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Melnik-Research-Group

Life at each moment permeates
the entire realm of phenomena
and is revealed in all phenomena.
(Nichiren,  1222–1282)

Life at the nanoscale; mathematics in biology and medicine

Biology and medicine provide many challenging problems that can be addressed efficiently only with adequately developed mathematical tools. As these disciplines are becoming increasingly quantitative, there is a need in the development of mathematical models for a vast variety of processes and phenomena, most of which have (spatial and/or temporal) multiscale characteristics. In a number of cases, our interests in this field lie with going down all the way to the building block of life, where we analyze RNA structures at the nanoscale. We develop mathematical models for such building blocks as well as for other biomolecules. Recall that the size of folded RNA structures, for example, is often measured by using the radius of gyration as being on the order of C N 1/3 Å, according to the Flory scaling law, with N being the number of nucleotides and certain constant C. RNA molecules that encode a specific protein product are known as messengers of RNA or mRNA. Although mRNA vary substantially in their size, it is fair to say that most proteins contain at least a hundred amino acid residues and that mRNA must have at least 300 nucleotides on the basis of the triplet code (100 x 3). This gives us certain ideas on the relative size of these molecules. The specific structures we are analyzing are known as RNA nanostructures. These building blocks in their symmetric versions have spatial dimensions lying in the range of 10 - 20 nm. In our research in this focus area we analyze objects from the nanoscale to meso- and to macro- scales. In addition to RNA nanostructures, we are also interested in a number of other areas of mathematical modeling in biology and medicine, including the dynamics of cell cycles, relaxation processes of DNA molecules, biomedical devices with smart materials, biological imaging, drug delivery techniques.

In this focus area, our research interests have included the following topics:

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Mathematical models for drug delivery and other biomedical applications

We are interested in the development of mathematical models for drug delivery, both fluid-based (or aerosole-based) and solid-state-based devices and systems. We analyzed several systems for drug delivery, including the pulmonary-drug-delivery-mode systems based on air-blast atomization which are very important for successful pharmacotherapy. In the heart of such medical systems operation lies the Venturi effect where we observe the reduction in fluid pressure as a result of a fluid flows through a geometrically confined section of a Venturi tube. A conservation-law-based mathematical model was developed and key control parameters, that can allow improvement in the efficiency of the system, were identified [ EJ-47 ]. Main characteristics of the system were analyzed numerically as functions or these parameters.

Other biomedical and biomechanical applications of interest include the development of mathematical models and efficient numerical methods for their solutions for the usage of smart materials. Materials such as shape memory alloys find a wide range of applications in orthodontics and dental biomechanics , optometry, orthopedic surgery, biologically-friendly replacements in body cavities, arteries and other vascular applications, implants, and in various surgery instruments. While there exists a vast amount of literature on constitutive and time-independent modelling of SMAs and SMA-based systems, our main focus over the years has been of the dynamics. This is quite important feature in biological and medical applications of SMAs. Two key technologies are related to this: smart materials and structures technology and nondestructive evaluation technology. The latter includes the development of non-invasive (or less invasive) medical diagnostics. We first proposed a reduction of the full 3D PDE-based models for materials with shape memory to a more tractable differential-algebraic systems [ EJ-33 ] (further discussed in [EJ-93 ] for both uncoupled and coupled problems). This allowed to treat both stress-induced and temperature-induced phase transformations in these materials and related hysteresis phenomena in a unified manner [ EJ-44 ], with further generalization to include hyperbolicity of heat conduction to follow [ EJ-49 ]. Conservative numerical approximations in this field of dynamic problems were developed and justified mathematically in [ EJ-53 , EJ-72 ]. Chebyshev's collocation and Chebyshev's spectral procedures were developed for these problems in [ EJ-100 , EJ-107 ] (also proceedings [ Cruz-2009 ]) which also included the Rayleigh dissipation term [ EJ-111 ]. A new hybrid optimization procedure was developed and applied in [ EJ-118 ]. We developed a Finite Element Method (FEM) based approach that incorporates the lattice kinetics, involving the order variables, and non-equilibrium thermodynamics [ EJ-114 ]. Its variants, including those applied to 3D problems, were discussed in detail in [ EJ-82 , EJ-92 , EJ-104 ] and our new Finite Volume Method procedure for these dynamic problems was reported in [ EJ-101 ]. We have also developed several efficient procedures for model reductions in this field based on the manifold reduction, the Proper Orthogonal Decomposition (POD), and the Galerkin projection techniques [ EJ-33 , EJ-55 , EJ-106 , EJ-141 ]. More details on our research into the dynamics of SMAs and SMA-based devices can be found here.

