Ivan Kryven

I am a principal inverstigator at van 't Hoff Institute for Molecular Sciences (HIMS, University of Amsterdam) For me statistical physics is the key to understanding why the whole is different from the sum of its parts: identifying phase transitions and emergent properties;  understanding how the combined effect of simple interactions and many degrees of freedom creates the never ending complexity we observe in our surroundings. Through the prism of these philosophy, I investigate most diverse topics: from kinetics of nano-materials and morphology of fractal colloids, to regulatory mechanisms of biological cells, and even the molecular ageing of oil paintings. My research is supported by NWO VENI program.

(PO 94720)
Science Park 904
1090 GS Amsterdam
Room number: C2.228


Publications      Supporting source code      arXiv.rog


  1. I. Kryven. Analytic results on the polymerisation random graph model, J Math Chem, 56 (1), 140-157, 2018, [MORE] [PDF]
    The step-growth polymerisation of a mixture of arbitrary-functional monomers is viewed as a time-continuos random graph process with degree bounds that are not necessarily the same for different vertices. The sequence of degree bounds acts as the only input parameter of the model. This parameter entirely defines the timing of the phase transition. Moreover, the size distribution of connected components features a rich temporal dynamics that includes: switching between exponential and algebraic asymptotes and acquiring oscillations. The results regarding the phase transition and the expected size of a connected component are obtained in a closed form. An exact expression for the size distribution is resolved up to the convolution power and is computable in subquadratic time. The theoretical results are illustrated on a few special cases, including a comparison with Monte Carlo simulations.
  2. A Torres-Knoop, I Kryven, V Schamboeck, PD Iedema. Modeling the free-radical polymerization of hexanediol diacrylate (HDDA): a molecular dynamics and graph theory approa Soft matter 14 (17), 3404-3414, 2018 [MORE] [PDF]
    In the printing, coating and ink industries, photocurable systems are becoming increasingly popular and multi-functional acrylates are one of the most commonly used monomers due to their high reactivity (fast curing). In this paper, we use molecular dynamics and graph theory tools to investigate the thermo-mechanical properties and topology of hexanediol diacrylate (HDDA) polymer networks. The gel point was determined as the point where a giant component was formed. For the conditions of our simulations, we found the gel point to be around 0.18 bond conversion. A detailed analysis of the network topology showed, unexpectedly, that the flexibility of the HDDA molecules plays an important role in increasing the conversion of double bonds, while delaying the gel point. This is due to a back-biting type of reaction mechanism that promotes the formation of small cycles. The glass transition temperature for several degrees of curing was obtained from the change in the thermal expansion coefficient. For a bond conversion close to experimental values we obtained a glass transition temperature around 400 K. For the same bond conversion we estimate a Young's modulus of 3 GPa. Both of these values are in good agreement with experiments.
  3. Y Orlova, I Kryven, PD Iedema. Automated reaction generation for polymer networks, Computers & Chemical Engineering, 112, 37-47, 2018 [MORE] [PDF]
    Most of the theoretical studies on polymer kinetics has been performed by manually reducing the chemical system to a few simple reaction mechanisms having a repeatable nature. Not being constrained by such reducibility, this work considers the polymerization as a product of a complex network of reactions that need not to be known in advance. Combining various ideas from graph theory, combinatorics and random graphs, we introduce a new modeling approach to complex polymerization that automatically constructs a reaction network, solves kinetic model, and retrieves such topological properties of the final polymer network as, for instance, distribution of molecular weight. In this way, the new approach acts as an intermediate layer that propagates the knowledge of the basic chemistry in order to capture and understand the complexity of the real world polymerizing systems.
  4. I. Kryven. Finite connected components in infinite directed and multiplex networks with arbitrary degree distributions, Physical Review E 96 (5), 052304, 2017 [MORE] [PDF]
    This work presents exact expressions for size distributions of weak and multilayer connected components in two generalizations of the configuration model: networks with directed edges and multiplex networks with an arbitrary number of layers. The expressions are computable in a polynomial time and, under some restrictions, are tractable from the asymptotic theory point of view. If first partial moments of the degree distribution are finite, the size distribution for two-layer connected components in multiplex networks exhibits an exponent -3/2 in the critical regime, whereas the size distribution of weakly connected components in directed networks exhibits two critical exponents −1/2 and −3/2.
