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A survey of formal approaches for spatial and hierarchical modelling
 

A survey of formal approaches for spatial and hierarchical modelling

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Computational models for the biological phenomenon of bone remodelling; investigation of spatial and multiscale formalisms.

Computational models for the biological phenomenon of bone remodelling; investigation of spatial and multiscale formalisms.

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    A survey of formal approaches for spatial and hierarchical modelling A survey of formal approaches for spatial and hierarchical modelling Presentation Transcript

    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions A survey of formal approaches for spatial and hierarchical modelling Research Thesis in Distributed Calculus and Coordination Candidate: Nicola Paoletti Supervisor: Prof. Emanuela Merelli Assistant supervisor: Dr. Diletta Romana Cacciagrano University of Camerino School of Science and Technology Master of Science Degree Course in Computer Science Academic Year 2009-2010 A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Outline 1 Introduction 2 Computational Systems Biology Mathematical approaches Computational approaches 3 Spatial Multiscale Paradigms Spatial P Systems Complex Automata Shape Calculus 4 Bone Remodelling Bone remodelling in CxA Bone remodelling in SP Bone remodelling in Shape Calculus 5 Conclusions A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Computational Systems Biology Systems Biology promotes the systematic and holistic study of biological entities; a living organism is regarded a network of interacting components which cannot be analysed separately. Computational Systems Biology aims to study, analyse and understand complex biological systems using mathematical and computational methodologies. Formal bio-models are based on experimental data and include qualitative (structure, components, interactions) and quantitative (parameters) information. Models are refined or redefined when inconsistencies occur in the phase of model validation. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Spatial and multiscale models Real biological systems are intrinsically multiscale: macroscopic structure is influenced by the microscopic components and interactions occur at different spatial and temporal scales. Spatial and physical information such as mass, position, occupancy, movement and shape determines interactions among biological entities. Several spatial and multiscale formalisms have been proposed to model biological systems The goal of this work is to give a survey of spatial and hierarchical models and evaluate them with concrete biological case-studies; here, we will focus on the process of Bone Remodelling. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Outline 1 Introduction 2 Computational Systems Biology Mathematical approaches Computational approaches 3 Spatial Multiscale Paradigms Spatial P Systems Complex Automata Shape Calculus 4 Bone Remodelling Bone remodelling in CxA Bone remodelling in SP Bone remodelling in Shape Calculus 5 Conclusions A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Mathematical approaches Continuity and discreteness Three main categories of mathematical models are distinguished: Continuous models: are typically used when the state of a biological system is determined by the concentration values of the various components. Discrete models: have associated discrete states; the temporal evolution of the system is regulated by discrete transitions. Hybrid models: combine continuous and discrete approaches. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Mathematical approaches Differential Equations Differential equations are continuous and deterministic. They can predict temporal changes over infinitesimal time intervals. Efficient numerical and analytical solving methods are available. Ordinary Differential Partial Differential Equations Equations (ODE) (PDE) One independent variable Several independent (time) variables (time,x,y,z,...) Dynamical systems Multidimensional systems Used to model systems of Used to model chemical reactions reaction-diffusion systems A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Mathematical approaches Stochastic Processes Stochastic processes describe biological systems characterized by discrete states and random transitions. Formally, a stochastic process is a family X (t) : t ∈ T of random variables, indexed by some set T . If T = Z+ it is called discrete time, while if T = R+ is called continuous time. Continuous Time Markov Chains (CTMC) are a particular class of continuous time stochastic processes and represent the dominant formalism for discrete and stochastic models. Gillespie’s method is an exact stochastic algorithm for the simulation of chemical reaction systems and represents the leading technique for describing biological and chemical kinetics. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Computational approaches Petri Nets (1/2) Petri Nets (PN) are a graphical and computational formalism for complex systems. They are discrete and non-deterministic. A PN is a weighted direct graph with two disjoint sets of nodes, places P and transitions T ; arcs are only of the form (pin , t) or (t, pout ), where pin , pout ∈ P and t ∈ T . Tokens model the multiplicity of a place and the weight of an arc indicates how many tokens are “consumed/produced” by a transition. Many variants of PN are available (coloured, hierarchical, stochastic, timed, hybrid). PN have been widely used in systems biology, especially for the modelling of metabolic processes. In [Hofest¨dt98], the a biocatalytic reaction of lactose is modelled with a self-modified Petri net. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Computational approaches Petri Nets (2/2) Example (Biocatalytic reaction of lactose) Each unit of lactose produces one unit of galactose and one unit of glucose. The reaction is enabled if the enzyme β-galactosidase is available. Arcs labelled with “β-galactosidase” have associated a weight equals to the concentration of the enzyme itself. Hence, it is consumed and produced during the reaction. R. Hofest¨dt and S. Thelen. a Quantitative modeling of biochemical networks. In Silico Biology, 1(1):39–53, 1998. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Computational approaches Hybrid Automata (1/2) Hybrid Automata (HA) describe systems in mixed discrete-continue fashion. Its main components are: The control graph G = (Q, E ), which models the discrete state space; nodes in Q represent discrete states and edges in E represent discrete transitions. A set X of real variables, which models the continuous states of the system. Continuous dynamics is expressed by differential equations. Invariants, guards, and resets which are defined over the variables in X . Real biological systems are characterized by both continuity and discreteness. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Computational approaches Hybrid Automata (2/2) A key example is [Lewis98], where the mechanism of delta-Notch signalling is implemented with HA. Example (Delta-Notch signalling) The protein delta binds and activates its receptor Notch. Then, the activation of Notch affects the production of Notch ligands, leading to: Lateral inhibition high Notch levels inhibit ligand production Lateral induction activation of Notch promotes ligand production J. Lewis. Notch signalling and the control of cell fate choices in vertebrates* 1. In Seminars in Cell and Developmental Biology, volume 9, pages 583–589. Elsevier, 1998. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Computational approaches Process Algebras (1/2) Process algebras (PA) are formal languages for the specification of concurrent and interacting systems. Processes performs input and output actions on channels and can be composed through operations of alternative, sequential and parallel composition. Bisimulation is an equivalence relation determining if two syntactically different processes are behaviourally equivalent. Semantics is given in terms of Labelled Transitions Systems (LTS), obtained from the process definition by a set of SOS rules. Model checking allows the formal verification of properties over the system. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Computational approaches Process Algebras (2/2) PAs model biological systems as networks of communicating and concurrent processes (e.g. processes → species, and actions → reactions) and many applications in systems biology have been proposed. The main advantages are: Complex biological systems can be defined by starting from its subcomponents Compositional reuse of models Extensions to basic PAs allow to describe aspects such as temporal dynamics, stochastic and probabilistic transitions, spatial and physical characteristics, reaction reversibility or compartmentalization. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Outline 1 Introduction 2 Computational Systems Biology Mathematical approaches Computational approaches 3 Spatial Multiscale Paradigms Spatial P Systems Complex Automata Shape Calculus 4 Bone Remodelling Bone remodelling in CxA Bone remodelling in SP Bone remodelling in Shape Calculus 5 Conclusions A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Spatial P Systems P Systems P Systems are bio-inspired computing devices, composed by a hierarchy of membranes each of them contains a multiset of objects. The root membrane is called skin membrane. Objects represent molecules and evolution rules of the form u → v model chemical reactions between reactant u and product v objects. Target messages specify whether the products of the reaction remain in the membrane or are moved out. The result of a successful (convergent) computation is the multiset of objects sent out the skin membrane. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Spatial P Systems Spatial P Systems A Spatial P System (SP) is composed of rectangular membranes and objects located in the two-dimensional space. Membrane positioning is subject to some intuitive constraints; a single position can contain an arbitrary number of ordinary objects, but only one mutually esclusive object. Evolution rules are of the form u → v or u1 − u2 → v1 − v2 (simultaneous application of two rules) Target messages can be of the form: vδp , δp ∈ Z2 : multisets of objects v in position p that remain in the same membrane are located to position p + δp; vini : products are sent in the nearest position of the child membrane i; vout : products are sent out in one of the nearest positions outside the membrane. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Spatial P Systems Syntax Definition (Spatial P System) A Spatial P system Π is a tuple V , E , µ, σ, W (1) , ..., W (n) , R1 , ..., Rn where V and E are disjoint alphabets of ordinary objects and mutually esclusive (ME) objects; µ ⊂ N × N describes the tree-structure of membranes; (i, j) ∈ µ implies that membrane j is child of membrane i; σ : {1, ..., n} → N 2 × (N + )2 describes position and dimension of membranes; (i) W (i) = {wx,y ∈ (V ∪ E )∗ | 0 ≤ x < wi , 0 ≤ y < hi }, with i = 1, ..., n indicates the objects at each position of i; Ri is the set of evolution rules of membrane i. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Spatial P Systems Example Example Membranes 4 and 5 are wrongly located, since adjacent edges are forbidden. Red arrows indicate the possible positions of the object a after an out rule. R. Barbuti, A. Maggiolo-Schettini, P. Milazzo, G. Pardini, and L. Tesei. Spatial P systems. Natural Computing, pages 1–14, 2010. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Complex Automata Complex Automata Complex Automata (CxA) are a formalism for multiscale complex systems, composed by a number of single-scale Cellular Automata (CA). Formally, a CxA is a graph (V , E ), where each vertex Ci ∈ V is a CA and each edge Eij is a coupling procedure between Ci and Cj . Coupling procedures describes how information is exchanged between the two vertices and edges act as communication channels. A. Hoekstra, J. Falcone, A. Caiazzo, and B. Chopard. Multi-scale modeling with cellular automata: The complex automata approach. Cellular Automata, pages 192–199, 2010. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Complex Automata Cellular Automata Definition (Cellular Automata) A CA is a tuple C = A(∆x, ∆t, L, T ), S, s 0 , R, G , F where A is the spatial domain of size L and made of cells of size ∆x. ∆t is the time step and T is the number of iterations. Hence, time scales range from ∆t and T , and spatial scales are between ∆x and L. S is the set of states; s 0 ∈ S is the initial state. R is the evolution rule. G is the topology describing the neighbourhood relation. F is the flux of information exchanged at each iteration between the system and its surroundings. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Complex Automata Evolution rules Each cell of a CA has associated an evolution rule consisting of three operations: Propagation: sends the local states to the neighbours that need it. ComputeBoundary: specifies the values of the variables that are defined by the external environment. Collision: executes the evolution rule for each cell, using the values obtained from propagation and computeBoundary. B. Chopard and M. Droz. Cellular automata modeling of physical systems. Cambridge University Press Cambridge, 1998. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Complex Automata Scale separation map The main CxA is decomposed through a Scale Separation Map (SSM), on which each single-scale subsystem occupies an area wrt its spatial (x-axis) and temporal (y-axis) scales. Given two processes A and B, five different interaction regions are identified, according to the position of B on the map relative to A. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Shape Calculus Shape Calculus The Shape Calculus is a spatial bio-inspired process algebra for describing 3D processes moving and interacting in the 3D space; it provides the theoretical basis for the BioShape spatial simulator. A 3D process is characterized by a shape (basic or complex) and by a behaviour specified in terms of a Timed CCS process. E. Bartocci, F. Corradini, M. Di Berardini, E. Merelli, and L. Tesei. Shape Calculus. A spatial calculus for 3D colliding shapes. Tech. Rep. 6, Department of Mathematics and Computer Science, University of Camerino (Jan 2010). A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Shape Calculus Spatial and physical features (1/2) Shapes have associated a velocity and a mass, and can be composed by binding on compatible channels exposed in their surface. Breaking of established bonds is implemented with two operations: strong-split (urgent) and weak-split (not urgent). The Shape Calculus incorporates a collision detection algorithm supporting both elastic collisions and inelastic collisions (bindings). The time domain is continuous, divided into small time steps ∆. At each step, collisions are resolved and the function steer updates the velocities of all the shapes in the system. The detection of a collision can break the timeline before ∆ has elapsed. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Shape Calculus Spatial and physical features (2/2) Example Figure (a) illustrates the binding of the processes S0 [B0 ] and S1 [B1 ] on channel b, · ; the resulting composed process is S0 [B0 ] b, W S1 [B1 ], where W denotes the common surface of contact Y ∩ Z ; eventually, a weak split occurs. Figure (b) shows an example of timeline; note that collisions break the timeline, while splits are resolved at the end of the time step ∆. (a) Binding and weak-split (b) Timeline and events A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Shape Calculus 3D Shapes Definition (Basic shape) A basic shape is a tuple σ = V , m, p, v where: V is a set of points representing either a sphere, a cone, a cylinder or a convex polyhedron; m is the mass of the shape; p is the centre of mass; v is the current velocity. Definition (3D Shape) The set S of 3D shapes is generated by the grammar S ::= σ|S X S, where σ is a basic shape and X is the common surface. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Shape Calculus 3D Processes Definition (Shape behaviours) The set B of shape behaviours is given by the grammar B ::= nil | α, X .B | ω(α, X ).B | ρ(L).B | (t).B | B + B | K where K is a process name, and (t) is the delay operator. Definition (3D Process) The set 3DP of 3D processes is generated by the grammar P ::= S[B] | P α, X P, where S ∈ S, B ∈ B, and α, X is a channel with X = ∅. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Outline 1 Introduction 2 Computational Systems Biology Mathematical approaches Computational approaches 3 Spatial Multiscale Paradigms Spatial P Systems Complex Automata Shape Calculus 4 Bone Remodelling Bone remodelling in CxA Bone remodelling in SP Bone remodelling in Shape Calculus 5 Conclusions A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions The bone remodelling process Old bone is continuously replaced by new tissue. Mechanical integrity of the bone is maintained. In healthy conditions: no global changes in morphology/mass. Pathological conditions can alter the equilibrium between bone resorption and bone formation. Osteoporosis is an example of negative remodelling: resorption prevails on formation, reducing bone density, so increasing the risk of spontaneous fractures. Bone Remodelling (BR) is a spatial multiscale phenomenon: macroscopic behaviour and microstructure strongly influence each other, and spatial properties are crucial in the functioning of the BR process. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone Remodelling scales Bone remodelling is an important case study in the perspective to compare the spatial and multiscale formalisms introduced. We study the BR process at two different levels: cellular and tissue. At tissue level, mechanical loading mainly affects BR: bone adapts its structure in response to the mechanical demands. At cellular level, the phenomenon is observed in the so-called Basic Multicellular Unit (BMU). A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Osteocytes (Oy ) are connected by a network of canaliculi in the mineralized part; stem cells, stromal cells and pre-osteoclasts (Pc ) circulate in the fluid part. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model A sudden stress causes a micro-fracture to appear; Oy s near the crack undergo apoptosis, while the other Oy s detect the strain and produce biochemical signals. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Lining cells (Lc ) pull away from bone matrix and merge with blood vessels. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Stromal cells generate pre-osteoblasts (Pb ). A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Pb s express the signal RANK-L attracting Pc s which have a RANK receptor on their surface. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Pc s enlarge and fuse into mature osteoclasts (Oc ). Oc s attach to bone surface, and create an acid environment to resorb the bone. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Resorption takes about two weeks; eventually, Oc s undergo apoptosis. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Pb s mature into osteoblasts (Ob ) which stop sending RANK-L. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Ob s line the resorbed cavity and mineralize it by secreting osteoids. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model Mineralization takes about 3-4 months. Some Ob s turn into Oy s, some into Lc s, and the rest undergo apoptosis A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Real BMU model The network of canaliculi connecting the Oy s is re-established; the microdamage has been repaired. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA CxA model (1/2) The model is composed of A “macro” CA, C1 modelling the tissue level as a lattice of BMU, and a “micro” CA for each cell i of C1 , C(i,2) modelling a BMU as a lattice of Oy s. It takes into account only mechanical stimuli, ignoring metabolic ones. Size of C1 is linearly determined by the size of C(i,2) , which depends on the density of Oy s. D. Cacciagrano, F. Corradini, and E. Merelli. Bone remodelling: a complex automata-based model running in BioShape. In ACRI 2010: The Ninth International Conference on Cellular Automata for Research and Industry, 2010. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA CxA model (2/2) Micro execution flow: the state of a cell j of C(i,2) at time j t(i,2) is given by m(i,2) (t(i,2) ), varying from 0 (fluid cell) to 1 (mineralized cell) and depending on the state of the cell i in C1 . A Meshless Cell Method (MCM) calculates the j mechanical stimulus F(i,2) (t(i,2) ) and mass is changed accordingly, until the internal equilibrium state is reached. Macro execution flow: the state of a cell i of C1 at time t1 i is determined by the density m1 (t1 ), varying from 0 (void) to i 1 (fully-mineralized). The stress field on i, F1 (t1 ) is computed by a global MCM analysis; each iteration of C1 corresponds to a complete simulation of C(i,2) , whose outputs modify each i i m1 ; hence, F1 (t1 ) needs to be updated until there no change in densities and in the stress field. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA Coupling scheme The two models are linked with the “micro-macro” coupling mechanism: a fast process on a small spatial scale (C(i,2) ) is coupled to a slow process on a large spatial scale (C1 ). The macro model takes input (mineralization values) from the micro model; this paradigm is called Hierarchical Model Coupling (HMC). A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA Macro and Micro CAs A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA Macro and Micro CAs A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA Macro and Micro CAs A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA Macro and Micro CAs A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA Macro and Micro CAs A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in CxA Macro and Micro CAs A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP SP model The model is composed of A “macro” SP, S1 modelling the tissue level, and a “micro” SP for each cell i of S1 , S(i,2) modelling a single BMU. D. Cacciagrano, F. Corradini, E. Merelli, and L. Tesei. Multiscale Bone Remodelling with Spatial P Systems. Membrane Computing and Biologically Inspired Process Calculi 2010, page 65, 2010. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Macro model (1/2) Each position p contains: An activator object a, determining if p corresponds to a surface cell A number j of objects c proportional to the mineralization density; a cell is on the surface if j ∈ [m, m + n) At most one g object which models a micro damage; the corresponding cell will be selected for remodelling. At most one h objects indicating that the cell is randomly selected for remodelling. The evolution rules are: r1 : c m a → b1 d1 , r2 : c n b1 → c n+m b, r3 : d1 → d, r4 : db → λ, r5 : db1 → c m f , r6 : fg → r , r7 : fh → r . A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Macro model (2/2) A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Macro model (2/2) A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Macro model (2/2) A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Macro model (2/2) A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Macro model (2/2) A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (1/2) The skin membrane is divided in two zones: A mineralized part where there are two ME objects: Oy (bone cell with an osteocyte) and C (bone cell with no osteocytes). A non-mineralized part where the child membrane 2 is located; membrane 2 models the connection with blood and marrow, and produces Pb s and Pc s once the starter object s has entered it. The initial configuration of the mineralized part depends on the mineralization degree computed at the higher level. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) The biochemical signal s spread over the fluid part and moves towards East until it enters membrane 2. Rule(s) s → sN sE sS s → sE s → sin2 s → sout A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) The biochemical signal s spread over the fluid part and moves towards East until it enters membrane 2. Rule(s) s → sN sE sS s → sE s → sin2 s → sout A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) The biochemical signal s spread over the fluid part and moves towards East until it enters membrane 2. Rule(s) s → sN sE sS s → sE s → sin2 s → sout A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) The biochemical signal s spread over the fluid part and moves towards East until it enters membrane 2. Rule(s) s → sN sE sS s → sE s → sin2 s → sout A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) A single s triggers the production of k Pc s and l Pb s from membrane 2; here k = l = 4; any other object s entering the membrane is inactivated by the presence of s . Rule(s) - Membrane 2 s → s (Pc )k (Pb )out out s s→s s → s sN s → s sS A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) Pc s move randomly and aggregate to form a mature Oc . Object Cn , with n < N OC denotes a conglomerate of n Pc s. N OC is the number of Pc s needed to form a grown Oc . Here, N OC = 4. Rule(s) Pc → Pc Pc → Pc N Pc → Pc S Pc → Pc O Pc → Pc E Pc h → Ch Pc h1 − Pc h2 → λ − Ch1 +h2 Ch − Pc → Ch+1 − λ Ch Pc → Ch+1 CN OC −1 − Pc → Oc 0 − λ CN OC −1 Pc → Oc 0 A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) Pc s move randomly and aggregate to form a mature Oc . Object Cn , with n < N OC denotes a conglomerate of n Pc s. N OC is the number of Pc s needed to form a grown Oc . Here, N OC = 4. Rule(s) Pc → Pc Pc → Pc N Pc → Pc S Pc → Pc O Pc → Pc E Pc h → Ch Pc h1 − Pc h2 → λ − Ch1 +h2 Ch − Pc → Ch+1 − λ Ch Pc → Ch+1 CN OC −1 − Pc → Oc 0 − λ CN OC −1 Pc → Oc 0 A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) The formed Oc s move towards West to the mineralized part. Rule(s) Oc → Oc W A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) Object Oci models an osteoclast which has destroyed i mineralized cells, with i ≤ N DC ; here N DC = 3. Rule(s) Oy − Oc z → Oc z+1 − λ C − Oc z → Oc z+1 − λ Oy − Oc N DC −1 →λ−o A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Micro model (2/2) Once absorbed N DC cells, the Oc dies, releasing an object o which represents the biochemical signal that will trigger the formation of Ob s for bone reconstruction. Rule(s) C − Oc N DC −1 →λ−o A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in SP Coupling scheme Two integration functions: f ↓ (top-down): if a cell i of S1 is subject to remodelling, the function puts the starter object s in S(i,2) . Moreover, f ↓ sets the initial configuration of S(i,2) according to the number of c objects in i. f ↑ (bottom-up): after the simulation of S(i,2) , it determines the number of c objects to be placed on the cell i of S1 . A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Shape Calculus model The system is decomposed in two different scales: BMU Scale: a BMU is represented as a network of 3D processes (Oy s, Lc s, Pb s, Pc s, Ob s, Oc s). Tissue Scale: tissue is modelled with bone and fluid cubes; surface cubes are decomposed in more complex shapes. This work broadly follows the model for Bone Remodelling defined in BioShape. F. Buti, D. Cacciagrano, F. Corradini, E. Merelli, M. Pani, and L. Tesei. Bone remodelling in BioShape. In CS2BIO 2010: Interactions between Computer Science and Biology, 1st International Workshop, 2010. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Macro model (1/2) The process involved are: Smin [Bmin ], a mineralized component with mass m + n Sfluid [Bfluid ], a fluid component with mass m Ssurf [Bsurf ], a surface component with mass m + δn δ denotes the mineralization density of a cell; in Ssurf , 0 < δ < 1, in Smin , δ = 1, while in Sfluid , δ = 0. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Macro model (2/2) Process Definitions def Tissue = Min Tissue Surf Tissue Fluid Tissue def (1) (1) (nmin ) (nmin ) Min Tissue = Smin [Bmin ] ... Smin [Bmin ] def (1) (1) (nsurf) surf (n ) Surf Tissue = Ssurf [Bsurf ] ... Ssurf [Bsurf ] def (1) (1) (nfluid ) fluid (n ) Fluid Tissue = Sfluid [Bfluid ] ... Sfluid [Bfluid ] A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (1/2) Processes in the mineralized part are: Soy [Boy ], a cell containing an Oy ; Soy is a cube with edge length l. Sc [Bc ], a cell without Oy s; Sc is a cube with edge length l. Slc [Blc ], a Lc ; Slc is a cube with edge length l. Ssig [Bsig ], the remodelling signal produced by a Oy ; Ssig is a little sphere with radius rsig . Processes in the fluid part are: Spb [Bpb ], a pre-osteoblast; Spb is a cube with edge length l. Spc [Bpc ], a pre-osteoclast; Spc is a cube with edge length lpc . Srec [Brec ], the receptor for the remodelling signal; Srec is a little sphere with radius rrec . A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) Mineralized cells are bound together, so implementing the network of canaliculi. Oy s activates remodelling by performing a can action which propagates towards the lining cells. A receptor is attached to each Pb and Pc . A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) A remodelling signal is attached to each Lc ; when a Lc binds with a mineralized cell on an exposed channel can, · , a weak split causes Ssig [Bsig ] to detach and move towards the fluid part. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) Signals collide and bind with receptors on channel asig , · , provoking another weak split which involves pre-osteoblasts and pre-osteoclasts. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) Pc s move randomly and aggregate by binding in order to form a full Oc ; each Pc has six free exposed channels. The number of free surfaces in a conglomerate of k Pc s varies from 3k to 4k + 2, depending on its structure. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) A grown Oc corresponds to the composition of n OC Pc s. Here, n OC = 8 and the Oc has 28 free surfaces. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) Once the osteoclast is formed, its velocity is updated so that it can reach the mineralized part in a time tOC . A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) A Oc can erode a single mineralized cell (or Lc ) for each of its free surfaces. A mineralized cell is absorbed when it binds to an Oc on a channel del, · ; then, a strong split breaks all the bonds of the cell which is send out of the BMU. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) A Oc can erode a single mineralized cell (or Lc ) for each of its free surfaces. A mineralized cell is absorbed when it binds to an Oc on a channel del, · ; then, a strong split breaks all the bonds of the cell which is send out of the BMU. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) After a time tOC , the Oc undergoes apoptosis; at the same time, Pb s turn into mature osteoblasts and reach the bone surface in a time tOB . A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) Each Ob attach to a bone cell and behaves as Boy or Bc , so replacing the consumed cells. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) The bone formation process lasts a time tOB , after which the remaining Ob s replace the absorbed lining cells. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Micro model (2/2) The original structure is re-established. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Bone remodelling in Shape Calculus Coupling scheme (i) (i) (i) (i) (i) (i) Each macro-component Smin [Bmin ], Ssurf [Bsurf ] or Sfluid [Bfluid ] have associated a process BMU of the micro model. We consider only surface components, as subject to remodelling. Two integration functions: f ↓ (top-down): computes the number of mineralized cells in the BMU model, which is proportional to the mass of the corresponding tissue cell m(Ssurf ), which in turn depends on the density δ. f ↑ (bottom-up): after the execution of the BMU model, f ↑ modifies m(Ssurf ) according to the new mineralization values of the lower level. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Outline 1 Introduction 2 Computational Systems Biology Mathematical approaches Computational approaches 3 Spatial Multiscale Paradigms Spatial P Systems Complex Automata Shape Calculus 4 Bone Remodelling Bone remodelling in CxA Bone remodelling in SP Bone remodelling in Shape Calculus 5 Conclusions A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Comparison of models (1/2) We consider the following criteria: Uniformity: implies the capability to apply well-defined and non ambiguous coupling schemes to link single-scale models; a good level of uniformity reduce the loss of information between different scales. Integration of different scales: CxA define integration schemes natively (as edges of the graph itself), and the SSM illustrates the single-scale processes in terms of their spatial and temporal scales and of their mutual coupling. SP systems and Shape Calculus don’t include a priori integration mechanisms. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Comparison of models (2/2) Faithfulness: measure the “distance” between the computational model and the real biological system described. All the models describe the tissue scale in a similar way. The BMU models in SP and Shape Calculus are able to express cellular dynamics (biochemical signals, Pb s and Pc s formation, . . .); the CxA model only considers mechanical stimuli. Spatial features: spatial information in a CA is limited to the cell size, the total size and the neighbourhood relation. SP systems implement compartmentalization, a 2D space, and movement of objects by the execution of evolution rules. In the Shape Calculus, processes are located in the 3D space, and have associated a shape, a mass, and a velocity ; it supports elastic collisions, inelastic collisions (binding) and splitting. A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Future work Validation of models with experimental data Collaboration with biologists and bioengineers to acquire the necessary information A theory for the verification of qualitative and quantitative properties in the Shape Calculus Model checking tools for BioShape A survey of formal approaches for spatial and hierarchical modelling University of Camerino
    • Introduction Computational Systems Biology Spatial Multiscale Paradigms Bone Remodelling Conclusions Thank you! A survey of formal approaches for spatial and hierarchical modelling University of Camerino