PAM: Particle Automata in Modeling of Multiscale Biological Systems WITOLD DZWINEL and RAFAL WCISŁO, AGH University of Science and Technology, Poland DAVID A. YUEN, China University of Geosciences, Wuhan, China; Minnesota Supercomputing Institute, Minneapolis, MN
SHEA MILLER, Ottawa Research and Development Centre, Agriculture and Agri-Food Canada, Ottawa, Canada
Serious problems with bridging multiple scales in the scope of a single numerical model make computer simulations too demanding computationally and highly unreliable. We present a new concept of modeling framework that integrates the particle method with graph dynamical systems, called the particle automata model (PAM). We assume that the mechanical response of a macroscopic system on internal or external stimuli can be simulated by the spatiotemporal dynamics of a graph of interacting particles representing fine-grained components of biological tissue, such as cells, cell clusters, or microtissue fragments. Meanwhile, the dynamics of microscopic processes can be represented by evolution of internal particle states represented by vectors of finite-state automata. To demonstrate the broad scope of application of PAM, we present three models of very different biological phenomena: blood clotting, tumor proliferation, and fungal wheat infection. We conclude that the generic and flexible modeling framework provided by PAM may contribute to more intuitive and faster development of computational models of complex multiscale biological processes.
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CCS Concepts: Computing methodologies → Model development and analysis; computing → Computational biology;
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Applied
Additional Key Words and Phrases: Particle automata model, modeling using particles, graph dynamical systems, blood flow, tumor growth, F. graminearum proliferation ACM Reference Format: Witold Dzwinel, Rafal Wcisło, David A. Yuen, and Shea Miller. 2016. PAM: Particle automata in modeling of multiscale biological systems. ACM Trans. Model. Comput. Simul. 26, 3, Article 20 (January 2016), 21 pages. DOI: http://dx.doi.org/10.1145/2827696
20 1. INTRODUCTION
The majority of biological processes have a very complex multiscale character. Moreover, for the most of them, the macroscopic scales (e.g., the tissue scale) are tightly coupled with microscopic processes (e.g., occurring at a molecular or cellular level). The nonlinear interactions across spatiotemporal scales make their modeling and numerical simulation both very demanding computationally and unreliable in the scope of classical modeling paradigms (e.g., Dzwinel [2012] and Weinan [2011]). To address this issue, we propose a cross-scale modeling method—the particle automata model (PAM). It combines diverse spatiotemporal scales in the scope of a computational framework of a generalized graph dynamical system (GDS) (e.g., Aledo et al. [2015a, 2015b]). In This research was financed by the Polish National Center of Science (NCN) project DEC 2013/10/M/ ST6/00531 and partially by AGH grant 11.11.230.124. Authors’ addresses: W. Dzwinel; email:
[email protected]; R. Wcisło; email:
[email protected]; D. A. Yuen; email:
[email protected]; S. Miller; email:
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[email protected]. c 2016 ACM 1049-3301/2016/01-ART20 $15.00 DOI: http://dx.doi.org/10.1145/2827696
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comparison to other GDS approaches, in PAM the graph architecture undergoes spatiotemporal evolution stimulated by particle (vertexes) interactions and Newtonian dynamics. In general, the GDS (also referred to as the generalized cellular automata (GCA) when automata are synchronously updated) are computation models that capture the graph (network) of interacting objects where each object behaves like a finite-state ¨ [2012] demonstrate the application of cellular automata on automaton. Marr and Hutt graphs for exploring the relationship between network architecture and dynamics from the perspective of pattern formation. This facilitates an evolutionary interpretation of real biological networks in the light of dynamical function. The integrated modeling frameworks for GDS were proposed in Kulman et al. [2011] and Rosenkrantz et al. [2015], who provide a powerful formalism to model and analyze agent-based systems (ABS). In the recent publications of Arendt and Blaha [2015] and Liu and Wei [2014], the GDS were employed in community detection in social networks and modeling some social behaviors. Many papers use GDS in studying nonlinear dynamics of biological systems, such as disease outbreaks [Eubank et al. 2004] and regulatory networks [Marr ¨ 2012]. The GDS methodology is tightly connected with dynamics on complex and Hutt networks [Strogatz 2001, 2014; Watts and Strogatz 1998]. In the context of biological systems, PAM is a kind of GDS in which the objects (graph vertexes) correspond to interacting and moving particles representing tissue fragments such as cells or clusters of cells. The particles are endowed with some inherent attributes defined by the vector of particle states. In particular, the particle network can be the nearest neighbor graph. The graph edges define interactions between particles and their neighbors, which directly stimulates the particle spatiotemporal dynamics. Simultaneously with particle dynamics, the particle states evolve in time according to a set of rules that correspond to microscopic processes occurring “inside” a particle and/or in its neighborhood. These microscopic rules directly affect the behavior of the entire system. In this article, we present the main concept and assumptions staying behind PAM. We show that PAM reflects a specific coarse-graining procedure in which the microscopic degrees of freedom are encapsulated inside a particle and manifest in changes of its state vector, consequently influencing local and global dynamics of the entire system. This way, the particle automata ensemble can simulate multiscale systems by using only one macroscopic spatiotemporal scale controlled by particle dynamics and a microscopic rule-based operator. We demonstrate the applicability of PAM in the modeling of complex multiscale biological systems. As a proof of concept, we present three examples. First, we describe the particle model (PM), which we used earlier in modeling of the blood-clotting process in capillaries [Boryczko et al. 2004]. We show that the microscopic process of clot formation can be mimicked by a simple finite-state automata rule integrated with blood flow modeled by using fluid particle dynamics. Then we describe two models: the model of cancer proliferation and the model of wheat colonization by a fungal pathogen, which present more advanced aspects of the PAM framework. We close the article with our conclusions. 2. PARTICLE AUTOMATA MODEL 2.1. Motivation
The model of interacting particles is one of the most popular and intuitive modeling ˜ [1998], Haile [1992], Spohn [2012], and paradigms (e.g., see Dzwinel [2012], Espanol Weinan [2011]). However, due to many difficulties with realistic representation of many multiscale phenomena in the scope of this paradigm, and surrealistic computational ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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Fig. 1. The diagrams demonstrating computationally irreducible systems described by microscopic models: the molecular dynamics (MD) and the nonequilibrium molecular dynamics (NEMD) (a); the system with clearly separable scales: MD, the fast multipole method (FMM), DPD and thermodynamically consistent DPD (TC-DPD), smoothed particle hydrodynamics (SPH), and partial differential equations (PDEs) (b); and the systems with strong cross-scales interactions (c). The spheres represent consecutive spatiotemporal scales, and arrows denote the couplings between them. In (c), the arrows are drawn arbitrarily and show possible cross-scale interactions.