  5. I Kryven, YR Zhao, KB McAuley, P Iedema. A Deterministic Model for Positional Gradients in Copolymers, Chemical Engineering Science 177 (23), 491–500, 2018, [MORE] [PDF]
    A deterministic modelling approach is developed to predict the internal structure of gradient copolymer chains. A key innovation of the modelling approach is the introduction of a positional variable that gives direct access to quantitative gradient characteristics: the ensemble average composition and the gradient deviation. This positional variable is used to develop multi-dimensional population balance equations that can be solved numerically to calculate gradient quality measures. The methodology is illustrated using the gradient copolymerisation of ethylene and 1-octene via coordinative chain transfer mechanism, which is representative of a variety of polymerisation schemes for gradient copolymers. Simulation results are validated with those obtained by stochastic simulations which, until now, were the only means of predicting detailed gradient quality.
  6. V. Schamboeck, I. Kryven, P. Iedema. Acrylate Network Formation by Free-Radical Polymerization Modeled Using Random Graphs, Macromolecular Theory and Simulations, 26 (6), 1700047, 2017, [MORE] [PDF]
    A novel technique is developed to predict the evolving topology of a diacrylate polymer network under photocuring conditions, covering the low-viscous initial state to full transition into polymer gel. The model is based on a new graph theoretical concept being introduced in the framework of population balance equations (PBEs) for monomer states (mPBEs). A trivariate degree distribution that describes the topology of the network locally is obtained from the mPBE, which serves as an input for a directional random graph model. Thus, access is granted to global properties of the acrylate network which include molecular size distribution, distributions of molecules with a specific number of crosslinks/radicals, gelation time/conversion, and gel/sol weight fraction. Furthermore, an analytic criterion for gelation is derived. This criterion connects weight fractions of converted monomers and the transition into the gel regime. Valid results in both sol and gel regimes are obtained by the new model, which is confirmed by a comparison with a “classical” macromolecular PBE model. The model predicts full transition of polymer into gel at very low vinyl conversion. Typically, this low-conversion network is very sparse, as becomes apparent from the predicted crosslink distribution.
  7. I. Kryven. General expression for component-size distribution in infinite configuration networks, Physical Review E 95, 052303, 2017 [MORE] [PDF]
    In the infinite configuration network the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. This work presents a simple equation that gives for an arbitrary degree distribution the corresponding size distribution of connected components. This equation is suitable for fast and stable numerical computations up to the machine precision. The analytical analysis reveals that the asymptote of the component size distribution is completely defined by only a few parameters of the degree distribution: the first three moments, scale, and exponent (if applicable). When the degree distribution features a heavy tail, multiple asymptotic modes are observed in the component size distribution that, in turn, may or may not feature a heavy tail.
  8. I. Kryven. Emergence of the giant weak-component in directed random graphs with arbitrary degree distributions, Physical Review, E 94, 012315, 2016 [MORE] [PDF]
    The weak component generalizes the idea of connected components to directed graphs. In this paper, an exact criterion for the existence of the giant weak component is derived for directed graphs with arbitrary bivariate degree distributions. In addition, we consider a random process for evolving directed graphs with bounded degrees. The bounds are not the same for different vertices but satisfy a predefined distribution. The analytic expression obtained for the evolving degree distribution is then combined with the weak-component criterion to obtain the exact time of the phase transition. The phase-transition time is obtained as a function of the distribution that bounds the degrees. Remarkably, when viewed from the step-polymerization formalism, the new results yield Flory-Stockmayer gelation theory and generalize it to a broader scope.
  9. I. Kryven, J. Duivenvoorden, J. Hermans, P. Iedema. Random graph approach to multifunctional molecular networks, Macromolecular Theory and Simulations 25 (5), 2016, 449--465. [MORE] [PDF]
    Formation of a molecular network from multifunctional precursors is modelled with a random graph process. The process does not account for spatial positions of the monomers explicitly, yet the Euclidean distances between the monomers are derived from the topological information by applying self-avoiding random walks. This allows favoring reactivity of monomers that are close to each other, and to disfavor the reactivity for monomers obscured by the surrounding. As a result, the model is applicable to large time scales. The phenomena of conversion-dependent reaction rates, gelation, microgelation, and structural inhomogeneity are predicted by the model. Resulting nonhomogeneous network topologies are analyzed to extract such descriptors as: size distribution, crosslink distances, and gel-point conversion. Furthermore, new to the molecular simulation community descriptors are suggested that are especially useful when explaining evolution of the gel as being a single molecule: local clustering coefficient, network modularity, cluster size distribution.
