prof. dr hab. Tomasz Lipniacki 

Doktorat
1998  Wirowa warstwa przyścienna w dynamice nadciekłego helu. Turbulencja kwantowa
 542 
Habilitacja
20070615  Anizotropowa turbulencja w nadciekłym helu i dynamika wirów dyskretnych 
Profesura
20120807  Nadanie tytułu naukowego profesora 
Promotor prac doktorskich
1.  20141007  Jaruszewicz Joanna (IBIB PAN)  The influence of noise characteristics on the relative stability of attractors in bistable biochemical systems  1074  
2.  20100128  Hat Beata  Wpływ ilości kopii genu na dynamikę sieci regulatorowych w komórce  634  
3.  20090922  Puszyński Krzysztof (PŚl)  Deterministyczne i stochastyczne modele ścieżek regulatorowych związanych z apoptozą  1038  
4.  2006  Paszek Paweł (RU)  Modeling stochasticity in gene regulation  1037 
Ostatnie publikacje
1.  Czerkies M., Korwek Z., Prus W., Kochańczyk M., JaruszewiczBłońska J., Tudelska K., Błoński S., Kimmel M.^{♦}, Brasier A.R.^{♦}, Lipniacki T., Cell fate in antiviral response arises in the crosstalk of IRF, NFκB and JAK/STAT pathways, Nature Communications, ISSN: 20411723, DOI: 10.1038/s41467017026408, Vol.9, pp.493, 2018 Streszczenie: The innate immune system processes pathogeninduced signals into cell fate decisions. How information is turned to decision remains unknown. By combining stochastic mathematical modelling and experimentation, we demonstrate that feedback interactions between the IRF3, NFκB and STAT pathways lead to switchlike responses to a viral analogue, poly(I:C), in contrast to pulselike responses to bacterial LPS. Poly(I:C) activates both IRF3 and NFκB, a requirement for induction of IFNβ expression. Autocrine IFNβ initiates a JAK/STATmediated positivefeedback stabilising nuclear IRF3 and NFκB in first responder cells. Paracrine IFNβ, in turn, sensitises second responder cells through a JAK/STATmediated positive feedforward pathway that upregulates the positivefeedback components: RIGI, PKR and OAS1A. In these sensitised cells, the ‘liveordie’ decision phase following poly(I:C) exposure is shorter—they rapidly produce antiviral responses and commit to apoptosis. The interlinked positive feedback and feedforward signalling is key for coordinating cell fate decisions in cellular populations restricting pathogen spread. Słowa kluczowe: cellular signalling networks, innate immunity, regulatory networks, stochastic modelling Afiliacje autorów:
 45p.  
2.  Kochańczyk M., Hlavacek W.S.^{♦}, Lipniacki T., SPATKIN: a simulator for rulebased modeling of biomolecular site dynamics on surfaces , BIOINFORMATICS, ISSN: 13674803, DOI: 10.1093/bioinformatics/btx456, Vol.33, No.22, pp.36673669, 2017 Streszczenie: Rulebased modeling is a powerful approach for studying biomolecular site dynamics. Here, we present SPATKIN, a generalpurpose simulator for rulebased modeling in two spatial dimensions. The simulation algorithm is a latticebased method that tracks Brownian motion of individual molecules and the stochastic firing of ruledefined reaction events. Because rules are used as event generators, the algorithm is networkfree, meaning that it does not require to generate the complete reaction network implied by rules prior to simulation. In a simulation, each molecule (or complex of molecules) is taken to occupy a single lattice site that cannot be shared with another molecule (or complex). SPATKIN is capable of simulating a wide array of membraneassociated processes, including adsorption, desorption and crowding. Models are specified using an extension of the BioNetGen language, which allows to account for spatial features of the simulated process. AVAILABILITY AND IMPLEMENTATION: The C ++ source code for SPATKIN is distributed freely under the terms of the GNU GPLv3 license. The source code can be compiled for execution on popular platforms (Windows, Mac and Linux). An installer for 64bit Windows and a macOS app are available. The source code and precompiled binaries are available at the SPATKIN Web site (http://pmbm.ippt.pan.pl/software/spatkin). Afiliacje autorów:
 45p.  
3.  Habibi I.^{♦}, Cheong R.^{♦}, Lipniacki T., Levchenko A.^{♦}, Emamian E.S.^{♦}, Abdi A.^{♦}, Computation and measurement of cell decision making errors using single cell data, PLOS COMPUTATIONAL BIOLOGY, ISSN: 1553734X, DOI: 10.1371/journal.pcbi.1005436, Vol.13, No.4, pp.e1005436117, 2017 Streszczenie: In this study a new computational method is developed to quantify decision making errors in cells, caused by noise and signaling failures. Analysis of tumor necrosis factor (TNF) signaling pathway which regulates the transcription factor Nuclear Factor κB (NFκB) using this method identifies two types of incorrect cell decisions called false alarm and miss. These two events represent, respectively, declaring a signal which is not present and missing a signal that does exist. Using single cell experimental data and the developed method, we compute false alarm and miss error probabilities in wildtype cells and provide a formulation which shows how these metrics depend on the signal transduction noise level. We also show that in the presence of abnormalities in a cell, decision making processes can be significantly affected, compared to a wildtype cell, and the method is able to model and measure such effects. In the TNF—NFκB pathway, the method computes and reveals changes in false alarm and miss probabilities in A20deficient cells, caused by cell’s inability to inhibit TNFinduced NFκB response. In biological terms, a higher false alarm metric in this abnormal TNF signaling system indicates perceiving more cytokine signals which in fact do not exist at the system input, whereas a higher miss metric indicates that it is highly likely to miss signals that actually exist. Overall, this study demonstrates the ability of the developed method for modeling cell decision making errors under normal and abnormal conditions, and in the presence of transduction noise uncertainty. Compared to the previously reported pathway capacity metric, our results suggest that the introduced decision error metrics characterize signaling failures more accurately. This is mainly because while capacity is a useful metric to study information transmission in signaling pathways, it does not capture the overlap between TNFinduced noisy response curves. Słowa kluczowe: Decision making, Radar, Probability distribution, Transcription factors, Signal processing, Signal transduction, Signaling networks, Statistical signal processing Afiliacje autorów:
 45p.  
4.  Kochańczyk M., Kocieniewski P., Kozłowska E.^{♦}, JaruszewiczBłońska J., Sparta B.^{♦}, Pargett M.^{♦}, Albeck J.G.^{♦}, Hlavacek W.S.^{♦}, Lipniacki T., Relaxation oscillations and hierarchy of feedbacks in MAPK signaling, Scientific Reports, ISSN: 20452322, DOI: 10.1038/srep38244, Vol.7, No.38244, pp.115, 2017 Streszczenie: We formulated a computational model for a MAPK signaling cascade downstream of the EGF receptor to investigate how interlinked positive and negative feedback loops process EGF signals into ERK pulses of constant amplitude but dosedependent duration and frequency. A positive feedback loop involving RAS and SOS, which leads to bistability and allows for switchlike responses to inputs, is nested within a negative feedback loop that encompasses RAS and RAF, MEK, and ERK that inhibits SOS via phosphorylation. This negative feedback, operating on a longer time scale, changes switchlike behavior into oscillations having a period of 1 hour or longer. Two auxiliary negative feedback loops, from ERK to MEK and RAF, placed downstream of the positive feedback, shape the temporal ERK activity profile but are dispensable for oscillations. Thus, the positive feedback introduces a hierarchy among negative feedback loops, such that the effect of a negative feedback depends on its position with respect to the positive feedback loop. Furthermore, a combination of the fast positive feedback involving slowdiffusing membrane components with slower negative feedbacks involving faster diffusing cytoplasmic components leads to local excitation/global inhibition dynamics, which allows the MAPK cascade to transmit paracrine EGF signals into spatially nonuniform ERK activity pulses. Słowa kluczowe: MAPK signaling, Oscillations, Mathematical modelling Afiliacje autorów:
 40p.  
5.  Tudelska K., Markiewicz J., Kochańczyk M., Czerkies M., Prus W., Korwek Z., Abdi A.^{♦}, Błoński S., Kaźmierczak B., Lipniacki T., Information processing in the NFκB pathway, Scientific Reports, ISSN: 20452322, DOI: 10.1038/s4159801716166y, Vol.7, pp.15926, 2017 Streszczenie: The NFκB pathway is known to transmit merely 1 bit of information about stimulus level. We combined experimentation with mathematical modeling to elucidate how information about TNF concentration is turned into a binary decision. Using KolmogorovSmirnov distance, we quantified the cell’s ability to discern 8 TNF concentrations at each step of the NFκB pathway, to find that input discernibility decreases as signal propagates along the pathway. Discernibility of low TNF concentrations is restricted by noise at the TNF receptor level, whereas discernibility of high TNF concentrations it is restricted by saturation/depletion of downstream signaling components. Consequently, signal discernibility is highest between 0.03 and 1 ng/ml TNF. Simultaneous exposure to TNF or LPS and a translation inhibitor, cycloheximide, leads to prolonged NFκB activation and a marked increase of transcript levels of NFκB inhibitors, IκBα and A20. The impact of cycloheximide becomes apparent after the first peak of nuclear NFκB translocation, meaning that the NFκB network not only relays 1 bit of information to coordinate the allornothing expression of early genes, but also over a longer time course integrates information about other stimuli. The NFκB system should be thus perceived as a feedbackcontrolled decisionmaking module rather than a simple information transmission channel. Słowa kluczowe: cellular signaling networks, innate immunity, stress signaling Afiliacje autorów:
 40p.  
6.  Varga A.^{♦}, Ehrenreiter K.^{♦}, Aschenbrenner B.^{♦}, Kocieniewski P., Kochańczyk M., Lipniacki T., Baccarini M.^{♦}, RAF1/BRAF dimerization integrates the signal from RAS to ERK and ROKα, Science Signaling, ISSN: 19450877, DOI: 10.1126/scisignal.aai8482, Vol.10, No.469, pp.eaai8482111, 2017 Streszczenie: Downstream of growth factor receptors and of the guanine triphosphatase (GTPase) RAS, heterodimers of the serine/threonine kinases BRAF and RAF1 are critical upstream kinases and activators of the mitogenactivated protein kinase (MAPK) module containing the mitogenactivated and extracellular signal–regulated kinase kinase (MEK) and their targets, the extracellular signal–regulated kinase (ERK) family. Either direct or scaffold protein–mediated interactions among the components of the ERK module (the MAPKKKs BRAF and RAF1, MEK, and ERK) facilitate signal transmission. RAF1 also has essential functions in the control of tumorigenesis and migration that are mediated through its interaction with the kinase ROKα, an effector of the GTPase RHO and regulator of cytoskeletal rearrangements. We combined mutational and kinetic analysis with mathematical modeling to show that the interaction of RAF1 with ROKα is coordinated with the role of RAF1 in the ERK pathway. We found that the phosphorylated form of RAF1 that interacted with and inhibited ROKα was generated during the interaction of RAF1 with the ERK module. This mechanism adds plasticity to the ERK pathway, enabling signal diversification at the level of both ERK and RAF. Furthermore, by connecting ERK activation with the regulation of ROKα and cytoskeletal rearrangements by RAF1, this mechanism has the potential to precisely coordinate the proper timing of proliferation with changes in cell shape, adhesion, or motility. Słowa kluczowe: MAPK pathway, kinase RAF, protein isoform, phosphorylation, mathematical modeling Afiliacje autorów:
 40p.  
7.  Białecki S., Kaźmierczak B., Lipniacki T., Polarization of concave domains by traveling wave pinning, PLOS ONE, ISSN: 19326203, DOI: 10.1371/journal.pone.0190372, Vol.12, No.12, pp.e0190372110, 2017 Streszczenie: Pattern formation is one of the most fundamental yet puzzling phenomena in physics and biology. We propose that traveling front pinning into concave portions of the boundary of 3dimensional domains can serve as a generic gradientmaintaining mechanism. Such a mechanism of domain polarization arises even for scalar bistable reactiondiffusion equations, and, depending on geometry, a number of stationary fronts may be formed leading to complex spatial patterns. The main advantage of the pinning mechanism, with respect to the Turing bifurcation, is that it allows for maintaining gradients in the specific regions of the domain. By linking the instant domain shape with the spatial pattern, the mechanism can be responsible for cellular polarization and differentiation. Afiliacje autorów:
 35p.  
8.  JaruszewiczBłońska J., Lipniacki T., Genetic toggle switch controlled by bacterial growth rate, BMC SYSTEMS BIOLOGY, ISSN: 17520509, DOI: 10.1186/s1291801704834, Vol.11, pp.117111, 2017 Streszczenie: Background Słowa kluczowe: Mathematical modeling, Stochastic simulations, Regulatory pathways, Bistability, DNA replication, Gene copy number Afiliacje autorów:
 35p.  
9.  Hat B., Kochańczyk M., Bogdał M.N., Lipniacki T., Feedbacks, bifurcations, and cell fate decisionmaking in the p53 system, PLOS COMPUTATIONAL BIOLOGY, ISSN: 1553734X, DOI: 10.1371/journal.pcbi.1004787, Vol.12, No.2, pp.e1004787128, 2016 Streszczenie: The p53 transcription factor is a regulator of key cellular processes including DNA repair, cell cycle arrest, and apoptosis. In this theoretical study, we investigate how the complex circuitry of the p53 network allows for stochastic yet unambiguous cell fate decisionmaking. The proposed Markov chain model consists of the regulatory core and two subordinated bistable modules responsible for cell cycle arrest and apoptosis. The regulatory core is controlled by two negative feedback loops (regulated by Mdm2 and Wip1) responsible for oscillations, and two antagonistic positive feedback loops (regulated by phosphatases Wip1 and PTEN) responsible for bistability. By means of bifurcation analysis of the deterministic approximation we capture the recurrent solutions (i.e., steady states and limit cycles) that delineate temporal responses of the stochastic system. Direct switching from the limitcycle oscillations to the “apoptotic” steady state is enabled by the existence of a subcritical Neimark—Sacker bifurcation in which the limit cycle loses its stability by merging with an unstable invariant torus. Our analysis provides an explanation why cancer cell lines known to have vastly diverse expression levels of Wip1 and PTEN exhibit a broad spectrum of responses to DNA damage: from a fast transition to a high level of p53 killer (a p53 phosphoform which promotes commitment to apoptosis) in cells characterized by high PTEN and low Wip1 levels to longlasting p53 level oscillations in cells having PTEN promoter methylated (as in, e.g., MCF7 cell line). Słowa kluczowe: Apoptosis, Cell cycle and cell division, DNA damage, DNA repair, Phosphorylation, Biochemical simulations, Cell cycle inhibitors, Transcription factors Afiliacje autorów:
 45p.  
