Concluding Remarks Volume 3 of the set of textbooks devoted to Circulatory and Ventilatory Sys- tems in the framework of Biomathematical and Biomechanical Modeling gives the basic knowledge on the first phases of signal transduction triggered by extracellular messengers. Signaling determines cell behavior. Numerous process can be modeled to quickly assess effects of parameters, all other agents remaining constant, once the mathematical model has been vali- dated. Advantage of mathematical models is to provide the complete quantity fields, whereas measurements are made in some points or corresponds to averages of ex- ploration windows of the field of the investigated variable. Mathematical description of large, complex biochemical reaction networks, in which molecules are nodes, and modeling of the dynamics of interactions (scaffold- ing, reaction, transcription, etc.) relies on computational simulations. A biochemical reaction network is defined by: (1) a set of variables — the state variables — that de- fine the state of the system and (2) rules of temporal changes and possible transport of involved variables. The network behavior can be analyzed in a single cells and extended to multicellular systems, especially in tumor models. Mathematical models of cascades of chemical reactions are aimed at describing evolution of molecular concentrations 1 using the mass action law or a reaction– diffusion equation set. A given reaction occurs with a certain probability. Stochastic models rely on a stochastic update of system variables. On the other hand, determin- istic models are carried out in systems with a large number of molecules, the time and variable states uniquely defining the state at the next time step. In a deterministic continuous formulation based on the mass action formalism, 2 molecular reactions 1. Typical molecular concentrations, i.e., continuous variables related to the number of molecules per unit volume, range from 0.1 nmol to 1 μmol. 2. The law of mass action states that the rate of a given chemical reaction is proportional to the product of the concentrations of reactants with eventual exponents that are stoichiometric coefficients for the reaction, in the absence of one-to-one stoichiometry. It is based on several assumptions: (1) well mixed media; and (2) low molecular concentrations; and (3) a probabil- ity of diffusing molecules to be available for the reaction proportional to the concentration. M. Thiriet, Signaling at the Cell Surface in the Circulatory and Ventilatory Systems, 809 Biomathematical and Biomechanical Modeling of the Circulatory and Ventilatory Systems 3, DOI 10.1007/978-1-4614-1991-4, © Springer Science+Business Media, LLC 2012 810 Concluding Remarks are described by differential equations defining the rate of change of molecular con- centrations. A rate constant (or affinity parameter) describes the occurrence rate of a reaction when reactants are close each other. In general, a mass action reaction can be written as: NR NP k f ∑ siRi −−−−→ ∑ s jPj, (11.1) i=1 j=1 where NR reactants Ri generate NP products Pj with a given stoichometry sk (relative quantities of reactants and products in the chemical reaction) with a forward rate 3 parameter k f (and a backward rate parameter kb when the reaction is reversible. The modeling procedure for a biochemical network incorporates several steps: (1) definition of the molecular players and their interactions; (2) description of the major molecular interactions by a mathematical model; (3) estimation of values of the model parameters (e.g., diffusion coefficients and reaction rates); and (4) analysis of the dynamical behavior. The dynamics of a continuous system can be described by a set of ordinary differ- ential equations (ODE). In general, this ODE set cannot be solved analytically and, thus require numerical integrations. On the other hand, reaction–diffusion models are based on partial differential equations (PDE). For example, an activator (act)– inhibitor (inh) reaction-diffusion model (Gierer-Meinhardt model) in morphogenesis is given by: ∂ 2 cact cact 2 = p + pact − dact cact + Dact ∇ cact , ∂t cinh ∂ cinh 2 2 = pa + pinh − dinhcinh + Dinh∇ cinh, (11.2) ∂t where cact is the concentration of a short-range, autocatalytic activator, cinh that of a long-range inhibitor, p the production rate, pact the production rate of the activator required to initiate the activator autocatalysis, pinh the low baseline production rate of the inhibitor, d the decay rate, and D the diffusion coefficient. Cells react to various types of external stimuli, in particular, mechanical stresses. In physiological systems associated with flows, the magnitude and direction of me- chanical stresses applied by the flowing fluid on the wetted surface of conduit wall (i.e., vascular endothelium with its glycocalyx or respiratory epithelium with the mucus layer and periciliary fluid) as well as within the vessel wall varies during the cardiac and breathing cycles. The heart generates an unsteady flow with a given frequency spectrum in a network of blood vessels characterized by complicated ar- chitecture