  10. I. Kryven, S. Rüoblitz, Ch. Schütte. Solution of the chemical master equation by radial basis functions approximation with interface tracking, BMC System Biology 2015 [MORE] [PDF]
    Background: The chemical master equation is the fundamental equation of stochastic chemical kinetics. This differential-difference equation describes temporal evolution of the probability density function for states of a chemical system. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level.
    Results: We introduce an algorithm that significantly reduces computational resources and is especially powerful when dealing with multi-modal distributions. The algorithm is built according to two key principles. Firstly, when performing time integration, the algorithm keeps track of the subset of states with significant probabilities (essential support). Secondly, the probability distribution that solves the equation is parametrised with a small number of coefficients using collocation on Gaussian radial basis functions. The system of basis functions is chosen in such a way that the solution is approximated only on the essential support instead of the whole state space. In order to demonstrate the effectiveness of the method, we consider four application examples: a) the self-regulating gene model, b) the 2-dimensional bistable toggle switch, c) a generalisation of the bistable switch to a 3-dimensional tristable problem, and d) a 3-dimensional cell differentiation model that, depending on parameter values, may operate in bistable or tristable modes. In all multidimensional examples the manifold containing the system states with significant probabilities undergoes drastic transformations over time. This fact makes the examples especially challenging for numerical methods.
    Conclusions: The proposed method is a new numerical approach permitting to approximately solve a wide range of problems that have been hard to tackle until now. A full representation of multi-dimensional distributions is recovered. The method is especially attractive when dealing with models
  11. I. Kryven, PD. Iedema. Deterministic modelling of copolymer microstructure: composition drift and sequence patterns,
    Macromolecular Reaction Engineering [Special Issue: Statistical Modeling Tools and Approaches for Polymerization Reaction Engineering], Volume 9, Issue 3, pages 285–306, 2015; [MORE] [PDF]
    A concept of a population balance is applied to study evolving structures of propagating linear copolymer chains. As a result, a general numerical toolbox was developed capable to handle various copolymerization mechanisms utilizing two-dimensional distributions as means of system description. The method was applied to free radical copolymerization of styrene–acrylonitrile explaining peculiar bimodality in chain length distribution observed for asymmetrical initial monomer ratios.
  12. I. Kryven, PD. Iedema. Transition into the gel regime for crosslinking radical polymerisation in a continuously stirred tank reactor,
    Chemical Engineering Science, Volume 126, Issue 14, Pages 296–308, 2015 [MORE] [PDF]
    Crosslinking radical polymerisation in a continuously stirred tank reactor has been studied by means of a four-dimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiradicals as dimensions. The model covers both pre-gel and gel regimes in a straightforward manner. Approximations on radial basis functions have been employed to reduce the size of the system with minimal information loss. The comparison with Monte Carlo simulations shows interesting and unexpected features.
  13. X. Cao, A. van Dam, B. de Leeuw, C. Geldhauser, J. Grasman, I. Kryven, D. Lahaye, L. Morelli, V. Rottschäfer, H. Zhou. Always Nice Weather in Europe. Proceedings SWI 2015 [MORE] [PDF]
    Weather forecasting relies on mathematical models that exhibit chaotic behavior. This renders the solution of these models very sensitive to errors in the model, to choices of the initial conditions and to rounding errors in the numerical solution procedure. Over the course of the past decade, various meteorological institutes in Europe have developed different at- mosphere models. Each of these models has its strengths and weaknesses. The principle behind the so-called Super Modeling approach is to merge these existing models into a single larger model to combine common strengths while overcoming individual weaknesses. This approach was initially proposed and developed by the KNMI in the Netherlands to improve the reliability of its weather forecasts. The task formulated for this Study Group problem was to reevaluate the Super Modeling approach and to formulate recommendations for its future development.
  14. I. Kryven, Topology evolution in macromolecular networks,
    Univeristy of Amsterdam, PhD Thessis, ISBN: 978-90-9028519-1, 2014; [MORE] [PDF]