10.  Korwek Z., Tudelska K.^{♦}, NałęczJawecki P.^{♦}, Czerkies M., Prus W., Markiewicz J., Kochańczyk M., Lipniacki T., Importins promote highfrequency NFκB oscillations increasing information channel capacity, Biology Direct, ISSN: 17456150, DOI: 10.1186/s130620160164z, Vol.11, No.61, pp.121, 2016 Streszczenie: BACKGROUND: Słowa kluczowe: Karyopherins, Nucleocytoplasmic transport, Negative feedback, Channel information capacity, Mathematical modelling Afiliacje autorów:
 35p.  
11.  Kellogg R.A.^{♦}, Tian C.^{♦}, Lipniacki T., Quake S.R.^{♦}, Tay S.^{♦}, Digital signaling decouples activation probability and population heterogeneity, eLife, ISSN: 2050084X, DOI: 10.7554/eLife.08931, Vol.4, pp.e08931126, 2015 Streszczenie: Digital signaling enhances robustness of cellular decisions in noisy environments, but it is unclear how digital systems transmit temporal information about a stimulus. To understand how temporal input information is encoded and decoded by the NFκB system, we studied transcription factor dynamics and gene regulation under dose and durationmodulated inflammatory inputs. Mathematical modeling predicted and microfluidic singlecell experiments confirmed that integral of the stimulus (or area, concentration × duration) controls the fraction of cells that activate NFκB in the population. However, stimulus temporal profile determined NFκB dynamics, celltocell variability, and gene expression phenotype. A sustained, weak stimulation lead to heterogeneous activation and delayed timing that is transmitted to gene expression. In contrast, a transient, strong stimulus with the same area caused rapid and uniform dynamics. These results show that digital NFκB signaling enables multidimensional control of cellular phenotype via input profile, allowing parallel and independent control of singlecell activation probability and population heterogeneity. Afiliacje autorów:
 45p.  
12.  Szymańska P.^{♦}, Martin K.R.^{♦}, MacKeigan J.P.^{♦}, Hlavacek W.S.^{♦}, Lipniacki T., Computational analysis of an autophagy/translation switch based on mutual inhibition of mTORC1 and ULK1, PLOS ONE, ISSN: 19326203, DOI: 10.1371/journal.pone.0116550, Vol.10, No.3, pp.e0116550134, 2015 Streszczenie: We constructed a mechanistic, computational model for regulation of (macro)autophagy and protein synthesis (at the level of translation). The model was formulated to study the systemlevel consequences of interactions among the following proteins: two key components of MTOR complex 1 (MTORC1), namely the protein kinase MTOR (mechanistic target of rapamycin) and the scaffold protein RPTOR; the autophagyinitiating protein kinase ULK1; and the multimeric energysensing AMPactivated protein kinase (AMPK). Inputs of the model include intrinsic AMPK kinase activity, which is taken as an adjustable surrogate parameter for cellular energy level or AMP:ATP ratio, and rapamycin dose, which controls MTORC1 activity. Outputs of the model include the phosphorylation level of the translational repressor EIF4EBP1, a substrate of MTORC1, and the phosphorylation level of AMBRA1 (activating molecule in BECN1regulated autophagy), a substrate of ULK1 critical for autophagosome formation. The model incorporates reciprocal regulation of mTORC1 and ULK1 by AMPK, mutual inhibition of MTORC1 and ULK1, and ULK1mediated negative feedback regulation of AMPK. Through analysis of the model, we find that these processes may be responsible, depending on conditions, for graded responses to stress inputs, for bistable switching between autophagy and protein synthesis, or relaxation oscillations, comprising alternating periods of autophagy and protein synthesis. A sensitivity analysis indicates that the prediction of oscillatory behavior is robust to changes of the parameter values of the model. The model provides testable predictions about the behavior of the AMPKMTORC1ULK1 network, which plays a central role in maintaining cellular energy and nutrient homeostasis. Afiliacje autorów:
 40p.  
13.  NałęczJawecki P.^{♦}, Szymańska P.^{♦}, Kochańczyk M., Miękisz J.^{♦}, Lipniacki T., Effective reaction rates for diffusionlimited reaction cycles, JOURNAL OF CHEMICAL PHYSICS, ISSN: 00219606, DOI: 10.1063/1.4936131, Vol.143, No.21, pp.215102112, 2015 Streszczenie: Biological signals in cells are transmitted with the use of reaction cycles, such as the phosphorylationdephosphorylation cycle, in which substrate is modified by antagonistic enzymes. An appreciable share of such reactions takes place in crowded environments of twodimensional structures, such as plasma membrane or intracellular membranes, and is expected to be diffusioncontrolled. In this work, starting from the microscopic bimolecular reaction rate constants and using estimates of the mean firstpassage time for an enzyme–substrate encounter, we derive diffusiondependent effective macroscopic reaction rate coefficients (EMRRC) for a generic reaction cycle. Each EMRRC was found to be half of the harmonic average of the microscopic rate constant (phosphorylation c or dephosphorylation d), and the effective (crowdingdependent) motility divided by a slowly decreasing logarithmic function of the sum of the enzyme concentrations. This implies that when c and d differ, the two EMRRCs scale differently with the motility, rendering the steadystate fraction of phosphorylated substrate molecules diffusiondependent. Analytical predictions are verified using kinetic Monte Carlo simulations on the twodimensional triangular lattice at the singlemolecule resolution. It is demonstrated that the proposed formulas estimate the steadystate concentrations and effective reaction rates for different sets of microscopic reaction rates and concentrations of reactants, including a nontrivial example where with increasing diffusivity the fraction of phosphorylated substrate molecules changes from 10% to 90%. Słowa kluczowe: Enzymes, Enzyme kinetics, Diffusion, Reaction rate constants, Membrane biochemistry Afiliacje autorów:
 35p.  
14.  Szymańska P.^{♦}, Kochańczyk M., Miękisz J.^{♦}, Lipniacki T., Effective reaction rates in diffusionlimited phosphorylationdephosphorylation cycles, PHYSICAL REVIEW E, ISSN: 15393755, DOI: 10.1103/PhysRevE.91.022702, Vol.91, pp.022702115, 2015 Streszczenie: We investigate the kinetics of the ubiquitous phosphorylationdephosphorylation cycle on biological membranes by means of kinetic Monte Carlo simulations on the triangular lattice. We establish the dependence of effective macroscopic reaction rate coefficients as well as the steadystate phosphorylated substrate fraction on the diffusion coefficient and concentrations of opposing enzymes: kinases and phosphatases. In the limits of zero and infinite diffusion, the numerical results agree with analytical predictions; these two limits give the lower and the upper bound for the macroscopic rate coefficients, respectively. In the zerodiffusion limit, which is important in the analysis of dense systems, phosphorylation and dephosphorylation reactions can convert only these substrates which remain in contact with opposing enzymes. In the most studied regime of nonzero but small diffusion, a contribution linearly proportional to the diffusion coefficient appears in the reaction rate. In this regime, the presence of opposing enzymes creates inhomogeneities in the (de)phosphorylated substrate distributions: The spatial correlation function shows that enzymes are surrounded by clouds of converted substrates. This effect becomes important at low enzyme concentrations, substantially lowering effective reaction rates. Effective reaction rates decrease with decreasing diffusion and this dependence is more pronounced for the lessabundant enzyme. Consequently, the steadystate fraction of phosphorylated substrates can increase or decrease with diffusion, depending on relative concentrations of both enzymes. Additionally, steady states are controlled by molecular crowders which, mostly by lowering the effective diffusion of reactants, favor the more abundant enzyme. Afiliacje autorów:
 35p.  
15.  Nienałtowski K., Włodarczyk M.^{♦}, Lipniacki T., Komorowski M., Clustering reveals limits of parameter identifiability in multiparameter models of biochemical dynamics, BMC SYSTEMS BIOLOGY, ISSN: 17520509, DOI: 10.1186/s1291801502058, Vol.9, pp.6519, 2015 Streszczenie: Background Afiliacje autorów:
 35p.  
16.  Bertolusso R.^{♦}, Tian B.^{♦}, Zhao Y.^{♦}, Vergara L.^{♦}, Sabree A.^{♦}, Iwanaszko M.^{♦}, Lipniacki T., Brasier A.R.^{♦}, Kimmel M.^{♦}, Dynamic cross talk model of the epithelial innate immune response to doublestranded RNA stimulation: Coordinated dynamics emerging from celllevel noise, PLOS ONE, ISSN: 19326203, DOI: 10.1371/journal.pone.0093396, Vol.9, No.4, pp.e93396121, 2014 Streszczenie: We present an integrated dynamical crosstalk model of the epithelial innate immune reponse (IIR) incorporating RIGI and TLR3 as the two major pattern recognition receptors (PRR) converging on the RelA and IRF3 transcriptional effectors. bioPN simulations reproduce biologically relevant geneand protein abundance measurements in response to time course, gene silencing and doseresponse perturbations both at the population and single cell level. Our computational predictions suggest that RelA and IRF3 are under auto and crossregulation. We predict, and confirm experimentally, that RIGI mRNA expression is controlled by IRF7. We also predict the existence of a TLR3dependent, IRF3independent transcription factor (or factors) that control(s) expression of MAVS, IRF3 and members of the IKK family. Our model confirms the observed dsRNA dosedependence of oscillatory patterns in single cells, with periods of 1–3 hr. Model fitting to time series, matched by knockdown data suggests that the NFκB module operates in a different regime (with different coefficient values) than in the TNFαstimulation experiments. In future studies, this model will serve as a foundation for identification of virusencoded IIR antagonists and examination of stochastic effects of viral replication. Afiliacje autorów:
 40p.  
17.  Jaruszewicz J., Kimmel M.^{♦}, Lipniacki T., Stability of bacterial toggle switches is enhanced by cellcycle lengthening by several orders of magnitude, PHYSICAL REVIEW E, ISSN: 15393755, DOI: 10.1103/PhysRevE.89.022710, Vol.89, No.2, pp.022710126, 2014 Streszczenie: Bistable regulatory elements are important for nongenetic inheritance, increase of celltocell heterogeneity allowing adaptation, and robust responses at the population level. Here, we study computationally the bistable genetic toggle switch—a small regulatory network consisting of a pair of mutual repressors—in growing and dividing bacteria. We show that as cells with an inhibited growth exhibit high stability of toggle states, cell growth and divisions lead to a dramatic increase of toggling rates. The toggling rates were found to increase with rate of cell growth, and can be up to six orders of magnitude larger for fast growing cells than for cells with the inhibited growth. The effect is caused mainly by the increase of protein and mRNA burst sizes associated with the faster growth. The observation that fast growth dramatically destabilizes toggle states implies that rapidly growing cells may vigorously explore the epigenetic landscape enabling nongenetic evolution, while cells with inhibited growth adhere to the local optima. This can be a clever population strategy that allows the slow growing (but stress resistant) cells to survive long periods of unfavorable conditions. Simultaneously, at favorable conditions, this stress resistant (but slowly growing—or not growing) subpopulation may be replenished due to a high switching rate from the fast growing population. Słowa kluczowe: Gene expression, Bistability, Stochastic processes, Genetic toggle switch, Cell growth and division Afiliacje autorów:
 35p.  
18.  Pękalski J.^{♦}, Żuk P.J.^{♦}, Kochańczyk M., Junkin M.^{♦}, Kellogg R.^{♦}, Tay S.^{♦}, Lipniacki T., Spontaneous NFκB Activation by Autocrine TNFα Signaling: A Computational Analysis, PLOS ONE, ISSN: 19326203, DOI: 10.1371/journal.pone.0078887, Vol.8, No.11, pp.e78887114, 2013 Streszczenie: NFκB is a key transcription factor that regulates innate immune response. Its activity is tightly controlled by numerous feedback loops, including two negative loops mediated by NFκB inducible inhibitors, IκBα and A20, which assure oscillatory responses, and by positive feedback loops arising due to the paracrine and autocrine regulation via TNFα, IL1 and other cytokines. We study the NFκB system of interlinked negative and positive feedback loops, combining bifurcation analysis of the deterministic approximation with stochastic numerical modeling. Positive feedback assures the existence of limit cycle oscillations in unstimulated wildtype cells and introduces bistability in A20deficient cells. We demonstrated that cells of significant autocrine potential, i.e., cells characterized by high secretion of TNFα and its receptor TNFR1, may exhibit sustained cytoplasmic–nuclear NFκB oscillations which start spontaneously due to stochastic fluctuations. In A20deficient cells even a small TNFα expression rate qualitatively influences system kinetics, leading to longlasting NFκB activation in response to a shortpulsed TNFα stimulation. As a consequence, cells with impaired A20 expression or increased TNFα secretion rate are expected to have elevated NFκB activity even in the absence of stimulation. This may lead to chronic inflammation and promote cancer due to the persistent activation of antiapoptotic genes induced by NFκB. There is growing evidence that A20 mutations correlate with several types of lymphomas and elevated TNFα secretion is characteristic of many cancers. Interestingly, A20 loss or dysfunction also leaves the organism vulnerable to septic shock and massive apoptosis triggered by the uncontrolled TNFα secretion, which at high levels overcomes the antiapoptotic action of NFκB. It is thus tempting to speculate that some cancers of deregulated NFκB signaling may be prone to the pathogeninduced apoptosis. Afiliacje autorów:
 40p.  
19.  Kochańczyk M., Jaruszewicz J., Lipniacki T., Stochastic transitions in a bistable reaction system on the membrane, JOURNAL OF THE ROYAL SOCIETY INTERFACE, ISSN: 17425689, DOI: 10.1098/rsif.2013.0151, Vol.10, No.84, pp.112, 2013 Streszczenie: Transitions between steady states of a multistable stochastic system in the perfectly mixed chemical reactor are possible only because of stochastic switching. In realistic cellular conditions, where diffusion is limited, transitions between steady states can also follow from the propagation of travelling waves. Here, we study the interplay between the two modes of transition for a prototype bistable system of kinase–phosphatase interactions on the plasma membrane. Within microscopic kinetic Monte Carlo simulations on the hexagonal lattice, we observed that for finite diffusion the behaviour of the spatially extended system differs qualitatively from the behaviour of the same system in the wellmixed regime. Even when a small isolated subcompartment remains mostly inactive, the chemical travelling wave may propagate, leading to the activation of a larger compartment. The activating wave can be induced after a small subdomain is activated as a result of a stochastic fluctuation. Such a spontaneous onset of activity is radically more probable in subdomains characterized by slower diffusion. Our results show that a local immobilization of substrates can lead to the global activation of membrane proteins by the mechanism that involves stochastic fluctuations followed by the propagation of a semideterministic travelling wave. Słowa kluczowe: multistability, Markov process, spatially extended system, kinetic Monte Carlo on the lattice, kinase autophosphorylation, cell signalling Afiliacje autorów:
 40p.  
20.  Crooks E.^{♦}, Kaźmierczak B., Lipniacki T., A spatially extended model of kinasereceptor interaction, SIAM JOURNAL ON APPLIED MATHEMATICS, ISSN: 00361399, DOI: 10.1137/110860926, Vol.73, No.1, pp.374400, 2013 Streszczenie: We perform a mathematical analysis of a spatially extended model describing mutual phosphorylation of cytosolic kinases and membrane receptors. The analyzed regulatory system is a part of signal transduction mechanisms, which enables communication of the cell with its extracellular environment or other cells. The mutual receptorkinase interaction is characteristic for immune receptors and Src family kinases. From the mathematical viewpoint, the considered system is interesting because it couples differential equations defined in a domain $\Omega$ and on its boundary $\partial\Omega$ via nonlinear Robin boundary conditions. Assuming a spherically symmetric framework, our approach is to consider an auxiliary problem in which the Robin boundary condition on the external boundary of the spherical shell $\Omega$ is replaced by a uniform Dirichlet boundary condition. This method allows us to find the stationary spherically symmetric solutions, both stable and unstable. Interestingly, numerical computations suggest also the existence of nonspherically symmetric unstable stationary solutions to the spherically symmetric problem. These conjectured solutions appear to lie between super and subsolutions that converge in time to two different stable spherically symmetric steady states. The study is completed by the proof of an existence theorem for a general version of the original system encompassing the earlier models. The theorem holds in domains of arbitrary shape, with a number of holes that may represent various organelles impenetrable to the considered kinases. The last result ensures that the problem in which the flux of active kinases (which replaces the source term) is determined by the Robintype boundary condition at the cell membrane is well posed. The considered generalized model may thus serve as a template for intracellular signal transduction analysis. Słowa kluczowe: reactiondiffusion systems, Robintype boundary conditions, bistability, molecular signal transduction Afiliacje autorów:
 40p.  
21.  Jaruszewicz J., Żuk P.J.^{♦}, Lipniacki T., Type of noise defines global attractors in bistable molecular regulatory systems, JOURNAL OF THEORETICAL BIOLOGY, ISSN: 00225193, DOI: 10.1016/j.jtbi.2012.10.004, Vol.317, pp.140151, 2013 Streszczenie: The aim of this study is to demonstrate that in molecular dynamical systems with the underlying bi or multistability, the type of noise determines the most strongly attracting steady state or stochastic attractor. As an example we consider a simple stochastic model of autoregulatory gene with a nonlinear positive feedback, which in the deterministic approximation has two stable steady state solutions. Three types of noise are considered: transcriptional and translational – due to the small number of gene product molecules and the gene switching noise – due to gene activation and inactivation transitions. We demonstrate that the type of noise in addition to the noise magnitude dictates the allocation of probability mass between the two stable steady states. In particular, we found that when the gene switching noise dominates over the transcriptional and translational noise (which is characteristic of eukaryotes), the gene preferentially activates, while in the opposite case, when the transcriptional noise dominates (which is characteristic of prokaryotes) the gene preferentially remains inactive. Moreover, even in the zeronoise limit, when the probability mass generically concentrates in the vicinity of one of two steady states, the choice of the most strongly attracting steady state is noise typedependent. Although the epigenetic attractors are defined with the aid of the deterministic approximation of the stochastic regulatory process, their relative attractivity is controlled by the type of noise, in addition to noise magnitude. Since noise characteristics vary during the cell cycle and development, such mode of regulation can be potentially employed by cells to switch between alternative epigenetic attractors. Słowa kluczowe: Gene expression, Bistability, Stochastic processes, Epigenetic attractors Afiliacje autorów:
 35p.  
22.  Bogdał M.N., Hat B., Kochańczyk M., Lipniacki T., Levels of proapoptotic regulator Bad and antiapoptotic regulator BclxL determine the type of the apoptotic logic gate, BMC SYSTEMS BIOLOGY, ISSN: 17520509, DOI: 10.1186/17520509767, Vol.7, pp.117, 2013 Streszczenie: Background Słowa kluczowe: Apoptosis, Cell survival, Signaling pathway, Bcl2 family, Bistability, Boolean logic, Ordinary differential equations Afiliacje autorów:
 35p.  
23.  Kocieniewski P., Lipniacki T., MEK1 and MEK2 differentially control the duration and amplitude of the ERK cascade response, PHYSICAL BIOLOGY, ISSN: 14783967, DOI: 10.1088/14783975/10/3/035006, Vol.10, pp.035006115, 2013 Streszczenie: The Raf/MEK/ERK cascade is one of the most studied and important signal transduction pathways. However, existing models largely ignore the existence of isoforms of the constituent kinases and their interactions. Here, we propose a model of the ERK cascade that includes heretofore neglected differences between isoforms of MEK. In particular, MEK1 is subject to a negative feedback from activated ERK, which is further conferred to MEK2 via heterodimerization. Specifically, ERK phosphorylates MEK1 at the residue Thr292, hypothetically creating an additional phosphatase binding site, accelerating MEK1 and MEK2 dephosphorylation. We incorporated these recently discovered interactions into a mathematical model of the ERK cascade that reproduces the experimental results of Catalanotti et al (2009 Nature Struct. Mol. Biol. 16 294–303) and Kamioka et al (2010 J. Biol. Chem. 285 33540–8). Furthermore, the model allows for predictions regarding the differences in the catalytic activity and function of the MEK isoforms. We propose that the MEK1/MEK2 ratio regulates the duration of the response, which increases with the level of MEK2 and decreases with the level of MEK1. In turn, the amplitude of the response is controlled by the total amount of the two isoforms. We confirm the proposed model structure performing a random parameter sampling, which led us to the conclusion that the sampled parameters, selected to properly reproduce wildtype (WT) cell behavior, to allow for qualitative reproduction of differences in behavior WT cells and cell mutants studied experimentally. Afiliacje autorów:
 25p.  
24.  Jaruszewicz J., Lipniacki T., Toggle switch: Noise determines the winning gene, PHYSICAL BIOLOGY, ISSN: 14783967, DOI: 10.1088/14783975/10/3/035007, Vol.10, pp.035007110, 2013 Streszczenie: Bistable regulatory elements enhance heterogeneity in cell populations and, in multicellular organisms, allow cells to specialize and specify their fate. Our study demonstrates that in a system of bistable genetic switch, the noise characteristics control in which of the two epigenetic attractors the cell population will settle. We focus on two types of noise: the gene switching noise and protein dimerization noise. We found that the change of magnitudes of these noise components for one of the two competing genes introduces a large asymmetry of the protein stationary probability distribution and changes the relative probability of individual gene activation. Interestingly, an increase of noise associated with a given gene can either promote or suppress the activation of the gene, depending on the type of noise. Namely, each gene is repressed by an increase of its gene switching noise and activated by an increase of its proteinproduct dimerization noise. The observed effect was found robust to the large, up to fivefold deviations of the model parameters. In summary, we demonstrated that noise itself may determine the relative strength of the epigenetic attractors, which may provide a unique mode of control of cell fate decisions. Słowa kluczowe: Gene expression, Bistability, Stochastic processes, Genetic toggle switch Afiliacje autorów:
 25p.  
25.  Barua D.^{♦}, Hlavacek W.S.^{♦}, Lipniacki T., A computational model for early events in B cell antigen receptor signaling: analysis of the roles of Lyn and Fyn, JOURNAL OF IMMUNOLOGY, ISSN: 00221767, DOI: 10.4049/jimmunol.1102003, Vol.189, pp.646658, 2012 Streszczenie: BCR signaling regulates the activities and fates of B cells. BCR signaling encompasses two feedback loops emanating from Lyn and Fyn, which are Src family protein tyrosine kinases (SFKs). Positive feedback arises from SFKmediated trans phosphorylation of BCR and receptorbound Lyn and Fyn, which increases the kinase activities of Lyn and Fyn. Negative feedback arises from SFKmediated cis phosphorylation of the transmembrane adapter protein PAG1, which recruits the cytosolic protein tyrosine kinase Csk to the plasma membrane, where it acts to decrease the kinase activities of Lyn and Fyn. To study the effects of the positive and negative feedback loops on the dynamical stability of BCR signaling and the relative contributions of Lyn and Fyn to BCR signaling, we consider in this study a rulebased model for early events in BCR signaling that encompasses membraneproximal interactions of six proteins, as follows: BCR, Lyn, Fyn, Csk, PAG1, and Syk, a cytosolic protein tyrosine kinase that is activated as a result of SFKmediated phosphorylation of BCR. The model is consistent with known effects of Lyn and Fyn deletions. We find that BCR signaling can generate a single pulse or oscillations of Syk activation depending on the strength of Ag signal and the relative levels of Lyn and Fyn. We also show that bistability can arise in Lyn or Cskdeficient cells. Afiliacje autorów:
 40p.  
26.  Kocieniewski P., Faeder J.R.^{♦}, Lipniacki T., The interplay of double phosphorylation and scaffolding in MAPK pathways, JOURNAL OF THEORETICAL BIOLOGY, ISSN: 00225193, DOI: 10.1016/j.jtbi.2011.11.014, Vol.295, pp.116124, 2012 Streszczenie: The MAPK cascades are principal kinase transduction pathways in eukaryotic cells. This family includes RAF/ERK, JNK, and p38 pathways. In the MAPK cascade, the signal is transmitted through three layers of sequentially activated kinases, MAP3K, MAP2K, and MAPK. The latter two kinases require dual phosphorylation for activation. The dual phosphorylation requirement has been implicated in bringing about bistability and switchlike responses in the cascade. MAPK signaling has been known to involve scaffolds—multidomain proteins that can assemble protein complexes; in this case the three MAPK components. Scaffolds are thought to increase the specificity of signaling by pairing enzymes and substrates. Scaffolds have been shown to biphasically control the response (the level of activated MAPK) and amplify it at a certain scaffold concentration range. In order to understand the interplay of scaffolding and multisite phosphorylation, in this study we analyze simplified MAPK signaling models in which we assume that either mono or double phosphorylation of MAP2K and MAPK is required for activation. We demonstrate that the requirement for double phosphorylation directs signaling through scaffolds. In the hypothetical scenario in which monophosphorylation suffices for kinase activity, the presence of scaffolds has little effect on the response. This suggests that double phosphorylation in MAPK pathways, although not as efficient as monophosphorylation, evolved together with scaffolds to assure the tighter control and higher specificity in signaling, by enabling scaffolds to function as response amplifiers. Słowa kluczowe: Kinase cascades, Rule based modeling, Molecular pathways evolution Afiliacje autorów:
 35p.  
27.  Żuk P.J.^{♦}, Kochańczyk M., Jaruszewicz J., Bednorz W.^{♦}, Lipniacki T., Dynamics of a stochastic spatially extended system predicted by comparing deterministic and stochastic attractors of the corresponding birth–death process, PHYSICAL BIOLOGY, ISSN: 14783967, DOI: 10.1088/14783975/9/5/055002, Vol.9, pp.055002112, 2012 Streszczenie: Living cells may be considered as biochemical reactors of multiple steady states. Transitions between these states are enabled by noise, or, in spatially extended systems, may occur due to the traveling wave propagation. We analyze a onedimensional bistable stochastic birth–death process by means of potential and temperature fields. The potential is defined by the deterministic limit of the process, while the temperature field is governed by noise. The stable steady state in which the potential has its global minimum defines the global deterministic attractor. For the stochastic system, in the low noise limit, the stationary probability distribution becomes unimodal, concentrated in one of two stable steady states, defined in this study as the global stochastic attractor. Interestingly, these two attractors may be located in different steady states. This observation suggests that the asymptotic behavior of spatially extended stochastic systems depends on the substrate diffusivity and size of the reactor. We confirmed this hypothesis within kinetic Monte Carlo simulations of a bistable reaction–diffusion model on the hexagonal lattice. In particular, we found that although the kinase–phosphatase system remains inactive in a small domain, the activatory traveling wave may propagate when a larger domain is considered. Słowa kluczowe: multistability, Markov process, spatially extended system, kinetic Monte Carlo on the lattice, cell signalling Afiliacje autorów:
 30p.  
28.  Hat B., Kaźmierczak B., Lipniacki T., B cell activation triggered by the formation of the small receptor cluster: a computational study, PLOS COMPUTATIONAL BIOLOGY, ISSN: 1553734X, DOI: 10.1371/journal.pcbi.1002197, Vol.7, No.10, pp.113, 2011 Streszczenie: We proposed a spatially extended model of early events of B cell receptors (BCR) activation, which is based on mutual kinasereceptor interactions that are characteristic for the immune receptors and the Src family kinases. These interactions lead to the positive feedback which, together with two nonlinearities resulting from the double phosphorylation of receptors and MichaelisMenten dephosphorylation kinetics, are responsible for the system bistability. We demonstrated that B cell can be activated by a formation of a tiny cluster of receptors or displacement of the nucleus. The receptors and Src kinases are activated, first locally, in the locus of the receptor cluster or the region where the cytoplasm is the thinnest. Then the traveling wave of activation propagates until activity spreads over the whole cell membrane. In the models in which we assume that the kinases are free to diffuse in the cytoplasm, we found that the fraction of aggregated receptors, capable to initiate B cell activation decreases with the decreasing thickness of cytoplasm and decreasing kinase diffusion. When kinases are restricted to the cell membrane  which is the case for most of the Src family kinases  even a cluster consisting of a tiny fraction of total receptors becomes activatory. Interestingly, the system remains insensitive to the modest changes of total receptor level. The model provides a plausible mechanism of B cells activation due to the formation of small receptors clusters collocalized by binding of polyvalent antigens or arising during the immune synapse formation. Afiliacje autorów:
 45p.  
29.  Kursa M.^{♦}, Bajer K.^{♦}, Lipniacki T., Cascade of vortex loops initiated by a single reconnection of quantum vortices, PHYSICAL REVIEW B, ISSN: 10980121, DOI: 10.1103/PhysRevB.83.014515, Vol.83, pp.01451515, 2011 Streszczenie: We demonstrate that a single reconnection of two quantum vortices can lead to the creation of a cascade of vortex rings. Our analysis involves localized induction approximation, highresolution BiotSavart and GrossPitaevskii simulations. The latter showed that the rings cascade starts on the atomic scale, with rings diameters orders of magnitude smaller than the characteristic line spacing in the tangle. Vortex rings created in the cascades may penetrate the tangle and annihilate on the boundaries. This provides an efficient decay mechanism for sparse or moderately dense vortex tangle at very low temperatures. Afiliacje autorów:
 35p.  
30.  Chylek L.A.^{♦}, Hu B.^{♦}, Blinov M.L.^{♦}, Emonet T.^{♦}, Faeder J.R.^{♦}, Goldstein B.^{♦}, Gutenkunst R.N.^{♦}, Haugh J.M.^{♦}, Lipniacki T., Posner R.G.^{♦}, Yang J.^{♦}, Hlavacek W.S.^{♦}, Guidelines for visualizing and annotating rulebased models, MOLECULAR BIOSYSTEMS, ISSN: 1742206X, DOI: 10.1039/c1mb05077j, Vol.7, pp.27792795, 2011 Streszczenie: Rulebased modeling provides a means to represent cell signaling systems in a way that captures sitespecific details of molecular interactions. For rulebased models to be more widely understood and (re)used, conventions for model visualization and annotation are needed. We have developed the concepts of an extended contact map and a model guide for illustrating and annotating rulebased models. An extended contact map represents the scope of a model by providing an illustration of each molecule, molecular component, direct physical interaction, posttranslational modification, and enzyme–substrate relationship considered in a model. A map can also illustrate allosteric effects, structural relationships among molecular components, and compartmental locations of molecules. A model guide associates elements of a contact map with annotation and elements of an underlying model, which may be fully or partially specified. A guide can also serve to document the biological knowledge upon which a model is based. We provide examples of a map and guide for a published rulebased model that characterizes early events in IgE receptor (FceRI) signaling. We also provide examples of how to visualize a variety of processes that are common in cell signaling systems but not considered in the example model, such as ubiquitination. An extended contact map and an associated guide can document knowledge of a cell signaling system in a form that is visual as well as executable. As a tool for model annotation, a map and guide can communicate the content of a model clearly and with precision, even for large models. Afiliacje autorów:
 30p.  
31.  Szopa P., Lipniacki T., Kaźmierczak B., Exact solutions to a spatially extended model of kinasereceptor interaction, PHYSICAL BIOLOGY, ISSN: 14783967, DOI: 10.1088/14783975/8/5/055005, Vol.8, No.5, pp.055005117, 2011 Streszczenie: B and Mast cells are activated by the aggregation of the immune receptors. Motivated by this phenomena we consider a simple spatially extended model of mutual interaction of kinases and membrane receptors. It is assumed that kinase activates membrane receptors and in turn the kinase molecules bound to the active receptors are activated by transphosphorylation. Such a type of interaction implies positive feedback and may lead to bistability. In this study we apply the Steklov eigenproblem theory to analyze the linearized model and find exact solutions in the case of nonuniformly distributed membrane receptors. This approach allows us to determine the critical value of receptor dephosphorylation rate at which cell activation (by arbitrary small perturbation of the inactive state) is possible. We found that cell sensitivity grows with decreasing kinase diffusion and increasing anisotropy of the receptor distribution. Moreover, these two effects are cooperating. We showed that the cell activity can be abruptly triggered by the formation of the receptor aggregate. Since the considered activation mechanism is not based on receptor crosslinking by polyvalent antigens, the proposed model can also explain B cell activation due to receptor aggregation following binding of monovalent antigens presented on the antigen presenting cell. Afiliacje autorów:
 30p.  
32.  Tay S.^{♦}, Hughey J.J.^{♦}, Lee T.K.^{♦}, Lipniacki T., Quake S.R.^{♦}, Covert M.W.^{♦}, Singlecell NFkB dynamics reveal digital activation and analogue information processing, NATURE, ISSN: 00280836, DOI: 10.1038/nature09145, Vol.466, pp.267271, 2010 Streszczenie: Cells operate in dynamic environments using extraordinary communication capabilities that emerge from the interactions of genetic circuitry. The mammalian immune response is a striking example of the coordination of different cell types1. Celltocell communication is primarily mediated by signalling molecules that form spatiotemporal concentration gradients, requiring cells to respond to a wide range of signal intensities2. Here we use highthroughput microfluidic cell culture3 and fluorescence microscopy, quantitative gene expression analysis and mathematical modelling to investigate how single mammalian cells respond to different concentrations of the signalling molecule tumournecrosis factor (TNF)α, and relay information to the gene expression programs by means of the transcription factor nuclear factor (NF)κB. We measured NFκB activity in thousands of live cells under TNFα doses covering four orders of magnitude. We find, in contrast to populationlevel studies with bulk assays2, that the activation is heterogeneous and is a digital process at the singlecell level with fewer cells responding at lower doses. Cells also encode a subtle set of analogue parameters to modulate the outcome; these parameters include NFκB peak intensity, response time and number of oscillations. We developed a stochastic mathematical model that reproduces both the digital and analogue dynamics as well as most gene expression profiles at all measured conditions, constituting a broadly applicable model for TNFαinduced NFκB signalling in various types of cells. These results highlight the value of highthroughput quantitative measurements with singlecell resolution in understanding how biological systems operate. Słowa kluczowe: Cell biology, Biophysics, Immunology, Genetics, Genomics Afiliacje autorów:
 40p.  
33.  Kaźmierczak B., Lipniacki T., Spatial gradients in kinase cascade regulation, IET SYSTEMS BIOLOGY, ISSN: 17518849, DOI: 10.1049/ietsyb.2010.0002, Vol.4, No.6, pp.348355, 2010 Streszczenie: The spatiotemporal kinetics of proteins and other substrates regulate cell fate and signaling. In this study, we consider a reaction–diffusion model of interaction of membrane receptors with a twostep kinase cascade. The receptors activate the ‘upstream’ kinase, which may diffuse over cell volume and activate the ‘downstream’ kinase, which is also diffusing. Both kinase species and receptors are inactivated by uniformly distributed phosphatases. The positive feedback, key to the considered dynamics, arises since the upstream kinase activates the receptors. Such a mutual interaction is characteristic for immune cell receptors. Based on the proposed model, we demonstrated that cell sensitivity (measured as a critical value of phosphatase activity at which cell maybe activated) increases with decreasing motility of receptorinteracting kinases and with increasing polarity of receptors distribution. These two effects are cooperating, the effect of receptors localisation close to one pole of the cell grows with the decreasing kinase diffusion and vanishes in the infinite diffusion limit. As the cell sensitivity increases with decreasing diffusion of receptorinteracting kinase, the overall activity of the downstream kinase increases with its diffusion. In conclusion, the analysis of the proposed model shows that, for the fixed substrate interaction rates, spatial distribution of the surface receptors together with the motility of intracellular kinases control cell signalling and sensitivity to extracellular signals. The increase of the cell sensitivity can be achieved by (i) localisation of receptors in a small subdomain of the cell membrane, (ii) lowering the motility of receptorinteracting kinase, (iii) increasing the motility of downstream kinases which distribute the signal over the whole cell. Afiliacje autorów:
 27p.  
34.  Kaźmierczak B., Lipniacki T., Regulation of kinase activity by diffusion and feedback, JOURNAL OF THEORETICAL BIOLOGY, ISSN: 00225193, DOI: 10.1016/j.jtbi.2009.03.016, Vol.259, pp.291296, 2009 Streszczenie: In living cells proteins motilities regulate the spatiotemporal dynamics of molecular pathways. We consider here a reaction–diffusion model of mutual kinase–receptor activation showing that the strength of positive feedback is controlled by the kinase diffusion coefficient. For high diffusion, the activated kinase molecules quickly leave the vicinity of the cell membrane and cannot efficiently activate the receptors. As a result, in a broad range of parameters, the cell can be activated only if the kinase diffusion coefficient is sufficiently small. Our simple model shows that change in the motility of substrates may dramatically influence the cell responses. Słowa kluczowe: Reaction–diffusion system, Signal transduction, Positive feedback, Kinase activation, Membrane receptors, Protein motility Afiliacje autorów:
 32p.  
35.  Hat B., Puszyński K.^{♦}, Lipniacki T., Exploring mechanisms of oscillations in p53 and NFkappaB systems, IET SYSTEMS BIOLOGY, ISSN: 17518849, DOI: 10.1049/ietsyb.2008.0156, Vol.3, pp.342355, 2009 Streszczenie: A number of regulatory networks have the potential to generate sustained oscillations of irregular amplitude, but well conserved period. Singlecell experiments revealed that in p53 and nuclear factor (NF)kB systems the oscillation period is homogenous in cell populations, insensitive to the strength of the stimulation, and is not influenced by the overexpression of p53 or NFkB transcription factors. We propose a novel computational method of validation of molecular pathways models, based on the analysis of sensitivity of the oscillation period to the particular gene(s) copy number and the level of stimulation. Using this method, the authors demonstrate that existing p53 models, in which oscillations are borne at a saddlenodeoninvariantcircle or subcritical Hopf bifurcations (characteristic for systems with interlinked positive and negative feedbacks), are highly sensitive to gene copy number. Hence, these models cannot explain existing experiments. Analysing NFkB system, the authors show the importance of saturation in transcription of the NFkB inhibitor IkBa. Models without saturation predict that the oscillation period is a rapidly growing function of total NFkB level, which is in disagreement with experiments. The study supports the hypothesis that oscillations of robust period are based on supercritical Hopf bifurcation, characteristic for dynamical systems involving negative feedback and time delay. We hypothesise that in the p53 system, the role of positive feedback is not sustaining oscillations, but terminating them in severely damaged cells in which the apoptotic programme should be initiated. Afiliacje autorów:
 27p.  
36.  Puszyński K.^{♦}, Bertolusso R.^{♦}, Lipniacki T., Crosstalk between p53 and NFkappa B systems: proand antiapoptotic functions of NFkappa B, IET SYSTEMS BIOLOGY, ISSN: 17518849, DOI: 10.1049/ietsyb.2008.0172, Vol.3, pp.356367, 2009 Streszczenie: Nuclear factors p53 and NFkB control many physiological processes including cell cycle arrest, DNA repair, apoptosis, death, innate and adaptive immune responses, and inflammation. There are numerous pathways linking these systems and there is a bulk of evidence for cooperation as well as for antagonisms between p53 and NFkB. In this theoretical study, the authors use earlier models of p53 and NFkB systems and construct a crosstalk model of p53–NFkB network in order to explore the consequences of the twoway coupling, in which NFkB upregulates the transcription of p53, whereas in turn p53 attenuates transcription of NFkB inhibitors IkBa and A20. We consider a number of protocols in which cells are stimulated by tumour necrosis factora (TNFa) (that activates NFkB pathway) and/or gamma irradiation (that activates p53 pathway). The authors demonstrate that NFkB may have both anti and proapoptotic roles. TNFa stimulation, preceding DNA damaging irradiation, makes cells more resistant to irradiationinduced apoptosis, whereas the same TNFa stimulation, when preceded by irradiation, increases the apoptotic cell fraction. The finding suggests that diverse roles of NFkB in apoptosis and cancer could be related to the dynamical context of activation of p53 and NFkB pathways. Afiliacje autorów:
 27p.  
37.  Lipniacki T., Hat B., Faeder J.R.^{♦}, Hlavacek W.S.^{♦}, Stochastic effects and bistability in T cell receptor signaling, JOURNAL OF THEORETICAL BIOLOGY, ISSN: 00225193, DOI: 10.1016/j.jtbi.2008.05.001, Vol.254, No.1, pp.110122, 2008 Streszczenie: The stochastic dynamics of T cell receptor (TCR) signaling are studied using a mathematical model intended to capture kinetic proofreading (sensitivity to ligand–receptor binding kinetics) and negative and positive feedback regulation mediated, respectively, by the phosphatase SHP1 and the MAP kinase ERK. The model incorporates protein–protein interactions involved in initiating TCRmediated cellular responses and reproduces several experimental observations about the behavior of TCR signaling, including robust responses to as few as a handful of ligands (agonist peptide–MHC complexes on an antigenpresenting cell), distinct responses to ligands that bind TCR with different lifetimes, and antagonism. Analysis of the model indicates that TCR signaling dynamics are marked by significant stochastic fluctuations and bistability, which is caused by the competition between the positive and negative feedbacks. Stochastic fluctuations are such that singlecell trajectories differ qualitatively from the trajectory predicted in the deterministic approximation of the dynamics. Because of bistability, the average of singlecell trajectories differs markedly from the deterministic trajectory. Bistability combined with stochastic fluctuations allows for switchlike responses to signals, which may aid T cells in making committed cellfate decisions. Słowa kluczowe: T cell activation, Mathematical model, Kinetic proofreading, Hysteresis, Ordinary differential equations, Stochastic simulations Afiliacje autorów:
 