and variable geometry both in space and time. Vessel geometry varies over short distances. The vascular network of curved blood vessels is composed of successive geometrical singularities, mainly branchings. The thoracic muscular cage cyclically inflates and deflates, thereby lowering and heightening the intrathoracic pressure, and hence dilating and collapsing lung alveoli and airways to inhale and exhale air. The respiratory tract is characterized by a large wetted surface inside a 3. The reaction constant of a reversible reaction is the ratio k f /kb. Concluding Remarks 811 small volume, especially in the nose and thorax. In the nose, turbinates allows heat and water exchange, but renders air currents less simple. In addition, the laryngeal constriction, the aperture of which varies during the ventilatory cycle, provokes air jet. The bronchial tree is a network of successive branchings at inspiration, or junc- tions at expiration, between short, more or less curved pipes of corrugated walls in large bronchi due to the presence of partial or complete cartilaginous rings. Therefore, blood and air streams correspond to time-dependent, three-dimension- al, developing, as they are conveyed in conduit entrance length, where boundary layer develop (Vol. 7). Moreover, blood vessels and airways are deformable. Changes in transmural pressure (the pressure difference between the pressure at the wetted sur- face of the lumen applied by the moving fluid on the deformable conduit wall and the pressure at the external wall side that depends on the activity on the neighbor organs) can also influence the shape of vessel cross-section, especially when it becomes neg- ative. In addition, in the arterial compartment, the change in cross-section shape can result from taper. More generally, possible prints of adjacent organs with more or less progressive constriction and enlargment, and adaptation to branching (transition zone) also give rise to three-dimensional flows. These flows are commonly displayed by virtual transverse currents, even if the vessel is considered straight. Furthermore, geometrical singularities influence flow pattern both upstream and downstream from it. Local changes in the direction of stress components can also be caused by flow separation and flow reversal during the cardiac and respiratory cycles. Flow separa- tion is set by an adverse pressure gradient when inertia forces and fluid vorticity are high enough, especially in branching segments. Due to its time-dependent feature, flow separation regions spread over a variable length during the flow cycle and can move. The location and variable size of the flow separation region depends on the flow distribution between branches that can vary during the flow cycle. Flow reversal occurs during the diastole of the left ventricle in elastic arteries, such as the aorta, and most of the muscular arteries, such as brachial and femoral arteries (but not in the carotid arteries), as well as during alternates from inspiratory decelerating flow phase and expiratory accelerating flow phase and conversely. In arteries, flow rever- sal can be observed either in a region near the wall, more or less wide with respect to the position of the local center of vessel curvature, or in the entire lumen. Consequently, the stress field experienced by the wall tissues are strongly vari- able both in time and space. Cellular sensors then process mechanical signals by ensemble averaging not only to raise the signal-to-noise ratio, but also to adequately adapt the local size of the conduit lumen, i.e., the local flow resistance to maintain either flow rate or pressure, only in the case of sustained, abnormal stress field. References Introduction 1. Lucretius (1997) De rerum natura (L I-685) [On the Nature of Things]. Garnier Flam- marion, Paris Chap. 1. Signal Transduction 2. Huang R, Martinez-Ferrando I, Cole PA (2010) Enhanced interrogation: emerg- ing strategies for cell signaling inhibition. Nature Structural and Molecular Biology 17:646–649 3. Lim WA (2010) Designing customized cell signalling circuits. Nature Reviews – Molecular Cell Biology 11:393–403 4. Manz BN, Groves JT (2010) Spatial organization and signal transduction at intercellu- lar junctions. Nature Reviews – Molecular Cell Biology 11:342–352 5. Alexander SPH, Mathie A, Peters JA (2009) Guide to Receptors and Channels (GRAC), 4th ed., British Journal of Pharmacology 158:S1–S254 (www3.interscience.wiley.com/journal/122684220/issue) 6. Liu YY, Slotine JJ, Barabási AL (2011) Controllability of complex networks. Nature 473:167–173 7. Rall TW, Sutherland EW (1958) Formation of a cyclic adenine ribonucleotide by tissue particles. Journal of Biological Chemistry 232:1065–1076 8. Sturm OE, Orton R, Grindlay J, Birtwistle M, Vyshemirsky V, Gilbert1 D, Calder1 M, Pitt A, Kholodenko B, W Kolch (2010) The mammalian MAPK/ERK pathway exhibits properties of a negative feedback amplifier. Science Signaling 3:ra90 9. Shimizu TS, Tu Y, Berg HC (2010) A modular gradient-sensing network for chemota- xis in Escherichia coli revealed by responses to time-varying stimuli.