    Governed by various intermolecular forces, molecular networks tend to evolve from simple to very complex formations that have random structure. This randomness in the connectivity of the basic units can still be captured employing distributional description of the state of the system; the evolution itself by particular stochastic processes, for instance Smoluchowski coagulation. The Smoluchowski coagulation equation can be extended to include collisions of orders distinct from 2, which allows developing a framework that covers various special cases being far beyond the scope of the original application of the coagulation equation. The combination of the population balance equation based on a generalization of the Smoluchowski coagulation and a meshless projection method based on Gaussian basis functions has been found to be a powerful tool allowing to solve many problems of prior unmanageable complexity. For instance, in the case of cross-linking polymerisation in bulk, the population balance model describing a particular set of reaction mechanisms has been present in literature for decades before it has been successfully and to a full extent solved by the numerical method. The other case studies address formation of various molecular networks in a few important cases: polymerisation of AB2 monomers, modification of linear chains into a branched polymer, copolymerisation with a composition drift, and coagulation/ coalescence of colloids of variable fractal dimension.
  15. I. Kryven, PD. Iedema. Transition into the gel regime for free radical crosslinking polymerisation in a batch reactor,
    Polymer, Volume 55, Issue 16, Pages 3475–3489, 2014; [MORE] [PDF]
    Crosslinking polymerization has been studied by means of a four-dimensional population balance model accounting for chain length, free pending double bonds, crosslinks, and multiradicals as dimensions. The model, for the first time and to a full extent resolves the crosslinking problem as formulated by Zhu et al. [1] and covers both pre-gel and gel regimes, in a straightforward manner. Approximations on radial basis functions have been employed to reduce the size of the system with minimal information loss. The model has been validated with data from an experimental crosslinking polymerization, Methyl Methacrylate with Ethylene Glycol Dimethacrylate. Non-trivial patterns in the time evolution of average quantities like crosslink densities, partly observed in prior studies [23], are naturally emerging from the model by computing marginals of the four-dimensional distribution possessing an interesting multimodal structure.
  16. I. Kryven, S. Lazzari, G. Storti. Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems,
    Macromolecular Theory and Simulations, Volume 55, Issue 16, Pages 3475–3489, 2014; [MORE] [PDF]
    A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation (PBE) based on cluster mass and fractal dimension is solved, employing a discretization method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusion-limited-cluster aggregation (DLCA) and reaction-limited-cluster-aggregation (RLCA) kernels and different coalescence rates. It turns out that in both DLCA and RLCA regimes, a faster coalescence leads to smaller sized and more compact clusters, whereas a slow coalescence promotes the formation of highly reactive
  17. I. Kryven, PD. Iedema. Topology Evolution in Polymer Modification,
    Macromolecular Theory and Simulations, Volume 23, Issue 1, pages 7–14, 2014 [MORE] [PDF] [JOURNAL COVER]
    A recent numerical method has opened new opportunities in multidimensional population balance modeling. Here, this method is applied to a full three-dimensional population balance model (PBM) describing branching topology evolution driven by chain end to backbone coupling. This process is typical for polymer modification reactions, e.g., in polyethylene, where initially linear polymer chains undergo hydrogen abstraction, and subsequent branching or scission. Topologies are distinguished by chain ends, number of branches, and number of reactive ends. The resulting time dependent trivariate distribution is utilized to extract various distributive properties of the polymer. The results exhibit excellent agreement with data from Monte Carlo simulations
  18. I. Kryven, PD. Iedema. Predicting multidimensional distributive properties of hyperbranched polymer resulting from AB2 polymerization with substitution, cyclization and shielding, Volume 54, Issue 14, Pages 3472–3484, 2013 [MORE] [PDF]
    A deterministic mathematical model for the polymerization of hyperbranched molecules accounting for substitution, cyclization, and shielding effect has been developed as a system of nonlinear population balances. The solution obtained by a novel approximation method shows perfect agreement with the analytical solution in limiting cases and provides, for the first time in this class of polymerization problems, full multidimensional results.
  19. I. Kryven, A. Berkenbos, P. Melo, DM. Kim. PD. Iedema. Modeling Crosslinking Polymerization in Batch and Continuous Reactors, Volume 7, Issue 5, pages 205–220, 2013 [MORE] [PDF]