While pulmonary and injection modes of drug delivery have remained their leading positions over the years, it has been recognized that new biological drugs, in particular those based on proteins and nucleic acids that promise to minimize side effects, require new delivery technologies. Indeed, the success of widely discussed RNA and DNA therapies is essentially dependent on such technologies. A key technology to have a revolutionary impact on this field is nanotechnology, a technology that leads a way to the development of efficient nansocale drug delivery techniques. Nanotechnology-based drug carriers have enormous potential in both, the delivery of small molecules and the delivery of such large biomolecules as proteins and nucleic acids to specific targeted areas within the body. Our focus in this area of nanomedicine is on RNA nanostructures as structures that have substantial potential in the development of these new techniques as well as in other areas of biomedical sciences. Nevertheless, very little is known about these structures and our efforts of our group have been directed towards a better understanding of of RNA nanostructure properties [ EJ-137 ]. Several new phenomena have been discovered, including the phenomenon of self-stabilisation of such RNA nanostructures. A hierarchy of new models for systematic studies of such structures have been developed in [ EJ-145 ].

Other nanostructure-mediated drug delivery methods include quantum dots/shells and carbone nanotubes. You can find more information on various properties of these structures by scrolling down this page and in other sections of this website (in particular, Research Focus Section γ and Research Focus Section ζ).

 

RNA nanostructures; multiple scales in life science and medicine; DNA modeling and dynamics of biomolecules; RNA interference/silencing; cell dynamics

One of the fundamental components of cells are deoxyribonucleic and ribonucleic nucleic acids (DNA and RNA, respectively). These essential building and functional blocks of life, along with three other main components, proteins, glycans and lipids, consist of molecules which in their turn consist of certain elements. Most of life forms on this planet are made of four basic elements (made of only one type of atom): carbon, hydrogen, nitrogen, and oxygen. DNA, for example, has four nitrogen-containing bases adenine (A), cytosine (C), guanine (G), and thymine (T). DNA molecules form genes found on chromosomes, and genetic frequencies in a population gives us a clue to evolution. It is believed that DNA is transcribed into a messenger RNA (mRNA) which is translated into amino acids. While DNA is a double-stranded biomolecule, RNA is single-stranded such that its messenger (mRNA) carries all instructions from the nucleus to all places where they are needed throughout the cell.

Our interests here include a range of multiscale problems such as cell dynamics, in particular under the influence of unexpected events or fluctuations [ EJ -130 ], as well as the dynamics of DNA biomolecules where we carried out the analysis of their relaxation under different conditions [ EJ-105 , EJ-109 ]. Several models were constructed for this purpose where an important component was the Smoluchowski equation. In terms of RNA, we are interested in RNA silencing phenomena and technologies associated with them [ EP-67 ], as well as in studying RNA nanostructures where we succeeded to make important steps towards a better understanding of these structures' properties [ EJ-137 , EJ-145 ]. Other problems of interest include mathematical models for RNA-based systems such as nucleic acid sensors and their components (e.g., riboswitches). These problems are part of our larger program in studying complex multiscale dynamics of biological systems and molecular transport via a better understanding of systems behaviour at smaller scales.

 

Imaging in biology and medicine, the use of quantum dots, carbon nanotubes, bionanotechnology

Quantum dots are at the forefront of many biotechnological innovations. Details of our research into quantum dots properties, new phenomena associated with such properties, and the development of new models for their studies can be found here. When quantum dots are used as biological tags and for bio-imaging, their optical properties are very important as they determine their sensitivity for the analysis of biological objects. Our interest in the applications of quantum dots and other low dimensional nanostructures in bionanotechnology is linked to the development of adequate mathematical modeling support for such applications that would allow a better prediction of these and other properties. As the role of quantum dots in encoding, microscopy in vivo applications, bioimaging, immunofluorescent labeling, and other biological applications is growing, the importance of such support will continue to increase.

Our interests in this field include also bioconjugated quantum dots and nanosensors based on low dimensional nanostructures. The latter include such examples as quantum dots based nanosensors for DNA detections and plasmonic nanorods/nanotubes for bio-sensing. By now, we studied properties of ZnO nanowires [ EJ-125 , EJ-139 ] which are important as components of promising nanosensors for biodetection and identification. We have also been looking at properties of Carbon Nanotubes (CNTs) based sensors with potential applications in biological sciences, as well as CNTs as an alternative to medical X-ray imaging where we were focusing on field emission properties of these low dimensional nanostructures. We reported the efficiency of the proposed models and techniques based on theoretical, computational, and experimental results [ EJ-112 ,EJ-116 , EJ-122 , EJ-131 ].

 

Structurally inhomogeneous systems, biomechanics, biological networks

Most biological systems and networks are characterized by structural inhomogeneity and the development of mathematical models and mathematical and computational tools for the analysis of such systems represents a challenge. In this context we are interested in the analysis of bio-polymers, composites, biologically inspired materials with memory, genetic and other biological networks, as well as proteins, DNA, and RNA nanostructures.

