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1
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Dynamics of the tumor-immune system competition - the effect of time delay

100%
Galach M.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 3
395-406
EN
The model analyzed in this paper is based on the model set forth by V.A. Kuznetsov and M.A. Taylor, which describes a competition between the tumor and immune cells. Kuznetsov and Taylor assumed that tumor-immune interactions can be described by a Michaelis-Menten function. In the present paper a simplified version of the Kuznetsov-Taylor model (where immune reactions are described by a bilinear term) is studied. On the other hand, the effect of time delay is taken into account in order to achieve a better compatibility with reality.
2
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Mathematical modelling of molecule evolution in protocells

100%
Myszor D., Cyran K.
International Journal of Applied Mathematics and Computer Science
|
2013
|
tom 23
|
nr 1
213-229
EN
In this article, we analyse the process of the emergence of RNA polynucleotides located in an enclosed environment, at an early stage of the RNA world. Therefore we prepared a mathematical model, composed of a set of differential equations, which simulates the behaviour of an early biological system bounded by a protocell membrane. There is evidence that enclosed environments were available on the primordial Earth. There are also experimental proofs that RNA strands can develop in these formations. The proposed model allows analysis of the influence of membrane permeability on the composition of internal material. It takes into account phenomena that lead to the elongation of an RNA strand (ligation), fission of molecules (phosphodiester bond breakage) and replication of polynucleotides. Results obtained from the model point out that the existence of protocells might support concentration of material and creation of longer molecules.
3
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Event-Based Proof of the Mutual Exclusion Property of Peterson’s Algorithm

88%
Ivanov I., Nikitchenko M., Abraham U.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
325-331
EN
Proving properties of distributed algorithms is still a highly challenging problem and various approaches that have been proposed to tackle it [1] can be roughly divided into state-based and event-based proofs. Informally speaking, state-based approaches define the behavior of a distributed algorithm as a set of sequences of memory states during its executions, while event-based approaches treat the behaviors by means of events which are produced by the executions of an algorithm. Of course, combined approaches are also possible. Analysis of the literature [1], [7], [12], [9], [13], [14], [15] shows that state-based approaches are more widely used than event-based approaches for proving properties of algorithms, and the difficulties in the event-based approach are often emphasized. We believe, however, that there is a certain naturalness and intuitive content in event-based proofs of correctness of distributed algorithms that makes this approach worthwhile. Besides, state-based proofs of correctness of distributed algorithms are usually applicable only to discrete-time models of distributed systems and cannot be easily adapted to the continuous time case which is important in the domain of cyber-physical systems. On the other hand, event-based proofs can be readily applied to continuous-time / hybrid models of distributed systems. In the paper [2] we presented a compositional approach to reasoning about behavior of distributed systems in terms of events. Compositionality here means (informally) that semantics and properties of a program is determined by semantics of processes and process communication mechanisms. We demonstrated the proposed approach on a proof of the mutual exclusion property of the Peterson’s algorithm [11]. We have also demonstrated an application of this approach for proving the mutual exclusion property in the setting of continuous-time models of cyber-physical systems in [8]. Using Mizar [3], in this paper we give a formal proof of the mutual exclusion property of the Peterson’s algorithm in Mizar on the basis of the event-based approach proposed in [2]. Firstly, we define an event-based model of a shared-memory distributed system as a multi-sorted algebraic structure in which sorts are events, processes, locations (i.e. addresses in the shared memory), traces (of the system). The operations of this structure include a binary precedence relation ⩽ on the set of events which turns it into a linear preorder (events are considered simultaneous, if e1 ⩽ e2 and e2 ⩽ e1), special predicates which check if an event occurs in a given process or trace, predicates which check if an event causes the system to read from or write to a given memory location, and a special partial function “val of” on events which gives the value associated with a memory read or write event (i.e. a value which is written or is read in this event) [2]. Then we define several natural consistency requirements (axioms) for this structure which must hold in every distributed system, e.g. each event occurs in some process, etc. (details are given in [2]). After this we formulate and prove the main theorem about the mutual exclusion property of the Peterson’s algorithm in an arbitrary consistent algebraic structure of events. Informally, the main theorem states that if a system consists of two processes, and in some trace there occur two events e1 and e2 in different processes and each of these events is preceded by a series of three special events (in the same process) guaranteed by execution of the Peterson’s algorithm (setting the flag of the current process, writing the identifier of the opposite process to the “turn” shared variable, and reading zero from the flag of the opposite process or reading the identifier of the current process from the “turn” variable), and moreover, if neither process writes to the flag of the opposite process or writes its own identifier to the “turn” variable, then either the events e1 and e2 coincide, or they are not simultaneous (mutual exclusion property).
4
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Single stranded-DNA detection: the role of Wip1 in ATR-dependent pathway

75%
Kurpas M., Jonak K., Puszynski K.
Mathematica Applicanda
|
2018
|
tom 46
|
nr 1
PL
Obszary pojedynczonciowego DNA (ssDNA) powstaja w komórkachw wyniku ekspozycji na stres na przykład- promieniowanie UVC-lub podczas naprawy podwójnoniciowych pekniec DNA. ATR (ataxia telangiectasia mutated and Rad3-related) jest odpowiedzialne za wykrywanie ssDNA. Niedawno wykazano kluczowa role fosfatazy Wip1, która inaktywuje główne elementy szlaków odpowiedzi na uszkodzenia DNA. Opracowalismy matematyczny model sciezki ATR i połaczylismygo z modelem szlaku supresora nowotworowego p53, odpowiadajacego za aktywacje genów zaangazowanych w reakcje komórki na uszkodzenie materiału genetycznego (naprawe DNA/apoptoze). Co wiecej, dodalismy fosfataze Wip1, jako główny czynnik odpowiedzialny za wyłaczenie sciezek sygnałowych uruchamianych w ramach reakcji na uszkodzenie. Uzyskane wyniki pokazuja, ze dzieki prawidłowo dobranej dawce UVC i wyciszeniu lub zablokowaniu aktywnosci Wip1, mozliwe jest skierowanie komórek nowotworowych na szlak apoptotyczny
EN
Single-stranded DNA (ssDNA) areas arise in cells as a result of exposure to stress agents - like UVC - or during repair of DNA double-strand breaks. ATR(ataxia telangiectasia mutated and Rad3-related) is responsible for detecting ssDNA. Recently, it has been shown that one of the most important components of cellular response to the damage is Wip1 phosphatase, which inactivates main elements of DNA damage response (DDR) pathways. We developed a mathematical model of ATR detector system, connected to p53 tumor suppressor responsible for activation of genes involved in cellular response to the damage (DNA repair/apoptosis). Moreover, we added Wip1 phosphatase, as a main agent responsible for turning o DDR. Our results show that the apoptotic threshold, where more than half of cells die, is equal to 20 J/m2. The threshold shifts when activity of the specied proteins involved in the pathway is blocked or reduced. Moreover with accurate dose of UVC and silenced or blocked Wip1, it may be possible to drive cancer cells to apoptotic pathway.
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