38.  Puszyński K.^{♦}, Hat B., Lipniacki T., Oscillations and bistability in the stochastic model of p53 regulation, JOURNAL OF THEORETICAL BIOLOGY, ISSN: 00225193, DOI: 10.1016/j.jtbi.2008.05.039, Vol.254, No.2, pp.452465, 2008 Streszczenie: The p53 regulatory pathway controls cell responses, which include cell cycle arrest, DNA repair, apoptosis and cellular senescence. We propose a stochastic model of p53 regulation, which is based on two feedback loops: the negative, coupling p53 with its immediate downregulator Mdm2, and the positive, which involves PTEN, PIP3 and Akt. Existence of the negative feedback assures homeostasis of healthy cells and oscillatory responses of DNAdamaged cells, which are persistent when DNA repair is inefficient and the positive feedback loop is broken. The positive feedback destroys the negative coupling between Mdm2 and p53 by sequestering most of Mdm2 in cytoplasm, so it may no longer prime the nuclear p53 for degradation. It works as a clock, giving the cell some time for DNA repair. However, when DNA repair is inefficient, the active p53 rises to a high level and triggers transcription of proapoptotic genes. As a result, small DNA damage may be repaired and the cell may return to its initial ‘‘healthy’’ state, while the extended damage results in apoptosis. The stochasticity of p53 regulation, introduced at the levels of gene expression, DNA damage and repair, leads to high heterogeneity of cell responses and causes cell population split after irradiation into subpopulations of apoptotic and surviving cells, with fraction of apoptotic cells growing with the irradiation dose. Słowa kluczowe: Signaling pathways, Positive and negative feedbacks, Mathematical modeling, Stochastic simulations, Apoptosis, p53, Mdm2, PTEN Afiliacje autorów:
 