load when a macroscopic system is simulated directly from its atomistic or molecular scale, the question about its computational reducibility should be decided. ˜ [1998], and Groot As shown in several papers (e.g., Dzwinel et al. [2006], Espanol and Warren [1997]), the dynamics of an ensemble of atoms can be coarse grained by interacting clusters of atoms and, going up to the scale, by interacting droplets or pieces of matter. As demonstrated in Figure 1, such coarse graining can decouple the consecutive spatiotemporal scales. Similarly to discrete wavelet transform (DWT) (e.g., Goswami and Chan [2011]), coarse graining splits the space time into layers (tiles) in which coarse-grained models of various resolution are defined. Unlike DWT, the boundaries between the layers are fuzzy, and a simple mathematical formalism of a layer-to-layer transformation does not exist. However, by using the DWT analogy, the averaged properties from the finer scales can be passed up to the coarser ones. They can manifest there as emergent global properties of the system or local features of its components, such as potential fields (as it is in the particle in cell (PIC) method [Birdsall 1991; Hockney and Eastwood 2010]) or collision operators (e.g., as it is in dissipative particle dynamics (DPD) [Groot and Warren 1997; Li et al. 2014; Pastorino ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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and Goicochea 2015]), respectively. However, the complexity of the particle collision operator in the coarser spatiotemporal scales (clusters of molecules, fluid particles, cells, lumps of fluid, droplets, tissue components, etc.) increases exponentially with the precision of reconstruction of the corresponding fine-grained system evolution [Dzwinel et al. 2006; Li et al. 2014; Serrano and Espanol 2001; Yaghoubi et al. 2015]. Consequently, the parameter space of the model inflates what results in overfitting. Anyway, as shown in several papers (e.g., Dzwinel et al. [2006], Dzwinel [2012], Israeli and Goldenfeld [2006], Magiera and Dzwinel [2014], Dzwinel and Magiera [2015], and Weinan [2011]), coarse graining is a very effective procedure for bridging spatiotemporal scales of many real-world systems provided that the scales can be separable. It allows for simulating real-world phenomena in the scale of interest defined by a computational setup attributed by microscopic features. Biological systems can be located between systems of both fully separable and inseparable scales. On the one hand, the strong dependence of macroscopic behavior on molecular processes makes them chaotic and unpredictable. On the other hand, many of microscopic biological processes are local and well separated, especially those tied to particular biological structures such as DNA chains, molecular films, or those encapsulated inside closed subsystems such as organelles, cells, and microvessels. Thus, the various phases of those physicochemical processes can be represented by internal states of corresponding discrete biological agents, which define their properties. For example, internal molecular processes occurring in a cell can define its state as newborn, mitotic, apoptotic, or hypoxic, for example. Moreover, the state vector of a particular discrete object (particle) may depend on the state vectors of other objects, such as those located in its nearest neighborhood in the Euclidean or abstract spaces. For example, signaling cells may modify both their own states and properties of the cells in their vicinity. Analogously, the financial condition of a company may change depending on both its internal status and the statuses of all cooperating partners. Therefore, the microscopic processes can be reflected in the coarser scales by both individual evolution of particle attributes (states) caused by an internal process (e.g., cell life cycle) and similar to the GDS [Aledo et al. 2015a, 2015b], local rules. Consequently, the time evolution of the vector state of discrete particles can influence overall system dynamics. On the other hand, the spatiotemporal evolution of the particle system may affect the evolution of their individual state vectors. It is worth mentioning that the set of rule-based microscopic models may correspond to various (sometimes remote) temporal scales. By using the DWT analogy, they represent “details” from various resolution levels. The PAM approach allows for coupling all of them within the GDS framework of spatial dynamics of particles on the coarsest approximation level. Moreover, the particle dynamics and its vector state evolution may occur in electrical, temperature, pressure, or concentration continuous fields. These fields are usually the result of some physical processes (e.g., diffusion of substances secreted by cells, such as signaling proteins) modified by the system environment and boundary conditions. The evolution of the internal states of biological objects, their spatiotemporal dynamics, and the dynamics of continuous fields are tightly coupled, creating an integrated and generalized framework for modeling of multiscale systems. 2.2. Particle Automata as an Evolving Graph Dynamical System
We regard a particle system as an undirected dynamic graph Gt = (Vt , Et ), where Vt , Et are the sets of particles and particle-particle interactions ei j ∈ Et , respectively. We assume that Vt = {v1 , v2 , . . . , v N }t , where Nt = #Vt is the number of particles in time t and undirected edge ei j = (vi , v j ) ∈ Et . For any vertex (particle) vi ∈ V, the neighborhood of vi , Ng(vi ), is the set {vi } ∪ {v j ∈ V|(vi , v j ) ∈ Et } that contains vi and ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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Fig. 2. The snapshots from 2D simulations of dynamics of particle systems represented by evolving graphs: the nearest neighbor graph that corresponds to the ensemble of Nt = 5 × 104 particles interacting with their closest neighbors (a) and the social network where vertexes (Nt = 1.1 × 106 particles) interact with their neighbors connected by the outgoing edges (b).
