Topic 1: 

Enzyme-catalysed reactions are critical in biochemistry, where the reaction rates can be accelerated by well over a millionfold compared with non-catalysed reactions. These reactions are typified by an enzyme E binding with its substrate S to form an enzyme-substrate complex C, which can be transformed into a product P and the original enzyme. In the ordinary differential equation (ODE for short) setting, Michaelis and Menten [Biochem. z, 1913] proposed an approximation under the quasi-steady-state (shortly, QSS) assumption that the ratio between the initial concentrations of enzyme and substrate is very small. The QSS approximate system includes three algebraic equations and only one ODE, where the species kinetics, called Michaelis-Menten (shortly, MM) kinetics, is fractional and bounded. MM kinetics has become one of the most used in catalytic reactions in the literature. In the PDE setting, rigorous investigation of MM kinetics and, more general, biomedical enzyme kinetics is still challenging, although the formal derivation of MM kinetics in the PDE setting has been studied in [Kalachev-Kaper-Kaper-Popovic-Zagaris, EJDE, 2007].

• Rigorous derivation of MM kinetics: In work [Tang-Tran], we rigorously derive MM kinetics in the slow diffusion regime, where the system with MM kinetics also includes a cross-diffusion equation. Although the result required the closeness condition of diffusion rates in high dimensions, it seems to be the first rigorous derivation of MM kinetics in the PDE setting.

• When Michaelis-Menten kinetics seems to no longer be applicable: In the regimes of non-slow-diffusion, the structure of the QSS approximate system changes dramatically compared with the one proposed by Michaelis and Menten. In other words, the Michaelis–Menten kinetics seem to no longer be applicable in these regimes. I am collaborating with J. Morgan (University of Houston), Q.B. Tang (University of Graz) to propose suitable kinetics in the non-slow-diffusion regimes and their rigorous proofs.

• Widely applied mechanisms in biomedicine: Linking to applications of catalyst enzymes in biomedicine, the enzyme mechanism (mentioned earlier) is fundamental and rarely happens in an isolated system but rather embedded in a more complex system. Therefore, the models for certain enzyme-inhibiting drugs are still open at the formal level and challenging in the PDE setting. Widely applied mechanisms of enzyme kinetics are certainly my research objectives.

Related papers:

• Bao Quoc Tang and Bao-Ngoc Tran. Rigorous derivation of Michaelis-Menten kinetics in the presence of diffusion, to appear in SIAM Journal of Mathematical Analysis, 2024.

Topic 2:

Fast reaction limits (FRLs for short) in reaction-diffusion systems have gained much attention for the last two decades. Under biological situations, the nonlinearities incorporate fast reactions related to some species and may include different scale reaction kinetics. Simplification or reduction of such fast reaction systems by considering the limit, as the relaxation time tends to 0, is widely applied in many fields, where the limit is called an FRL. Studies of FSLs come back to work [L. Evans, Houston J. Math, 1980], in which solutions to a singular system converge to solutions of a system with nonlinear diffusion. Many results reveal that FRLs arise interesting types of limiting systems with different structures, such as nonlinear diffusion equations, Stefan free boundary problems, cross-diffusion systems, motion of interfaces, systems involving Young measures, etc. However, the study of FRLs is still not well understood, and moreover, the convergence rate analysis for this kind of limit has only been studied in several works.

• Analysis for fast reaction limits and convergence rates: In my result [Morgan-Soresina-Tang-Tran], we study a plant growth model involving toxicity with fast switching between two sub-species of biomass. In this study, a triangular cross-diffusion system is derived from a fast reaction-diffusion system by investigating the FRL arising from the fast switching. A study of the uniqueness and regularity of the triangular cross-diffusion system, where the Moser-Alikakos iteration technique has been exploited, is the main ingredient for our result of the convergence rate.

• Geometrical theory of dynamical systems: The idea of combining abstract theories of dynamical systems in Banach spaces and fast reaction limits in PDEs was discussed by C. Kuehn (TU. Munich) and Q.B. Tang (University of Graz) in 2021, and turned into an important part in their joint projects (that I work for). Since the singular term in a fast reaction limit problem forms by an algebraic equation of the species densities/concentrations, the critical manifold, where the system dynamics take place, can be obtained via the implicit function theorem in Banach spaces. In [Desvillettes-Kuehn-Sulzbach-Tang-Tran], we construct an approximation for the critical manifold, called the slow manifold, which is constituted in a generalised Fenichel theory. Our crucial point is analysing the geometrical properties of these manifolds.