    A new pseudo-distribution approach is applied to the modeling of crosslinking copolymerization of vinyl and divinyl monomer and compared to Monte Carlo (MC) simulations. With the number of free pending double bonds as the main distribution variable, a rigorous solution of the three leading moments of the molecular size distribution becomes possible. Validation takes place with data of methyl methacrylate with ethylene glycol dimethacrylate. Well within the sol regime perfect agreement is found, but near the gelpoint larger discrepancies do appear. This is probably due to the existence of multiradicals that are not taken into account in the population balance approaches.
  20. T. van der Aalst, D. Denteneer, H. Döring, MH. Duong, RJ. Kang, M. Keane, J. Kool, I. Kryven, T. Meyfroyt, T. Müller, G. Regts, J. Tomczyk The random disc thrower problem, Proceedings SWI 2013, Pages 59-78, 2013 [MORE] [PDF]
    We describe a number of approaches to a question posed by Philips Research, described as the "random disc thrower" problem. Given a square grid of points in the plane, we cover the points by equal-sized planar discs according to the following random process. At each step, a random point of the grid is chosen from the set of uncovered points as the centre of a new disc. This is an abstract model of spatial reuse in wireless networks. A question of Philips Research asks what, as a function of the grid length, is the expected number of discs chosen before the process can no longer continue? Our main results concern the one-dimensional variant of this problem, which can be solved reasonably well, though we also provide a number of approaches towards an approximate solution of the original two-dimensional problem. The two-dimensional problem is related to an old, unresolved conjecture ([6]) that has been the object of close study in both probability theory and statistical physics. Keywords: generating functions, Markov random fields, random sequential adsorption, Rényi’s parking problem, wireless networks
  21. V. Kukharskyy, Ya. Savula, I. Kryven Modified method of residual-free bubbles for solving the advection-diffusion problem with high Peclet number, Series Appl. Math. and Informatics. Visnyk of the Lviv University, Volum 20, Pages 85-94, 2013 [MORE]
    The numerical scheme based on the residual free bubbles approach and the hierarchy basis discretization for advection-diffusion problems with dominated advection is introduced. Among properties of the developed method are better stability to oscillations for hight Peclet numbers, better time performance, and simplicity in parallel computing application. The computational analysis of convergence and time performance on an example of a 2D symmetric advection-diffusion problem for 10e10 Peclet number is performed.
  22. I. Kryven, PD. Iedema. A Novel Approach to Population Balance Modeling of Reactive Polymer Modification Leading to Branching, Volume 22, Issue 2, pages 89–106, 2013 [MORE] [PDF] [JOURNAL COVER]
    The mathematical treatment of polymer modification systems, described by population balances containing convolution is discussed. The two-dimensional case (molecular weight vs. number of branch points) was considered by utilizing approximations of distributions, expanding them in terms of Gaussian basis functions. Three branching reactions were addressed: chain backbone to chain end point coupling; three-functional coupling of chain ends; and crosslinking. The results were compared to those of Monte Carlo (MC) simulations. Good agreement was observed, although the quality of a distribution as generated by the numerical approach is much better in view of the strong scatter in the MC data.
  23. J.B. van den Berg, R. Castro, J. Draisma, J. Evers, M. Hendriks, O. Krehel, I. Kryven, K. Mora, B. Szabó, P. Zwiernik. Non-imaging Optics for LED-Lighting, Proceedings of SWI 2012, Editor MAA. Boon, ISBN 978-90-6464-630-0, Pages 70-103, 2013 [MORE] [PDF]
    In this report, several methods are investigated to rapidly compute the light intensity function, either in the far field or on a finite-distance screen, of light emanating from a light fixture with a given shape. Different shapes are considered, namely polygonal and (piecewise) smooth. In the first case, analytic methods are sought to circumvent the use of Monte Carlo methods and ray-tracing with large sample size. In the second case, refinements of the Monte Carlo method (notably using a bootstrap procedure) are devised to minimize the number of samples needed for a good approximation of the intensity function.
  24. W. Kager, I. Kryven, K. Myerscough, T. van Opstal, T. Rot. Statistical Modelling of Pre-Impact Velocities in Car Crashes, Proceedings SWI 2011, Pages 133-148, 2013 [MORE] [PDF]
    The law wants to determine if any party involved in a car crash is guilty. The Dutch court invokes the expertise of the Netherlands Forensic Institute (NFI) to answer this question. We discuss the present method of the NFI to determine probabilities on pre-impact car velocities, given the evidence from the crash scene. A disadvantage of this method is that it requires a prior distribution on the velocities of the cars involved in the crash. We suggest a different approach, that of statistical significance testing, which can be carried out without a prior. We explain this method, and apply it to a toy model. Finally, a sensitivity analysis is performed on a simple two-car collision model.