all neighbors of vi . The particle network evolves in Euclidean space. Let us define the particle automata as a generalized GDS. Definition 2.1 (Particle Automata). Particle automata is a GDS represented by a tuple Tt = (Gt , R, A, , , V0 ) evolving in discretized time t, where —Gt = (Vt , Et ) is an undirected graph, whose nodes vt = (rt , at ) ∈ Vt are called particles. —V0 = {v0 = (r0 , a0 )} is the graph initial configuration. —rt ∈ R are coordinates of a particle vt in a geographical space R (here, R = R3 is the Euclidean space). —∀vt ∈ Vt , rt : R → R is the local collision/translation operator, which defines the particle spatial evolution in discretized time with timestep t and rt+t = rt .
(1)
—at ∈ A is the particle state vector, where A is a set of subsets of states representing state vector coordinates. —∀at ∈ Vt , at : A Ng(vt ) → A is the local transition operator, which defines temporal evolution of the particle state vector in discretized time with timestep τ and at+τ = at .
(2)
The respective attributes from vector state ai may evolve in various temporal scales, and they can correspond to multiple scales of many microscopic cellular and molecular processes. For the sake of simplicity, we assume that the ratios of the greatest timestep to the smaller ones are the integers. By using the PAM approach, one can model not only the dynamics of systems with short-range interactions but also the networks with long-range connections, such as social networks. For example, in Figure 2, we show the snapshots from 2D simulations of particle systems represented by these two types of evolving graphs. This way, the particle automata ensemble can simulate multiscale systems with only one “distinguished” spatiotemporal scale controlled by the operator and many microscopic processes simulated by the rule-based operator. We understand the “rule” as a simple if_then_else relation or more complicated function (e.g., of cellular automata type) that modifies the vector state of vertexes in the subsequent timesteps. It may ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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depend on the current vector state, the vector states of directly connected neighbors, and/or the values of density fields in the vicinity of the graph vertexes. The other assumptions are as follows: (1) The force component F = FC + F D + F B of the operator , such as in DPD [Groot and Warren 1997; Hoogerbrugge and Koelman 1992] or in the fluid particle model ˜ 1998], consists of the conservative FC , dissipative F D, and random (FPM) [Espanol F B (e.g., the Brownian) components. The conservative factor corresponds to the cumulative impact from two-body central interactions between the particles (vertexes) connected by the edge ei j = 0. The dissipative and fluctuation components of are responsible for controlling the amount of kinetic energy in the system—that is, F D alone freezes the system and F B melts it down. According to the fluctuationdissipation theorem [Groot and Warren 1997; Weber 1956], these two components of the collision operator are responsible for the “temperature” of the entire system. The temperature can reflect stochastic instability of the particle system caused by the coarse-grained degrees of freedom. (2) The particle attribute operator acts only on these components of state vector at , which evolve with time t according to rule-based principles involving the particle’s neighborhood or its internal scenario (e.g., cell mitosis, apoptosis). It means that the following state at+1 of a particle vi depends on its current state and/or the states of the neighboring particles v j , provided that ei j = 0. (3) Additionally, the operator follows the time evolution of some “concentration fields” in R3 , such as concentration of diffusive substances, and assigns the values of these fields as particle’s state vector attributes. In this case, the fields can be obtained solving some reaction-diffusion equations. (4) The particles Vt representing Gt nodes and its edges Et can proliferate or die. (5) The particle system can be bounded or unbounded—that is, various periodic boundary conditions can be applied (e.g., Dzwinel et al. [1991] and Hockney and Eastwood [2010]). Summing up, the entire PAM of a complex system consists of the following tightly coupled components: —The ensemble of interacting particles, which represents the coarsest spatiotemporal scale. —The GDS of automata, which mimics the evolution of hidden microscopic scales of the system. —The varying concentration fields resulting from continuous processes, such as diffusion, advection, and fluid flow, which go with the system evolution. In the following sections, as a proof of concept of PAM, we present its applications in the modeling of selected biological processes. 3. BIOLOGICAL APPLICATION OF THE PARTICLE AUTOMATA MODEL
In this section, we briefly describe three computational models of complex biological phenomena developed by using the particle automata paradigm, which combines particle dynamics with rule-based microscopic models. The models differ in both the scope in which the GDS automata rules are exploited and the context in which they were used. We discuss the models of —blood clotting in capillary vessels, —cancer dynamics, and —wheat colonization by the Fusarium graminearum fungal pathogen. ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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First, we briefly describe the model of blood clotting in microcapillaries, which was developed a relatively long time ago [Boryczko et al. 2003, 2004; Dzwinel et al. 2003] but still continues to be developed (e.g., Guy et al. [2007], Soares et al. [2013], and Storti et al. [2014]). This model is described to demonstrate the main assumptions of the FPM, which we use as a computational framework of PAM. Moreover, in this blood dynamics setup, we implemented a simple automata rule (Algorithm 1), which mimics the microscopic model of fibrinogen aggregation. The second example (based on Wcisło et al. [2009, 2010]) presents a much more complicated PAM of cancer proliferation in the presence of the process of angiogenesis. The last model is a metaphor of fungal wheat infection. This is a new application of PAM that is developed for a different setup than the previous two examples. 3.1. Particle Model of Blood Clotting
