Effect of Buffers with Multiple Binding Sites on Calcium Waves.

The existence and properties of intracellular waves of increased free cytoplasmic calcium concentration (calcium waves) are strongly affected by the binding and unbinding of calcium ions to a multitude of different buffers in the cell. These buffers can be mobile or immobile and, in general, have multiple binding sites that are not independent. Previous theoretical studies have focused on the case when each buffer molecule binds a single calcium ion. In this study, we analyze how calcium waves are affected by calcium buffers with two non-independent binding sites, and show that the interactions between the calcium binding sites can result in the emergence of new behaviors. In particular, for certain combinations of kinetic parameters, the profiles of buffer molecules with one calcium ion bound can be non-monotone.

Interplay between ceRNA and Epigenetic Control of microRNA: Modelling Approaches with Application to the Role of Estrogen in Ovarian Cancer.

MicroRNAs (miRNAs) play an important role in gene regulation by degradation or translational inhibition of the targeted mRNAs. It has been experimentally shown that the way miRNAs interact with their targets can be used to explain the indirect interactions among their targets, i.e., competing endogenous RNA (ceRNA). However, whether the protein translated from the targeted mRNAs can play any role in this ceRNA network has not been explored. Here we propose a deterministic model to demonstrate that in a network of one miRNA interacting with multiple-targeted mRNAs, the competition between miRNA-targeted mRNAs is not sufficient for the significant change of those targeted mRNA levels, while dramatic changes of these miRNA-targeted mRNAs require transcriptional inhibition of miRNA by its target proteins. When applied to estrogen receptor signaling pathways, the miR-193a targets E2F6 (a target of estrogen receptor), c-KIT (a marker for cancer stemness), and PBX1 (a transcriptional activator for immunosuppressive cytokine, IL-10) in ovarian cancer, such that epigenetic silencing of miR-193a by E2F6 protein is required for the significant change of c-KIT and PBX1 mRNA level for cancer stemness and immunoevasion, respectively, in ovarian cancer carcinogenesis.

Flux-augmented bifurcation analysis in chemical reaction network systems.

The dynamics of biochemical reaction networks are considered to be responsible for biological functions in living systems. Since real networks are immense and complicated, it is difficult to determine which reactions can cause a significant change of dynamical behaviors, namely, bifurcations. Also to what extent numerical results of network systems depend on the chosen kinetic rate parameters is not known. In this paper, an analytical setting that splits the information of the dynamics into the network structure and reaction kinetics is introduced. This setting possesses a factorization structure for some class of network systems which allows one to determine which subnetworks are responsible for the occurrence of a bifurcation. Subsequently, the bifurcation criteria are reformulated in a manner that allows the efficient determination of relevant reactions for bifurcations.

Traveling waves for a three-component reaction-diffusion model of farmers and hunter-gatherers in the Neolithic transition.

The Neolithic transition began the spread of early agriculture throughout Europe through interactions between farmers and hunter-gatherers about 10,000 years ago. Archeological evidences indicate that the expanding velocity of farming into a region occupied by hunter-gatherers is roughly constant all over Europe. In the late twentieth century, from the contribution of the radiocarbon dating, it could be found that there are two types of farmers: one is the original farmer and the other is the converted farmer which is genetically hunter-gatherers but learned agriculture from neighbouring farmers. Then this raises the following questions: Which farming populations play a key role in the expansion of farmer populations in Europe? and what is the fate of hunter-gatherers (e.g., become extinct, or live in lower density, or live in agricultural life-style)? We consider a three-component reaction-diffusion system proposed by Aoki, Shida and Shigesada, which describes the interactions among the original farmers, the converted farmers, and the hunter-gatherers. In order to resolve these two questions, we discuss traveling wave solutions which give the information of the expanding velocity of farmer populations. The main result is that two types of traveling wave solutions exist, depending on the growth rate of the original farmer population and the conversion rate of the hunter-gatherer population to the converted farmer population. The profiles of traveling wave solutions indicate that the expansion of farmer populations is determined by the growth rate of the original farmer and the (maximal) carrying capacity of the converted farmer, and the fate of hunter-gatherers is determined by the growth rate of the hunter-gatherer and the conversion rate of the hunter-gatherer to the converted farmer. Thus, our results provide a partial answer to the above two questions.