We analyzed the phenomenon of thermal spiking which is important in many areas of applications in medicine, in particular in polymerized materials for artificial joints, orthodontics and dental practices. We developed a new model for this phenomenon, as well as an efficient method for its numerical solution, and presented results on thermal spiking on an example of bone cement [ EJ-45 , EJ-59 ]. We showed that mathematically such problems are reducible to a coupled system of ODEs and PDE. Other authors used such mixed coupled models for other biological and medical applications, e.g. in vascular remodelling in biological tissues. The techniques we developed may proved to be quite useful in a number of other areas too. Our other works on polymers and composite materials included the development of efficient algorithms for initial topological optimization of complex polymeric and composite systems [ EJ-58 ], hyperbolic time relaxation models for polymeric materials [ EJ-78 ], relaxation of DNA and the influence of internal viscosity in associated problems [ EJ-105 , EJ-109 , EJ-126 ].

Many systems that we have to deal with in this focus areas are characterized by complex dynamic behaviour as well as by coupled multiscale spatial interactions. An important example is provided by nonlinear thermoelasticity which has an increasing range of applications in life sciences in general and in biomechanics in particular. A number of our contributions were devoted to the development of mathematical models and efficient numerical methods for dynamic problems in bio-inspired materials such as shape memory alloys (SMAs). These materials find a wide range of applications in orthodontics and dental biomechanics , optometry, orthopedic surgery, biologically-friendly replacements in body cavities, arteries and other vascular applications, implants, and in various surgery instruments. While there exists a vast amount of literature on constitutive and time-independent modelling of SMAs and SMA-based systems, our main focus over the years has been of the dynamics. This is quite important feature in biological and medical applications of SMAs. Two key technologies are related to this: smart materials and structures technology and nondestructive evaluation technology. The latter includes the development of non-invasive (or less invasive) medical diagnostics. We first proposed a reduction of the full 3D PDE-based models for materials with shape memory to a more tractable differential-algebraic systems [ EJ-33 ] (further discussed in [EJ-93 ] for both uncoupled and coupled problems). This allowed to treat both stress-induced and temperature-induced phase transformations in these materials and related hysteresis phenomena in a unified manner [ EJ-44 ], with further generalization to include hyperbolicity of heat conduction to follow [ EJ-49 ]. Conservative numerical approximations in this field of dynamic problems were developed and justified mathematically in [ EJ-53 , EJ-72 ]. Chebyshev's collocation and Chebyshev's spectral procedures were developed for these problems in [ EJ-100 , EJ-107 ] (also proceedings [ Cruz-2009 ]) which also included the Rayleigh dissipation term [ EJ-111 ]. A new hybrid optimization procedure was developed and applied in [ EJ-118 ]. We developed a Finite Element Method (FEM) based approach that incorporates the lattice kinetics, involving the order variables, and non-equilibrium thermodynamics [ EJ-114 ]. Its variants, including those applied to 3D problems, were discussed in detail in [ EJ-82 , EJ-92 , EJ-104 ] and our new Finite Volume Method procedure for these dynamic problems was reported in [ EJ-101 ]. We have also developed several efficient procedures for model reductions in this field based on the manifold reduction, the Proper Orthogonal Decomposition (POD), and the Galerkin projection techniques [ EJ-33 , EJ-55 , EJ-106 , EJ-141 ]. More details on our research into the dynamics of SMAs and SMA-based devices can be found here.

Examples of other structurally inhomogeneous biological systems we have been focusing in include also RNA nanostructures [ EJ-137 , EJ-145 ] and complex biological networks, dynamics of which has been treated as a multiscale process in the systems biology framework [ EJ-37 ]. Further development of these ideas led to a hierarchy of mathematical models for cell cycle dynamics accounting for stochastic fluctuations [ EJ-130 ]. The multiscale character of evolutionary problems and an intrinsic link between microscopic and macroscopic models for evolution via the concept of perturbed generalized dynamic systems has been discussed in detail in [ EJ-120 ].  This concept has other potential applications ranging from control [ EJ-21 ]to the development of evolutionary algorithms in different areas of science, engineering, and even finance [ EP-T-2010], and to the analysis of complex networks.

Coupled effects are of crucial importance in studying proteins and biological systems that contain proteins. We studied the dynamic collagen piezoelectricity in human cornea including the effect of dehydration. Collagen is a structural protein that is a basic building block of human connective tissues (as well as bones). A constitutive model was proposed for the numerical characterization of cornea based on the available experimental data and a new model was derived based on the conservation law for the dynamic piezoelectricity supplemented by the time-dependent equation for the electromagnetic field [ EJ-99 ]. Numerical results presented here demonstrate promising applications of the developed model in aiding refractive surgery and a better understanding of regenerative processes in cornea.

 

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