Hemorrhage over a vast tissue volume caused by blood clotting in capillary vessels is very feasible under multiple g jet accelerations or violent deceleration during a car accident. In Boryczko et al. [2003], we presented a microscopic model of blood, valid over short time scales, which allows for simulating such shocking events. We assumed that the system, which consists of a capillary, blood plasma, fibrinogen, and red blood cells (RBCs), is made of particles vi = (ri , ai ) of various types. The attributes of a discrete particle are as follows: particle type, mass, moment of inertia, and translational and angular momenta. The capillary, filled with plasma and RBCs, is made of motionless particles. Its diameter is of the size of a single RBC (i.e., approximately 10μm). Each fluid (plasma) particle represents coarse-grained clusters of molecules, whereas RBCs are the structures made of particles interacting via elastic forces [Boryczko et al. 2003]. Two particles, vi and v j , interact with each other by a collision operator, Fi j , defined as a sum of constituent forces: central and noncentral conservative FC , dissipative F D, and Brownian F B. The collision operator is defined as ˜ 1998]: follows [Espanol Fi j = FC
+F D
+F B, 1 ˜ B(ri j ), i + ω j ) +F Fi j = −F(ri j )ei j −γ [A(ri j 1 + B(ri j )ei j ei j ] ◦ vi j + ri j × (ω 2
(3)
where F(ri j ) is the module of central conservative force, such as in DPD [Hoogerbrugge and Koelman 1992]; γ is a scaling factor of dissipative component F D; ωi is the angular velocity of particle vi ; ei j is the unit vector; and ri j = ||ri −r j || is the separation distance between particles vi and v j . A(ri j ) and B(ri j ) are the weighting functions, whereas F B is the Brownian component. The value of Fi j is equal to 0 if the separation distance ri j between two particles i and j exceeds a cut-off radius Rcut . The total force Fi acting on a particle i is the sum of forces between i and other particles located within the sphere O(ri , Rcut ). The dynamics of the particle system obeys the Newtonian laws of motion: ˙i = P Fi j (ri , vi , ωi ), j;ri j Rbrake then number of bounds(i) ← number of bounds(i) − 1; number of bounds( j) ← number of bounds( j) − 1; actualize list of neighbors(i); actualize list of neighbors( j); a(i) ← a( j) ← 0; // change a state vector a end end end
model of tumor proliferation and wheat infection, in which the tumor and fungus cells demonstrate considerably richer individual behavior. 3.2. Tumor Proliferation
Tumor proliferation is a very nonlinear and heterogeneous process occurring in multiple spatiotemporal scales. It develops in five main phases: oncogenesis, avascular ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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growth, angiogenesis, vascular growth, and metastasis. Oncogenesis represents a cascade of biological processes occurring at molecular level stimulated by serious mutationbased disruptions in DNA repair system. In a larger spatiotemporal scale, an avascular tumor develops in the absence of blood supply. For tumors of size of a few millimeters in diameter, hypoxic cells from the avascular tumor mass produce and release many signaling proteins called tumor angiogenic factors (TAFs). They diffuse throughout the surrounding tissue and, by hitting vasculature, trigger a cascade of events leading to vascularization of the tumor. In the vascular stage, the tumor has access to virtually unlimited resources, so it can grow beyond any limits. Moreover, it acquires a means of transport for cancerous cells that penetrate into the vasculature and can form metastases in any part of the host organism. A plethora of computer models of both the separate subprocesses of cancer proliferation and more general multiscale approaches exist. The main modeling techniques are presented in many books and overviews [Adam and Bellomo 2012; Barillot et al. 2012; Cristini and Lowengrub 2010; Deisboeck and Stamatakos 2011]. However, most methods used for simulating tumor growth assume the coarse-graining scenario shown in Figure 1(b). Meanwhile, just the cross-scale influence of microscopic processes occurring inside a single cell and its nearest neighborhood on the overall tumor dynamics can be crucial for its growth dynamics. For example, some of internal cell processes, such as those influencing the cell life cycle, depend on its local environmental conditions stimulated by cancerogenesis. The diversity of possible situations will lead to the development of a heterogeneous tumor that promotes the cancer cells having the highest survival abilities (e.g., see Smith [2013]). These malignant cells decide about the fatal prognoses of tumor proliferation. Therefore, the tissue cells—especially tumor cells—have to be treated as independent, partly autonomous objects that reveal very sophisticated behavior. Thus, in comparison to the blood clotting model presented earlier, the complexity of the model of cancer dynamics must be considerably higher. We present such the model in Wcisło et al. [2009]. Next, we briefly describe it in terms of the PAM framework. Let us assume that a small piece of tissue is made of an ensemble of Nt particles vi = (ri , ai , t). Each particle represents a single tissue cell with a fragment of extracellular matrix (ECM). As shown in Figure 4, for the sake of simplicity, we assume that the blood vessel is constructed