigorous example, consider the dynamical system shown in figure 2
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| : <math> \dot{x} = -x^2 + \varepsilon </math>
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| where ε is the control parameter. At first, when ε is greater than 0, the system has one stable fixed point and one unstable fixed point. As ε decreases the fixed points move together, briefly collide into a semi-stable fixed point at ε = 0, and then cease to exist when ε < 0. [[File:Saddle node bifurcation diagram.png|thumb | fig.3. Bifurcation diagram for a saddle node bifurcation]] In this case, because the behavior of the system changes significantly when the control parameter ε is 0, 0 is a [[bifurcation point]]. This example bifurcation is called the [[saddle-node bifurcation]] and its [[bifurcation diagram]] (this time for <math> \dot{x} = x^2 + \varepsilon </math>) is shown in figure 3.
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| Other types of bifurcations are also important in dynamical systems, but the saddle node bifurcation is more important in biology. The reason for this is that biological systems are real and include small [[stochastic]] variations. For example, adding a very small term, 0 < h << 1 to a [[pitchfork bifurcation]] yields a stable fixed point and a saddle node bifurcation<ref name = "Strogatz" /> (figure 3). Similarly, a small error term collapses a [[transcritical bifurcation]] to two saddle-node bifurcations (figure 4).
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| [[File:Pitchfork Bifurcation with a small error term.png|thumb | fig.4. Bifurcation diagram for a pitchfork bifurcation without imperfection (left) and with a small imperfection term (right).]]
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| Combined saddle node bifurcations in a system can generate [[multistability]]. [[Bistability]] (a special case of multistability) is an important property in many biological systems often produced by network architecture that contains [[positive feedback]] interactions and [[ultrasensitivity|ultra-sensitive element]]s. Bistable systems are [[hysteretic]], that is, their behavior depends on the history of the input.<ref name = "Angeli">David Angeli, James E. Ferrell, Jr., and Eduardo D.Sontag. Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems. PNAS February 17, 2004 vol. 101 no. 7 1822-1827</ref> A hysteretic network can produce different output values for the same input value depending on its state (produce by the history of the input), a property crucial for switch-like control of cellular processes.<ref name = "Angeli" />
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| ==Examples==
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| [[File:Multistability in lactose utilization of E. coli.png|300px | thumb | fig.5. GFP expression in individual cells induced by GAL promoter activation follows a bimodal distribution (left). GFP expression as a function of TMG (lactose analogue) concentration shows bistability with two bifurcation points]]
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| Networks with bifurcation in their dynamics control many important transitions in the [[cell cycle]]. The [[Biochemical switches in the cell cycle#The G1/S switch|G1/S]], [[Biochemical switches in the cell cycle#The G2/M switch|G2/M]], and [[Biochemical switches in the cell cycle#Metaphase-anaphase switch|Metaphase–Anaphase]] transitions all act as [[biochemical switches in the cell cycle]].
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| In [[population ecology]], the dynamics of [[food web]] interactions networks can exhibit [[Hopf bifurcation]]s. For instance, in an aquatic system consisting of a [[primary producer]], a mineral resource, and an herbivore, researchers found that patterns of equilibrium, cycling, and extinction of populations could be qualitatively described with a simple nonlinear model with a Hopf Bifurcation.<ref name = "Fussmann">Gregor F. Fussmann, Stephen P. Ellner, Kyle W. Shertzer, and Nelson G. Hairston Jr. Crossing the Hopf Bifurcation in a Live Predator–Prey System. ''Science''. 17 November 2000: 290 (5495), 1358–1360. {{doi|10.1126/science.290.5495.1358}}</ref>
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| [[Galactose]] utilization in [[budding yeast]] (S. cerevisiae) is measurable through [[Green fluorescent protein|GFP]] expression induced by the GAL promoter as a function of changing galactose concentrations. The system exhibits bistable switching between induced and non-induced states.<ref name = "Song">Song C, Phenix H, Abedi V, Scott M, Ingalls BP, et al. 2010 Estimating the Stochastic Bifurcation Structure of Cellular Networks. ''PLoS Comput Biol'' 6(3): e1000699. {{doi|10.1371/journal.pcbi.1000699}}</ref>
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| Similarly, [[lactose]] utilization in [[E. coli]] as a function of thyo-methylgalactoside (a lactose analogue) concentration measured by a GFP-expressing lac promoter (figure 5) exhibits hysteresis and bistability.<ref name = "Ozbudak">Ertugrul M. Ozbudak, Mukund Thattai, Han N. Lim, Boris I. Shraiman & Alexander van Oudenaarden. Multistability in the lactose utilization network of Escherichia coli. ''Nature''. 2004 Feb 19 ;427(6976):737–40</ref>
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| ==See also==
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| *[[Biochemical switches in the cell cycle]]
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| *[[Dynamical Systems]]
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| *[[Dynamical systems theory]]
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| *[[Bifurcation Theory]]
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| *[[Cell cycle]]
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| *[[Theoretical Biology]]
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| *[[Computational Biology]]
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| *[[Systems Biology]]
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| *[[Cellular model]]
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| ==References==
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| {{Reflist}}
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| [[Category:Bifurcation theory]]
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Howdy and welcome. My title is Korey Westhoff. Just one of my beloved hobbies is ice skating and I am attempting to make it a career. My spouse and I selected to reside in North Dakota and my spouse and children enjoys it. I am currently a manufacturing and distribution officer but I program on shifting it. Go to my web-site to find out far more: http://wiki.acceed.de/index.php/Benutzer:RachellTQEN
My web site :: Zapatillas Asics