Fast reaction-diffusion systems with non-linear diffusion and higher-order interactions and methods for the convergence rate of FRLs (particularly, convergence rate estimates in L\infty-in-time spaces) are developing into a broader class of fast reaction-diffusion systems. Moreover, since the geometrical theory of dynamical systems has just been studied in one dimension, we continue to investigate in higher dimensions.

Related papers:

• Jeff Morgan, Cinzia Soresina, Tang Quoc Bao, and Bao-Ngoc Tran. Plant growth involves toxicity, diffusion, and fast switching, in preparation.

• Laurent Desvillettes, Christian Kuehn, Jan-Eric Sulzbach, Tang~Quoc Bao, and Bao-Ngoc Tran. Slow manifolds for PDEs and fast reaction limit, in preparation.

Topic 3:

Chemotaxis systems, in which the movement of species is biased along spatial gradients of chemicals secreted by other species, have fascinated mathematicians for decades. However, in many cases, studies of these systems are still challenging because of the unavailability of strong regularity information, or more clearly, facing low regularity [Lankeit-Winkler, Jahresbericht der Deutschen Mathematiker-Vereinigung, 2020].

• Global existence and regularity: As mentioned above, studying the global existence and regularity of solutions to chemotaxis systems remains difficult, especially those including nonlinear diffusion and cross-diffusion. In this topic, one idea from work [Morgan-Soresina-Tang-Tran] is mainly extended to obtain an unconditional result [Reisch-Tran-Yang] on the global existence of a unique classical solution to a competitive chemotaxis system, including two prey and one predator species. 

Currently, I am very interested in chemotaxis systems with nonlinear diffusion and cross-diffusion, such as the doubly-degenerate nutrient taxis systems.

• Fast signal diffusion limits and convergence rates: Recently, the biological situation in chemotaxis systems that chemicals diffusive much faster than self-diffusion of all species has been studied rigorously for the last several years, for example, [Wang-Winker-Xiang, Calc. Var. 2019], [Ogawa-Suguro, Math. Ann., 2023], where a parabolic-parabolic chemotaxis system with slow chemical evolution can be reduced to the respective parabolic-elliptic system. The limit as this evolution disappears is called the fast signal diffusion limit. 

In [Reisch-Tran-Yang], the analysis for this kind of singular limit and L\infty-in-time convergence rates have been studied carefully. My second result is presented in [Bui-Huynh-Tang-Tran], we recover the classical Keller-Segel system from a chemotaxis system with indirect signalling by considering fast signal diffusion limits generated from multiple regimes of parameters.

Related papers:

• Cordula Reisch, Bao-Ngoc Tran, and Juan Yang. Global existence, fast signal diffusion limit, and L\infty-in-time convergence rates in a competitive chemotaxis system. arXiv:2405.17392, 48 pages, 2024.

• Le Trong Thanh Bui, Thi Kim Loan Huynh, Quoc Bao Tang, and Bao-Ngoc Tran. Fast signal diffusion limits generated from biologically relevant parameter regimes in chemotaxis systems with indirect signalling, in preparation.

And more...

I am interested in deriving reaction-diffusion equations from kinetic equations of Boltzmann's type, which was studied in [Bisi-Desvillettes, JSP, 2006]. Currently, I am collaborating with Andrea Bondesan (University of Parma) to recover the Michealis-Menten kinetics from rescaled Boltzmann equations. 

On the other hand, I work with Bui Le Trong Thanh (Vietnam National University) and Nguyen Nhut Hung (Nong Lam University) on the analysis of PDEs for multi-species population systems with fractional cross-diffusion, which has been studied recently [Jüngel-Zamponi, JDE, 2022] to describe large-distance interactions. 

Inverse initial-value problems intrigued me during my PhD studies. Therefore, I am currently collaborating with Nguyen Huy Tuan  (Ho Chi Minh University of Banking), in [Nguyen-Tran], to establish the existence and regularity of solutions for this type of problem in fractional Sobolev spaces.

Related papers:

• Huy Tuan Nguyen and Bao-Ngoc Tran. Inverse initial-value problems for time-fractional diffusion equations in fractional Sobolev spaces, in minor revision, 2023.

Full Publications  

Please see MathSciNet

https://mathscinet.ams.org/mathscinet/publications-search?query=auid%3A1277587

or in my Curriculum Vitae

https://drive.google.com/file/d/1ehOF9LzWN8EQtez7GQKZzgG1kHgPPE9g/view?usp=sharing