of endothelial tubelike “particles”—EC tubes—made of two particles connected by a rigid spring. Consequently, three types of interactions— particle-particle, particle-tube, and tube-tube—have to be considered. The forces between particles mimic mechanical repulsion from squashed cells, attraction due to cell adhesiveness, and depletion interactions between the ECM and the cell. The collision operator has a general form similar to Equation (3). The vessel and cell dynamics are simulated by using the Newtonian equations of motion (see Equation (4)). The entire system is confined in a cubical box under constant external pressure. The vector of states and attributes ai = (a1 , a2 , . . . , ak) for every cell consists of the particle type {tumor cell (TC), normal cell (NC), EC-tube}; cell life cycle state {newly formed, mature, in hypoxia, after hypoxia, apoptosis, necrosis}; cell size; cell age lifetime in hypoxia; and values of continuous fields, such as concentrations of TAF, oxygen, and total pressure exerted on particle i from its nearest neighbors. The cell states change in time according to their own clocks (cell life cycle states, cell size, time spent in hypoxia state) and the dynamics of continuum concentration fields. The latter are computed by solving continuous reaction-diffusion equations (e.g., see Cristini and Lowengrub [2010]) and employing the virial theorem for calculating local pressure [Hockney and Eastwood 2010]. The blood pressure in vessels is calculated by using the Kirchhoff law [Wcisło et al. 2009].
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Fig. 4. Tissue particles and tubelike EC-tube particles made of two spherical “vessel particles.”
Fig. 5. The snapshots from PAM simulation of two phases of tumor growth. (a) In the avascular phase, one can see the tumor blob consisting of tumor cells in various stage of hypoxia (oxygen deficiency). The cells closer to the blood vessels are better oxygenated. The normal tissue cells are invisible in this figure. (b) In the vascular phase, the new sprouts and newly formed vessels are created. One can see the tumor cells in various stages of hypoxia and normal cells (white particles). In both simulations, the dynamics of about 105 particles were simulated.
After the initialization phase, in subsequent timesteps we calculate: forces acting on particles, new particle positions, the diffusion fields of active substances (i.e., nutrients, TAFs, pericytes), and the pressure in blood vessels. Simultaneously, we change the states of individual cells, whose evolution is controlled by the factors mentioned previously. In Algorithm 2, we present selected automata rules used in our PAM. As shown in Figure 5, our model of tumor proliferation allows for investigating the tumor dynamics in both avascular and angiogenic stages of growth. The life cycle of individual cells—from mitosis to its apoptosis (or hypoxia)—changes the number of ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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ALGORITHM 2: The pseudocode describing some rules of the cells state vector. if state = Alive and type = Normal and nei tumor cnt > nei normal cnt then // normal, alive cell will change state to apoptosis if majority of its neighbors is tumor state ← Apoptosis; end if state = Hypoxia then // cell secretes TAF in hypoxia produce TAF(); end if state = Necrosis then // cell consumes oxygen if not in necrotic state consume O2 (); end if state = Alive and age > time to apoptosis then // old cell changes its state to apoptotic state ← Apoptosis; end if state = Alive and O2 concentration < O2 hypoxia then // cell changes its state to hypoxic if oxygen concentration drops below defined threshold state ← Hypoxia; end if (state = Apoptosis or state = Hypoxia) and state age > time to necrosis then // cell changes its state to necrotic after defined time in hypoxia or apoptosis state ← Necrosis; end if state = Necrosis and state age > time in necrosis then // cell is removed from simulation after defined time in necrotic state remove cell(); end
particles in the system. Consequently, the increase of tumor size in a restricted volume can influence global properties of the system, such as internal pressure, which triggers a cascade of both microscopic and macroscopic events. For example, purely mechanical processes, such as vessel remodeling, result in tumor heterogeneity and development of most malignant tumor cells. In the PAM, the local particle interactions and local automata rules, together with Newton laws of motion, decide about the system dynamics. However, in colonies of organisms such as the coral reef, bacterial biofilm, and human civilization, the long distance interactions interfere with those of short-range character. Similar situations can be observed for the following PAM. 3.3. Fusarium Graminearum Infection
The fungal pathogen F. graminearum is the cause of many devastating cereal diseases, such as Gibberella ear rot of maize, Fusarium head blight (FHB), and scab of wheat and barley. This kind of infections cause significant crop and food quality losses. The micrographs from Figure 6 show the F. graminearum colony in two different setups. It develops weblike structures of long threads of connected Fusarium cells. As shown in the right panel of Figure 6, the plant colonization process is more globally and locally constrained than the previous modeling examples. This is mainly due to the particular structure of the plant and the sparse localization of nutrient sources, as well ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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Fig. 6. Micrographs demonstrating the F. graminearum network expanding in a Petri dish (left panel) and the cross section (right panel) of rachis of Chinese Spring wheat cultivar, showing in green the fungus colony growing in the vascular bundles (arrows).