How matter becomes fractal on the way from discrete to continuum?

in collaboration with Α. Τorres-Κnoop

Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum concept which is used in continuum mechanics. But what exactly happens at the transition of the scale from discrete to continuous? Even the most basic discrete models for liquids, as for instance the hard sphere model, do exhibit non-trivial fluctuations of density on the small scales. By employing molecular dynamics, we demonstrate that, as the scale increases, the cross-linked polymer materials transiently feature fractal structure before reaching the state of continuum.

Random graphs explain the supramolecualr gelation

in collaboration with V. Lakshminarayanan

Conventional linear polymers are long chains made of many repeated units, the monomers, that are connected by covalent bonds; whereas the supramolecular polymers, make use of reversible interactions that hold their chains together. In contrast to supramolecular polymerisation, little is known about the mechanism of supramolecular gelation -- the polymerisation process that eventually leads to higher order chain structures. Gelation constitutes one more important point in which conventional polymers and supramolecular polymers are essentially different. Conventional polymer gels stem from hyper-branched polymers, cross-linked networks or inter-entangled systems of polymer rings, but can never be formed from exclusively linear chains, whereas purely linear supramolecular polymers often exhibit a gel-like behaviour in experiments. This project relates the phenomenon of the supramolecular gelation to the emergence of the giant component in the underlaing, effective, network.

Generalised configuration model for random graphs

in collaboration with V.Schamboeck

In the infinite configuration model the links between nodes are assigned randomly with the only restriction that the degree distribution has to match a predefined function. Size distribution of connected components denotes probability that a randomly chosen node is part of a connected component of finite size n. Connected components in the infinite configuration network can be of finite or infinite size. Depending upon a specific context behind the network, the size distribution of connected components may summarise important features of the modelled system, for instance as it often happens in chemistry, where connected components represent irregular molecular structures; epidemiology, where configuration model is widely used in modelling disease outbreaks; or linguistics, where connected components are proved to be useful when studying sentence similarity graphs and structure of natural languages. This brief list of application case studies is far from being exhaustive.

By exploiting the fact that there exists a relationship between sizes of connected components and specially designed random walks we obtain analytical expressions for the size distribution of connected components and its asymptotes.

Auto-generated complex reaction netowrks

in collaboration with Y.Orlova

Reaction network is a bipartite graph in which nodes represent reactions and species whereas directed edges represent the participation of a species in respective reactions. An ingoing edge indicates that a species acts as a reactant; an outgoing edge indicates that a species acts as a product of a reaction. In this way, the nodes with in-degree zero are classified as the input species, and the nodes with out-degree zero — output species.

We have developed an in-house code that allows us to construct reaction networks for a wide range of complex chemical and biological systems by iterating the reaction rules encoding the basic principles of organic chemistry. The resulting networks are not random and typically count thousands of nodes and reactions. The main function of these networks is reaction kinetics — a specific type of nonlinear diffusion that corresponds to how the mass of the species is converted while flowing form the input to output nodes. Some of the output species are important from the application point of view, the others are merely byproducts. Do the generated networks feature some peculiar structure? Is there a centrality measure that gives a chemically meaningful ranking of species and/or reactions? Naturally, we like to understand what part of the network is responsible for the control of the important output nodes.

Numerical Methods for Chemical Master Equations

in collaboration with Ch. Schuette & S. Roeblitz

The chemical master equation is the fundamental equation describing stochastics of chemical and biological systems. This differential-difference equation describes the temporal evolution of the probability density function for system states. A state of the system, usually encoded as a vector, represents the number of entities or copy numbers of interacting species, which are changing according to a list of possible reactions. It is often the case, especially when the state vector is high-dimensional, that the number of possible states the system may occupy is too large to be handled computationally. One way to get around this problem is to consider only those states that are associated with probabilities that are greater than a certain threshold level - the essential support.