39.  Lipniacki T., Puszyński K.^{♦}, Paszek P.^{♦}, Brasier A.R.^{♦}, Kimmel M.^{♦}, Single TNFalpha trimers mediating NFkappaB activation: Stochastic robustness of NFkappaB signaling, BMC BIOINFORMATICS, ISSN: 14712105, DOI: 10.1186/147121058376, Vol.8, pp.376400, 2007 Streszczenie: Background Afiliacje autorów:
 
40.  Fujarewicz K.^{♦}, Kimmel M.^{♦}, Lipniacki T., Świerniak A.^{♦}, Adjoint systems for models of cell signalling pathways and their application to parameter fitting, IEEEACM TRANSACTIONS ON COMPUTATIONAL BIOLOGY AND BIOINFORMATICS, ISSN: 15455963, DOI: 10.1109/tcbb.2007.1016, Vol.4, pp.322335, 2007 Streszczenie: The paper concerns the problem of fitting mathematical models of cell signaling pathways. Such models frequently take the form of sets of nonlinear ordinary differential equations. While the model is continuous in time, the performance index used in the fitting procedure involves measurements taken at discrete time moments. Adjoint sensitivity analysis is a tool which can be used for finding the gradient of a performance index in the space of parameters of the model. In the paper, a structural formulation of adjoint sensitivity analysis called the Generalized Backpropagation Through Time (GBPTT) is used. The method is especially suited for hybrid, continuousdiscrete time systems. As an example, we use the mathematical model of the NFkB regulatory module, which plays a major role in the innate immune response in animals. Słowa kluczowe: Biology and genetics, modeling, ordinary differential equations, parameter learni Afiliacje autorów:
 