as the elongated shape of a single fungi cell and its directional reproduction mechanism [Boswell et al. 2007; Miller et al. 2004; Meˇskauskas et al. 2004]. As shown in Figures 6 and 7, all of these factors favor a directional type of growth. The Fusarium colony spreads mainly through vascular bundles, also penetrating the closest neighborhood (see Figure 7(c)). When it finds the part of plant that is rich in nutrients, the growth type changes from a linear to an extensive (undirected) one. It is very similar to that from the micrograph shown in the left panel of Figure 6. This way, F. graminearum is able to completely exploit and destroy invaded plant fragments. The widespread colony had to develop other (faster) mechanisms of nutrients distribution than diffusion. Otherwise, the motionless colony section placed in the distant and exploited plant area would die from starvation. The transduction mechanism uses the topological properties of the colony and its long web threads. It allows for transporting the nutrients throughout the colony body from nutrient-rich to nutrient-poor plant sections. Most fungal and plant cells, except those close to the penetrating hyphal tip cells, are motionless. Therefore, the PAM of F. graminearum differs considerably from that of tumor proliferation and blood clotting. The evolution of the particle attributes prevails that of cell dynamics. According to the definition of PAM, the simulated system (i.e., the plant cells and Fusarium colony) is made of a set of particles vi = (ri , ai , t). The vector of attributes ai is defined by the particle type {plant cell, wall cell, Fusarium}, current cell state {tip, active, inactive, spore, dead}, size, and concentration of nutrients. According to Boswell et al. [2007], the tip and active cells are involved in nutrient uptake, threads branching, and nutrient translocation. Particularly, the tip cells are responsible for colony growth and, together with active cells, secrete the enzymes and toxins. Moreover, they penetrate mechanical barriers (e.g., capillary walls) and disarm the plant’s immunological system. The inactive cells are the cells that are no longer directly involved in translocation, branching, or nutrient uptake. The spores are reproductive entities that are adapted for dispersal and survival for extended periods of time in unfavorable conditions. The state of each cell changes with concentration of diffusive substances and total pressure exerted by its closest neighbors. The plant cell is represented by a single particle (spherical cell), whereas the Fusarium cell (due to its elongated rodlike shape) is made of two particles separated by a distance li , similar to EC-tubes from the cancer ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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Fig. 7. (a) Fluorescence micrograph showing green fungus in an infected wheat floret. The fungus body is denoted by arrows. (b) Snapshots from PAM simulation of F. graminearum growth in rachis. The arrow shows the fungus breaking throughout the plant wall. The dynamics of about 105 particles were simulated.
model. The interaction F(ri j ) between these two particles is harmonic and very stiff in order to maintain the constant length li of Fusarium cells. To reduce the computational load, we assume that all plant cells are motionless and interact only with F. graminearum tubelike cells. The repulsive forces acting on the plant cells are ignored and assumed to be dissipated in the plant body. Thus, we can define only sphere-tube (plant cell–fungal cell) and tube-tube (fungal cell–fungal cell) interactions. The plant cell walls can be degraded by DON toxins and acid secreted by Fusarium cells, and, finally, their interior can be consumed. The sphere-tube interactions are represented by the force acting between a plant particle and the two particles, which build the Fusarium cell body. The particle-particle interactions represented by the force F(ri j ) mimic both the mechanical repulsion from neighboring cells and attraction due to cell adhesiveness and depletion forces. The mechanical repulsion is approximated by Hooke’s law. We assume that the attractive tail of the interaction force has similar character but is less rigid than its repulsive part. The tube-tube (fungi-fungi) interactions are of diverse characters. Typically, the Fusarium cells create both chain and anastomosing web structures. We assume that the cell located at the tip of the growing web attracts another Fusarium cell stronger than other cells. Similarly, the particles that form the nodes of the Fusarium thread or branching sites are firmly glued. The heuristic formulas for various types of fungi-fungi interactions are enumerated in Boswell et al. [2007]. Unlike in the tumor model, we assume a much simpler time evolution of a particle ensemble: mi
dVi = −a · ∇(di j ) − λvi , dt
dri = vi , dt
ri j = (ri − r j ) · (ri − r j )T ,
(6)
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where λ is a friction coefficient. This set of equations of motion is solved numerically by using the direct leap-frog numerical scheme (e.g., see Haile [1992] and Hockney and Eastwood [2010]). The Fusarium cells form the threadlike structures shown in Figures 6 and 7, which can be represented by a disconnected graph Gt = (Vt , Et ). The attributes of particle i are updated according to the state of cells in its neighborhood and prescribed finite-state automata rules. For example, new Fusarium cells can appear, cell function may change (tip → active → inactive), and their mass, stiffness, and size can evolve due to cell growth or degradation. Next, we collect the most important biological mechanisms simulated by using simple automata rules and thresholding relations. 3.3.1. Anastomosis. The tip is attracted to the active cell and can create a loop. 3.3.2. Branching. Active Fg cells can create a new colony branch in the node k (a new graph edge and node) with the following probability:
p2 (k) = c2 s f (k, t) · t.