Propagation direction of traveling waves for a class of bistable epidemic models.

Traveling waves of a reaction-diffusion (RD) system connecting two spatially uniform stable equilibria are termed as bistable waves. Due to the uniqueness of a bistable wave in RD systems, it is difficult to determine its propagation direction, and there are very few analytical results on this subject. In this study, we propose an approach to give a complete characterization of the propagation direction of bistable waves for a class of bistable epidemic models arising from the spread of a cholera epidemic. Moreover, this characterization also gives a parameter threshold above which the epidemic disease eventually tends to extinction, and below which the epidemic outbreak happens.

E2F6 functions as a competing endogenous RNA, and transcriptional repressor, to promote ovarian cancer stemness.

Ovarian cancer is the most lethal cancer of the female reproductive system. In that regard, several epidemiological studies suggest that long-term exposure to estrogen could increase ovarian cancer risk, although its precise role remains controversial. To decipher a mechanism for this, we previously generated a mathematical model of how estrogen-mediated upregulation of the transcription factor, E2F6, upregulates the ovarian cancer stem/initiating cell marker, c-Kit, by epigenetic silencing the tumor suppressor miR-193a, and a competing endogenous (ceRNA) mechanism. In this study, we tested that previous mathematical model, showing that estrogen treatment of immortalized ovarian surface epithelial cells upregulated both E2F6 and c-KIT, but downregulated miR-193a. Luciferase assays further confirmed that microRNA-193a targets both E2F6 and c-Kit. Interestingly, ChIP-PCR and bisulphite pyrosequencing showed that E2F6 also epigenetically suppresses miR-193a, through recruitment of EZH2, and by a complex ceRNA mechanism in ovarian cancer cell lines. Importantly, cell line and animal experiments both confirmed that E2F6 promotes ovarian cancer stemness, whereas E2F6 or EZH2 depletion derepressed miR-193a, which opposes cancer stemness, by alleviating DNA methylation and repressive chromatin. Finally, 118 ovarian cancer patients with miR-193a promoter hypermethylation had poorer survival than those without hypermethylation. These results suggest that an estrogen-mediated E2F6 ceRNA network epigenetically and competitively inhibits microRNA-193a activity, promoting ovarian cancer stemness and tumorigenesis.

Structural bifurcation analysis in chemical reaction networks.

In living cells, chemical reactions form complex networks. Dynamics arising from such networks are the origins of biological functions. We propose a mathematical method to analyze bifurcation behaviors of network systems using their structures alone. Specifically, a whole network is decomposed into subnetworks, and for each of them the bifurcation condition can be studied independently. Further, parameters inducing bifurcations and chemicals exhibiting bifurcations can be determined on the network. We illustrate our theory using hypothetical and real networks.

A mathematical model of bimodal epigenetic control of miR-193a in ovarian cancer stem cells.

Accumulating data indicate that cancer stem cells contribute to tumor chemoresistance and their persistence alters clinical outcome. Our previous study has shown that ovarian cancer may be initiated by ovarian cancer initiating cells (OCIC) characterized by surface antigen CD44 and c-KIT (CD117). It has been experimentally demonstrated that a microRNA, namely miR-193a, targets c-KIT mRNA for degradation and could play a crucial role in ovarian cancer development. How miR-193a is regulated is poorly understood and the emerging picture is complex. To unravel this complexity, we propose a mathematical model to explore how estrogen-mediated up-regulation of another target of miR-193a, namely E2F6, can attenuate the function of miR-193a in two ways, one through a competition of E2F6 and c-KIT transcripts for miR-193a, and second by binding of E2F6 protein, in association with a polycomb complex, to the promoter of miR-193a to down-regulate its transcription. Our model predicts that this bimodal control increases the expression of c-KIT and that the second mode of epigenetic regulation is required to generate a switching behavior in c-KIT and E2F6 expressions. Additional analysis of the TCGA ovarian cancer dataset demonstrates that ovarian cancer patients with low expression of EZH2, a polycomb-group family protein, show positive correlation between E2F6 and c-KIT. We conjecture that a simultaneous EZH2 inhibition and anti-estrogen therapy can constitute an effective combined therapeutic strategy against ovarian cancer.