41.  Bobrowski A.^{♦}, Lipniacki T., Piechór K., Rudnicki R.^{♦}, Asymptotic behavior of distributions of mRNA and protein levels in a model of stochastic gene expression, JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ISSN: 0022247X, DOI: 10.1016/j.jmaa.2006.11.043, Vol.333, No.2, pp.753769, 2007 Streszczenie: The paper is devoted to a stochastic process introduced in the recent paper by Lipniacki et al. [T. Lipniacki, P. Paszek, A. MarciniakCzochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression, J. Theor. Biol. 238 (2006) 348–367] in modelling gene expression in eukaryotes. Starting from the full generator of the process we show that its distributions satisfy a (Fokker–Plancktype) system of partial differential equations. Then, we construct a c0 Markov semigroup in L1 space corresponding to this system. The main result of the paper is asymptotic stability of the involved semigroup in the set of densities. Słowa kluczowe: Piecewise deterministic process, Stochastic gene expression, Semigroups of operators, Feller semigroups, Dual semigroups, Markov semigroups, Asymptotic stability Afiliacje autorów:
 
42.  Lipniacki T., Kimmel M.^{♦}, Deterministic and stochastic models of NFkB pathway, CARDIOVASCULAR TOXICOLOGY, ISSN: 15307905, DOI: 10.1007/s120120079003x, Vol.7, No.4, pp.215234, 2007 Streszczenie: In the article, we discuss the state of art and perspectives in deterministic and stochastic models of NFκB regulatory module. The NFκB is a transcription factor controlling various immune responses including inflammation and apoptosis. It is tightly regulated by at least two negative feedback loops involving IκBα and A20. This mode of regulation results in nucleustocytoplasm oscillations in NFκB localization, which induce subsequent waves of NFκB responsive genes. Single cell experiments carried by several groups provided comprehensive evidence that stochastic effects play an important role in NFκB regulation. From modeling point of view, living cells might be considered noisy or stochastic biochemical reactors. In eukaryotic cells, in which the number of protein or mRNA molecules is relatively large, stochastic effects primarily originate in regulation of gene activity. Transcriptional activity of a gene can be initiated by transactivator molecules binding to the specific regulatory site(s) in the target gene. The stochastic event of gene activation is amplified by transcription and translation, since it results in a burst of mRNA molecules, and each copy of mRNA then serves as a template for numerous protein molecules. Another potential source of variability can be receptors activation. At lowdose stimulation, important in celltocell signaling, the number of active receptors can be low enough to introduce substantial noise to downstream signaling. Stochastic modeling confirms the large variability in cell responses and shows that no cell behaves like an “average” cell. This high celltocell variability can be one of the weapons of the immune defense. Such nondeterministic defense may be harder to overcome by relatively simple programs coded in viruses and other pathogens. Słowa kluczowe: Mathematical modeling, Systems biology, Immune response, NFκB, Stochastic regulation, Stochastic robustness, Single cell Afiliacje autorów:
 
43.  Hat B., Paszek P.^{♦}, Kimmel M.^{♦}, Piechór K., Lipniacki T., How the number of alleles influences gene expression, JOURNAL OF STATISTICAL PHYSICS, ISSN: 00224715, DOI: 10.1007/s1095500692184, Vol.128, pp.511533, 2007 Streszczenie: The higher organisms, eukaryotes, are diploid and most of their genes have two homological copies (alleles). However, the number of alleles in a cell is not constant. In the S phase of the cell cycle all the genome is duplicated and then in the G2 phase and mitosis, which together last for several hours, most of the genes have four copies instead of two. Cancer development is, in many cases, associated with a change in allele number. Several genetic diseases are caused by haploinsufficiency: Lack of one of the alleles or its improper functioning. In the paper we consider the stochastic expression of a gene having a variable number of copies. We applied our previously developed method in which the reaction channels are split into slow (connected with change of gene state) and fast (connected with mRNA/protein synthesis/decay), the later being approximated by deterministic reaction rate equations. As a result we represent gene expression as a piecewise deterministic timecontinuous Markov process, which is further related with a system of partial differential hyperbolic equations for probability density functions (pdfs) of protein distribution. The stationary pdfs are calculated analytically for haploidal gene or numerically for diploidal and tetraploidal ones. We distinguished nine classes of simultaneous activation of haploid, diploid and tetraploid genes. This allows for analysis of potential consequences of gene duplication or allele loss. We show that when gene activity is autoregulated by a positive feedback, the change in number of gene alleles may have dramatic consequences for its regulation and may not be compensated by the change of efficiency of mRNA synthesis per allele. Słowa kluczowe: stochastic gene expression, feedback regulation, diploid genes, haploinsufficiency, piecewise deterministic timecontinuous Markov process Afiliacje autorów:
 
44.  Lipniacki T., Paszek P.^{♦}, Brasier A.R.^{♦}, Luxon B.A.^{♦}, Kimmel M.^{♦}, Stochastic regulation in early immune response, BIOPHYSICAL JOURNAL, ISSN: 00063495, DOI: 10.1529/biophysj.104.056754, Vol.90, No.3, pp.725742, 2006 Streszczenie: Living cells may be considered noisy or stochastic biochemical reactors. In eukaryotic cells, in which the number of protein or mRNA molecules is relatively large, the stochastic effects originate primarily in regulation of gene activity. Transcriptional activity of a gene can be initiated by transactivator molecules binding to the specific regulatory site(s) in the target gene. The stochasticity of activator binding and dissociation is amplified by transcription and translation, since target gene activation results in a burst of mRNAs molecules, and each copy of mRNA then serves as a template for numerous protein molecules. In this article, we reformulate our model of the NFκB regulatory module to analyze a single cell regulation. Ordinary differential equations, used for description of fast reaction channels of processes involving a large number of molecules, are combined with a stochastic switch to account for the activity of the genes involved. The stochasticity in gene transcription causes simulated cells to exhibit large variability. Moreover, none of them behaves like an average cell. Although the average mRNA and protein levels remain constant before tumor necrosis factor (TNF) stimulation, and stabilize after a prolonged TNF stimulation, in any single cell these levels oscillate stochastically in the absence of TNF and keep oscillating under the prolonged TNF stimulation. However, in a short period of ∼90 min, most cells are synchronized by the TNF signal, and exhibit similar kinetics. We hypothesize that this synchronization is crucial for proper activation of early genes controlling inflammation. Our theoretical predictions of single cell kinetics are supported by recent experimental studies of oscillations in NFκB signaling made on single cells. Afiliacje autorów:
 