(7)
Branching can occur only once in the node. 3.3.3. Fusarium DON Toxins. DON is an inhibitor of protein synthesis and thus stops defense mechanisms [Miller et al. 2004]. The Fg cell secretes don f (k, t) toxins. The amount of secreted toxins is proportional to DON activator substances and nutrient concentration s f (k, t) in neighboring plant cell i. Both wall-degrading substances and DON spread due to diffusion. 3.3.4. Maintenance. An amount of food is needed per unit of time and length of the Fg cell. In every timestep, the concentration of nutrients in the Fg cell will decrease as
s f (k, t) − c1 tx.
(8)
If s f (k, t) < w f , the Fusarium cell becomes inactive or produces a spore. The spore undergoes slow linear degradation, and finally it is removed from the system. 3.3.5. Nutrition.
Uptake. The amount of food the Fg cell k drains out from attacked plant cell i in time on the unit length is as follows: s f (k, t) t. (9) se (i, t) uptake (k) = c3 s f (k, t) · Si s f (k, t) Sk
The summation goes through all attacked plant cells in the Fg cell neighborhood Sk. The attacked cell redistributes nutrients proportionally to each of Fusarium cells in its nearest neighborhood Si . Depletion. The decrease in nutrient concentration in the Fusarium nearest neighborhood is si (k, t) t c4 > c3 . se (i, t) (10) depletion (k) = −c4 s f (k, t) · Si si (k, t) Sk
If se (i, t) < we (we is a threshold), the Fusarium cell becomes inactive. After some degradation time, it is removed from the system. 3.3.6. Secretion of Substances Degrading the Cellular Membranes. The Fg cell secretes a f (k, t) cell wall degradation substances (enzymes). The amount of substances is proportional to the nutrient concentration s f (k, t). When the amount of nutrients in plant cells attacked by Fg cells drops below a threshold, the cell dries out and dies. ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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By comparing tumor cells in the PAM of cancer dynamics to Fusarium cells, the latter can “communicate” not only with their nearest neighbors. The nutrients can be translocated directionally from active to the tip cells or the colony body. We recognize two mechanisms of nutrients dispersion: —Active. This occurs only in the tip cell k1 (sprout) direction. An additional amount of nutrients is transported from the neighboring Fusarium active cell k2 (it is not a tip cell). This amount is proportional to the nutrient concentration in k2 , s f (k, t)—that is, active (k2 → k1 ) = c5 s f (k2 , t) · t. (11) The same amount is deducted from k2 . —Passive. This is caused by diffusion in the hyphal network. Diffusion between Fusarium cells is modeled via nutrient exchange processes between neighboring active (and inactive) cells: Diff (k2 → k1 ) = c6 (s f (k1 , t) − s f (k2, t)) · (t/x 2 ).
(12)
In this article, considering that we concentrate on the definition of a PAM method in the context of F. graminearum growth, the parameters c1 through c6 were matched coarsely using published data (mainly Boswell et al. [2002, 2007] and Boenisch and ¨ Schafer [2011]) and observations of micrographs from laboratory experiments. As shown in Figure 7(b) and (c), due to the degradation influence of Fusarium tip cells on capillary walls and other host tissue, the model is able to simulate realistic penetration properties of the colonization process, reproducing both vertical and lateral Fusarium invasion scenarios. As shown in Figure 7, the comparison of simulation results with the fluorescence micrographs from laboratory experiments show encouraging qualitative agreement between the two. 4. DISCUSSION AND CONCLUSIONS
In this work, we define a new modeling approach—PAM—which combines the interacting PM with rule-based GDS. PAM is more generic version of the complex automata model (CxA), which we presented in Dzwinel [2012]. In that work, we focused on systems with clearly separable scales (see Figure 1(b)) and various approaches to coarse graining of particle-based and cellular automata models [Magiera and Dzwinel 2014; Dzwinel and Magiera 2015]. In this work, we concentrate on generalization of dynamical systems represented by graphs such as GDS. Unlike other GDS approaches, in PAM the graph architecture undergoes spatiotemporal evolution stimulated by Newtonian dynamics of interacting particles. We emphasize on PAM applicability in simulating biological systems with not only local and separable spatiotemporal scales but also long-range and strong cross-scale interactions. In this context, we present a novel PAM of fungal infection and its spread in wheat. We compare it to other rule-based models that we developed earlier, such as blood-clotting and cancer proliferation models. We demonstrate that by representing a biological system as a generalized GDS in which microscopic processes are simulated by automata rules on graph vertexes, while its overall (macroscopic) architecture is driven by the interacting PM, we can create a robust modeling framework for simulating multiscale processes. In general, the automata rules may correspond to microscopic processes occurring in multiple temporal scales. Thus, they can be treated in a similar way as the “details” in DWT. Consequently, the reality can be approximated by the sum of “details” from the rulebased part of the model and the particle-based coarse-grained system dynamics. The great advantage of PAM is its reasonable computational complexity in spatiotemporal scales of processes occurring in microscopic tissue (approximately 10−9 m3 ), which ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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Fig. 8. Diagram presenting the speedup with the number of cells simulated for the PAM tumor model implemented in a CUDA environment on a GPU (Nvidia GeForce GTX 295, 240 threads) compared to a serial code run on an Intel Xeon
[email protected]. The upper plot shows the speedup for interaction calculations (PM), whereas the lower plot shows the overall speedup of the code.