Do calcium buffers always slow down the propagation of calcium waves?

Calcium buffers are large proteins that act as binding sites for free cytosolic calcium. Since a large fraction of cytosolic calcium is bound to calcium buffers, calcium waves are widely observed under the condition that free cytosolic calcium is heavily buffered. In addition, all physiological buffered excitable systems contain multiple buffers with different affinities. It is thus important to understand the properties of waves in excitable systems with the inclusion of buffers. There is an ongoing controversy about whether or not the addition of calcium buffers into the system always slows down the propagation of calcium waves. To solve this controversy, we incorporate the buffering effect into the generic excitable system, the FitzHugh-Nagumo model, to get the buffered FitzHugh-Nagumo model, and then to study the effect of the added buffer with large diffusivity on traveling waves of such a model in one spatial dimension. We can find a critical dissociation constant (K = K(a)) characterized by system excitability parameter a such that calcium buffers can be classified into two types: weak buffers (K ∈ (K(a), ∞)) and strong buffers (K ∈ (0, K(a))). We analytically show that the addition of weak buffers or strong buffers but with its total concentration b(0)(1) below some critical total concentration b(0,c)(1) into the system can generate a traveling wave of the resulting system which propagates faster than that of the origin system, provided that the diffusivity D1 of the added buffers is sufficiently large. Further, the magnitude of the wave speed of traveling waves of the resulting system is proportional to √D1 as D1 --> ∞. In contrast, the addition of strong buffers with the total concentration b(0)(1) > b(0,c)(1) into the system may not be able to support the formation of a biologically acceptable wave provided that the diffusivity D1 of the added buffers is sufficiently large.

Traveling waves in the discrete fast buffered bistable system.

We study the existence and uniqueness of traveling wave solutions of the discrete buffered bistable equation. Buffered excitable systems are used to model, among other things, the propagation of waves of increased calcium concentration, and discrete models are often used to describe the propagation of such waves across multiple cells. We derive necessary conditions for the existence of waves, and, under some restrictive technical assumptions, we derive sufficient conditions. When the wave exists it is unique and stable.

Are buffers boring? Uniqueness and asymptotical stability of traveling wave fronts in the buffered bistable system.

Traveling waves of calcium are widely observed under the condition that the free cytosolic calcium is buffered. Thus it is of physiological interest to determine how buffers affect the properties of calcium waves. Here we summarise and extend previous results on the existence, uniqueness and stability of traveling wave solutions of the buffered bistable equation, which is the simplest possible model of the upstroke of a calcium wave. Taken together, the results show that immobile buffers do not change the existence, uniqueness or stability of the traveling wave, while mobile buffers can eliminate a traveling wave. However, if a wave exists in the latter case, it remains unique and stable.

The asymptotic behavior of solutions of the buffered bistable system.

In this paper, we study a model for calcium buffering with bistable nonlinearity. We present some results on the stability of equilibrium states and show that there exists a threshold phenomenon in our model. In comparing with the model without buffers, we see that stationary buffers cannot destroy the asymptotic stability of the associated equilibrium states and the threshold phenomenon. Moreover, we also investigate the propagation property of solutions with initial data being a disturbance of one of the stable states which is confined to a half-line. We show that the more stable state will eventually dominate the whole dynamics and that the speed of this propagation (or invading process) is positive.