I study projection methods that represent the probability density function with a small number of basis functions. Furthermore, the system of basis functions is reconsidered on every time step in such a way that the solution is approximated only on the essential support of the probability density function instead of the whole state space.

The method is especially usefull when dealing with models that yield solutions of a complex structure, for instance, featuring multi-stability or supported on domains with complex topology. The figure on the right illustrates application of the method to the chemical master equation describing the process of binary cell differentiation. [MORE]

[FULL IMAGE]             [ANIMATION ~ 20Mb]

Random graph models explain molecular networks

Graph theory has always been a source of inspiration for describing many chemical systems. When it comes to large molecular structures, e.g. hyperbranched or cross-linked polymers, random graph models have been right in the center of attention. These models allow to capture the essence of the polymerisation: temporal assemblage of monomers (vertexes) into polymer (the network). However, existing random graph models do not capture all nuances that are so common in the molecular world. For instance, a node A may prefer to obtain a new connection with a node B that is already linked to A by a short path. As a direct consequence of such a rule, the system obtains many triangles, that are almost not observable in the standard Erdős–Rényi model. A reverse effect also may take place: a new connection with a node A may be less probable if A is surrounded by a 'dense' network, according to a certain centrality measure.

One of the remarkable features of such models is their tendency to exhibit high modularity, i.e nodes can be partitioned into clusters, highly interconnected inside, but with few inter-cluster edges. The figure on the left shows the cluster analysis on a network with a local connection preference. The colors indicates cluster membership. The node sizes denote centrality: the degree of being surrounded by the neighbours that may obstruct new connections.

Multiscale view on composition drift and sequence patterns in linear polymers

Due to the stochastic nature of polymer systems, the distributional description became a widely established tool. From this perspective, a polymer molecule is viewed as a single entity having various properties (e.g. size, branching index, fraction of comonomers, etc.); while the distribution represents an ensemble of these molecules. However, a polymer molecule is a complex system in itself, and such an approach may often be too coarse.

This project breaks down the paradigm by introducing distributional description on multiple levels of scales: to represent an ensemble of polymer molecules and to represent a single molecule as an ensemble of monomers. Even information on sequential inclusion of comonomers is captured by considering positional distributions: a probability of inclusion of certain monomer type as a function of position or probability of having a sequence of a given pattern starting at given position. Analysis of these new types of distributional properties reveals a tendency for polymer composition being statistically dependent on the position - a phenomena referred to as a "composition drift".


Kinetic model for coagulation and coalesence of fractal colloids

A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation based on cluster mass and fractal dimension is solved, employing a discretisation method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusion-limited-cluster aggregation (DLCA) and reaction-limited-cluster-aggregation (RLCA) kernels and different coalescence rates.


Phase transition in cross-linked molecular networks

Gelation is a drastic transformation of physical properties of the material that can be observed during crolss-linking polymerisation. This process is linked to one of the central phenomena in random graph theory - emergence of the giant component. The project aims in building a bridge between graph theory and polymer chemistry in order to combine the strong sides of the two worlds. Numerical approach allows here to consider the chemical system with least simplifications possible.

[MORE: POLYMER 2015] [MORE: CES 2015]

Effects of spatial configuration in hyperbranched polymers

Steric hindrance occurs when some groups within a molecule shield the rest of groups: making the shilded groups unavailable for chemical reactions. Although steric hindrance is sometimes a problem it can also be a very useful tool, and is often exploited by chemists to change the reactivity pattern of a molecule. Population balance approach allows to construct a model accounting for this phenomenon. Since the geometric shape of a molecule is dictated by random processes, the effect of geometry can be obtained only in terms of propagating uncertainty rather than a precise number.


Meshless projection for population balance equation

The stochastic approach has brought a refreshing way to describe the physical reality of the World by mathematical language. However, the new models that account for stochasticity, have emerged as a very complex mathematical objects. In polymer chemistry stochastic models, namely population balance equations (PBE), provide a means to solve a non-trivial and frequently self-contradictory task: describing a random system compactly while preserving properties of interest at the same time. PBEs are difficult to solve exactly and one is forced to seek an approximation.

This project focuses on a new numerical method utilizing Gaussian radial basis functions to find approximation to the exact solution of PBEs. A particular attention is devoted to the PBEs that describe evolution of a random network, and thus contain convolutional terms.

[MORE: MTS 2014] [MORE: MTS 2013]