45.  Lipniacki T., Paszek P.^{♦}, MarciniakCzochra A.^{♦}, Brasier A.R.^{♦}, Kimmel M.^{♦}, Transcriptional stochasticity in gene expression, JOURNAL OF THEORETICAL BIOLOGY, ISSN: 00225193, DOI: 10.1016/j.jtbi.2005.05.032, Vol.238, pp.348367, 2006 Streszczenie: Due to the small number of copies of molecular species involved, such as DNA, mRNA and regulatory proteins, gene expression is a stochastic phenomenon. In eukaryotic cells, the stochastic effects primarily originate in regulation of gene activity. Transcription can be initiated by a single transcription factor binding to a specific regulatory site in the target gene. Stochasticity of transcription factor binding and dissociation is then amplified by transcription and translation, since target gene activation results in a burst of mRNA molecules, and each mRNA copy serves as a template for translating numerous protein molecules. In the present paper, we explore a mathematical approach to stochastic modeling. In this approach, the ordinary differential equations with a stochastic component for mRNA and protein levels in a single cells yield a system of firstorder partial differential equations (PDEs) for twodimensional probability density functions (pdf). We consider the following examples: Regulation of a single autorepressing gene, and regulation of a system of two mutual repressors and of an activator–repressor system. The resulting PDEs are approximated by a system of many ordinary equations, which are then numerically solved. Słowa kluczowe: Gene regulation, Transcription, Stochasticity, Probability density function, Transporttype equations Afiliacje autorów:
 
46.  Lipniacki T., Dynamics of superfluid 4 He: twoscale approach, EUROPEAN JOURNAL OF MECHANICS BFLUIDS, ISSN: 09977546, DOI: 10.1016/j.euromechflu.2006.04.004, Vol.25, pp.435458, 2006 Streszczenie: The paper explores the two scale approach to the incompressible dynamics of superfluid 4He. The fluid is described by a system of equations: Navier–Stokes and Euler equations for the macroscopic normal and superfluid velocity fields respectively. The two equations couple via a mutual friction force exerted on superfluid (quantum) vortices by the normal component. The magnitude of this force, calculated via the analysis of dynamics of quantum vortices in the microscopic scale, is proportional to the value of the counterflow (relative velocity of two helium components) and to the density of quantum vortices. The latter is determined by the generalized Vinen equation, adequate for flows having a net macroscopic vorticity. The generalized equation includes the drift of the anisotropic vortex tangle caused by Magnus force. The derived system of equations is applied first to the analysis of steady state solution of rotating turbulence, and then to the numerical analysis of formation of plane Couette flow between two infinite parallel material surfaces z=0 and z=D. For t<0 both surfaces and the fluid are at rest, then at t=0 one material surface starts moving along the x axis with velocity V0. The viscosity forces cause the motion of normal component and the counterflow which make the linelength density grow, and the two components become coupled by the mutual friction. The fact that superfluid velocity tends to match with the normal velocity makes that ωs≠0, which implies polarization and drift of the tangle. At a given temperature the dynamics depends solely on DV0. Słowa kluczowe: Dynamics of 4He, Quantum turbulence, Anisotropy of quantum tangle, Rotating turbulence Afiliacje autorów:
 
47.  Paszek P.^{♦}, Lipniacki T., Brasier A.R.^{♦}, Bing Tian^{♦}, Nowak D.E.^{♦}, Kimmel M.^{♦}, Stochastic effects of multiple regulators on expression profiles in eukaryotes, JOURNAL OF THEORETICAL BIOLOGY, ISSN: 00225193, DOI: 10.1016/j.jtbi.2004.10.023, Vol.233, pp.423433, 2005 Streszczenie: The stochastic nature of gene regulation still remains not fully understood. In eukaryotes, the stochastic effects are primarily attributable to the binary nature of genes, which are considered either switched “on” or “off” due to the action of the transcription factors binding to the promoter. In the time period when the gene is activated, bursts of mRNA transcript are produced. In the present paper, we investigate regulation of gene expression at the single cell level. We propose a mechanism of gene regulation, which is able to explain the observed distinct transcription profiles assuming the number of coregulatory activities, without attempting to identify the specific proteins involved. The model is motivated by our experiments on NFκBκBdependent genes in HeLa cells. Our experimental data shows that NFκBκBdependent genes can be stratified into three characteristic groups according to their expression profiles: early, intermediate and late having maximum of expression at about 1, 3 and 6 h, respectively, from the beging of TNF stimulation. We provide a tractable analytical approach, not only in the terms of expected expression profiles and their moments, which corresponds to the measurements on the cell population, but also in the terms of single cell behavior. Comparison between these two modes of description reveals that single cells behave qualitatively different from the cell population. This analysis provides insights useful for understanding of microarray experiments. Słowa kluczowe: Stochastic gene regulation, Expression profiles, Single cell simulations, NFκκB Afiliacje autorów:

Prace konferencyjne
1.  Bogdał M.N., Hat B., Kochańczyk M., Lipniacki T., Gates to apoptosis, XVIII National Conference Applications of Mathematics in Biology and Medicine, 20120923/0927, Krynica Morska (PL), pp.16, 2012 Streszczenie: p53 is the key transcription factor controlling cellular responses to oncogenic stimulation and DNA da mage. Its activity is tightly controlled by numerous feedback loops. In response to DNA damage, p53 promotes expression of proteins, which suppress cell cycle and activate DNA repair. If the damage is irreparable or the repair takes too long, the programmed cell death (apoptosis) is initiated. In the current study we analyze the apoptotic module, a part of our larger p53 pathway model. In the model, the apoptosis is triggered due to the suppression of Akt activity and/or elevated level of p53 killer.p53 killer, i.e. p53 form phosphorylated at Ser46 (in addition to Ser15 and Ser20), promotes synthesis of pro apoptotic protein Bax. In healthy cells Bax is inactive due to binding to Bcl2, another member of Bcl2 family proteins. Suppression of Akt activity leads to the dissociation of Bax:Bcl2 complexes and release of Bax. Therefore, two signals may lead to the accumulation of free Bax: one coming from elevated level of p53 killer, the other resulting from decreased level of active Akt. We demonstrated that, depending on parameters, the apoptosis can be controlled by the logic gate ‘AND’ as well as gate ‘OR’. In the first case both signals are required simultaneously, while in the latter case any of the two signals suffices for the initiation of apoptosis. Afiliacje autorów:
 
2.  Jaruszewicz J., Żuk P.J.^{♦}, Lipniacki T., Probability density functions in bistable gene activation Model with two types of noise, 16th National Conference on Applications of Mathematics in Biology and Medicine, 20100914/0918, Krynica (PL), pp.4752, 2010 Streszczenie: The aim of this study is to demonstrate that in dynamical systems with underlying bistability the type of noise qualitatively influences the stationary probability distribution (SPD). Specifically, we consider a simplified model of gene expression with the nonlinear positive feedback, which in the deterministic approximation has two stable steady state solutions. Two types of noise are considered; transcriptional  due to the limited number of protein molecules, and gene switching noise  due to gene activation and inactivation. In the limit of zero noise, the SPD generically concentrates in the decreasing vicinity of one of the two stable steady states. We demonstrated that for a range of parameters the SPD corresponding to the system with transcriptional noise only concentrates around a different steady state than SPD corresponding to the system with gene switching noise only. Słowa kluczowe: Gene expression, Bistability, Stochastic processes, Epigenetic attractors Afiliacje autorów:
 
3.  Kocieniewski P., Lipniacki T., The interplay of double phosphorylation and scaffolding in the MAPK pathways, 16th National Conference on Applications of Mathematics in Biology and Medicine, 20100914/0918, Krynica (PL), pp.6570, 2010 Streszczenie: The MAPK cascade is a principal kinase transduction pathway in eukaryotic cells. It transmits signals through three layers of sequentially activated kinases, RAF, MEK and ERK. The latter two kinases require dual phosphorylation for activation. Another property of MAPK signalling is its involvement of scaffolds  multidomain proteins that can assemble protein complexes; in this case the three MAPK components. In this study we analyze analytically and numerically four heuristic models of MAPK signalling in the presence or absence of a scaffold considering both real MEK and ERK kinases and their hypothetical isoforms that require only monophosphorylation. Based on this analysis we will demonstrate that double phosphorylation enforces signaling through scaffolds, which increases both versatility and specificity of the regulation. Afiliacje autorów:
 
4.  Żuk P.J.^{♦}, Lipniacki T., Probability density functions in bistable kinase activation model, 16th National Conference on Applications of Mathematics in Biology and Medicine, 20100914/0918, Krynica (PL), pp.121126, 2010 Streszczenie: We consider a kinase autoactivation model in which the number of activated kinases follows the timecontinuous Markov process. In the deterministic approximation the process is described by the single nonlinear ordinary differential equation, which may have two stable steady states. We found that for sufficiently large number of kinases, the stationary probability distribution given by the Markov process concentrates in the vicinity of the two stable steady states of the deterministic approximation. However, if the number of kinases diverges to the infinity (zero noise limit), the stationary probability distribution concentrates (generically) in only one of the two steady states. Afiliacje autorów:

Abstrakty konferencyjne
1.  Lipniacki T., Innate immune responses at single cell level, III National Conference of Nano and Micromechanics, 20120704/0706, Warszawa (PL), pp.710, 2012 