cannot be described by continuum models. Moreover, as shown in Figure 8 and in Wcisło et al. [2010, 2013] and Worecki and Wcisło [2012], PAM simulations can be considerably accelerated because the method can be efficiently parallelized both on multicore CPU and GPU processors. The simplicity and the clear scheme of multiple scales bridging are also advantages of PAM over other modeling approaches. The PAM is scalable in the context of both the system size and modeling fidelity. The latter means that the PAM can be easily extended, including additional fine-grained processes, provided that they can be expressed in terms of particle states and/or complexity of the particle collision operator. The computational complexity scales roughly proportionally with the number of spatiotemporal scales (states) simulated. Thus, the model can be computationally efficient for a reasonable number of particles and particle states. For example, tumors up to 1mm in diameter can be simulated on a laptop computer equipped with a standard GPU card (up to 1 million particles can be simulated with a few particle states) and theoretically two orders of magnitude greater on medium-ranged MPI/GPU servers [Grinberg et al. 2011; Tang and Karniadakis 2013]. The PM, which defines the system spatial resolution and stays on top of the hierarchy of spatiotemporal scales, is responsible for the system dynamics and its mechanical properties. They are modified by microscopic processes represented by evolving particle states and simple automata rules. The ability to mimic mechanical interactions of active biological systems such as blood cells, tumor, and fungal colonies with the rest of tissue shows that PAM can reproduce realistic 3D dynamics of complex biological processes. This way, the PAM approach merges the advantages of both PM and cellular automata paradigms. In the models presented here, only basic principles of cancer and fungal growth were taken into account. However, the addition of more sophisticated ACM Transactions on Modeling and Computer Simulation, Vol. 26, No. 3, Article 20, Publication date: January 2016.
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processes to the framework of PAM, such as heterogeneous tumor cells, more angiogenic factors, immunological mechanisms, and toxin effects, appears to be straightforward. Our modeling approach also has some important drawbacks. Next, we discuss only the most serious ones. The PM represents the coarsest scale of the PAM. If a particle mimics a cell, the size of tissue modeled is limited by the number of particles N that currently can be simulated. The linear size of 3D object increases with N 1/3 . Thus, PAM applicability to simulate large systems, such as a tumor of diameter greater than a few millimeters, is limited by at least O(N) computational and memory complexities [Dzwinel 2012]. Therefore, to simulate the tissue scale larger than 1cm, the continuum level of the description should be considered or a new coarse-graining procedure developed (e.g., in which a particle can represent a tissue fragment instead of a single cell). Since this work focuses on PAM formulation, less attention was paid to its calibration. The parameters of models presented here were matched coarsely using data published earlier, rule of thumb, and visual comparisons. This is a consequence of serious difficulties with data assimilation to all models based on particle interactions caused by the lack of a viable formal procedure, which is able to represent mechanical properties of the tissue in terms of forces and their parameters. Because they are chosen mainly using rule of thumb, the PAM in its current form can rather be applied as a useful qualitative metaphor of multiscale growth than a quantitative and predictive tool. Further generalization of the PAM, more rigorous integration of interacting PM with rule-based GDS and continuous models, and above all, releasing it from various constraints imposed by physics, would allow for discovering even more areas of PAM application. The interactions between a pair of objects cannot only be physical and cannot concern only the nearest neighbors. For example, people and robots can communicate remotely, and the result of such interactions cannot be described by simple laws of motion. Moreover, the particles can be used in function optimization or as the universal solver for finding a global minimum of multimodal functions [Dzwinel 1997]. In machine learning, the particles can correspond to feature vectors in low-dimensional embeddings, whereas their state vectors may consist of the class labels, the class membership of their k-nearest neighbors, or text annotations. By using particle-based multidimensional scaling, it is possible to develop interactive visual classifiers [Dzwinel and Wcisło 2015]. Summing up, although the PAM approach is not yet a “silver bullet” in the modeling of complex multiscale phenomena, it is an interesting concept and an alternative for existing continuous-discrete modeling strategies. ACKNOWLEDGMENTS We are very grateful to Dr. Arkadiusz Dudek, Professor of Medicine at the Division of Hematology/Oncology, University of Illinois, for his collaboration in development of the cancer model. We also thank Dr. Krzysztof Boryczko, Professor of Computer Science at the AGH Department of Computer Science for providing us with the figure of blood clotting simulation.
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