MATHEMATICAL PHYSIOLOGY
POPULATION MODELS FOR PRECISION MEDICINE ΨΨ
Collaboration with Mike Reed and Fred Nijhout (Duke)
Ordinary differential equation (ODE) models can be used to analyze the biochemical mechanics that are based on known physiology and biochemistry. Typically the literature contains a wide range of values for each parameter, such as Km, Ki, and Vmax values for the enzymes and transporters. This is natural because the Km depend on pH and temperature, Vmax values depend on gene expression levels, and often there is a difference between in vivo and in vitro results. Not only will these values differ between different people, but they will differ in time in a single individual.
The large biological variation between individuals is often considered a nuisance by modelers. This variation must be taken into account in precision medicine if one wants to design intervention strategies (for example, drug treatment) that are tailored to the individual. Population models allow one to do just that. For each parameter p in the ODE model, one randomly selects a new value in the range .5p_o to 1.5p_o where p_o is the average value. One does this independently for each parameter in the model. Then the model is run to steady state while the concentrations, the velocities of all the reactions, and the parameters that were selected are recorded. This creates one ``virtual person.'' Repeating this process 10,000 times creates a database of 10,000 virtual persons with the phenotypic variation that one expects to find in the cell metabolism in the human population.
A database of virtual individuals and can be used to simulate populations with known genetic polymorphisms. These population models are a new approach to understanding and coping with biological variation. Once this database has been created, the normal tools of statistical analysis can be used to find interesting correlations between variables. However, since the database is based on variation of parameters in a deterministic model, the underlying models can also be used to determine the mechanistic reason for the observed correlations. I have applied these population models to understand the efficacy of selective serotonin reuptake inhibitors (SSRIs) for depression; this application was summarized in our review paper [1].
[1] F. Nijhout, F. Sadre-Marandi, J. Best, M. Reed, Systems biology of phenotypic robustness and plasticity, Integr. Comp. Biol., 57 (2017), pp.171-184. (PDF)
Collaboration with Mike Reed and Fred Nijhout (Duke)
Ordinary differential equation (ODE) models can be used to analyze the biochemical mechanics that are based on known physiology and biochemistry. Typically the literature contains a wide range of values for each parameter, such as Km, Ki, and Vmax values for the enzymes and transporters. This is natural because the Km depend on pH and temperature, Vmax values depend on gene expression levels, and often there is a difference between in vivo and in vitro results. Not only will these values differ between different people, but they will differ in time in a single individual.
The large biological variation between individuals is often considered a nuisance by modelers. This variation must be taken into account in precision medicine if one wants to design intervention strategies (for example, drug treatment) that are tailored to the individual. Population models allow one to do just that. For each parameter p in the ODE model, one randomly selects a new value in the range .5p_o to 1.5p_o where p_o is the average value. One does this independently for each parameter in the model. Then the model is run to steady state while the concentrations, the velocities of all the reactions, and the parameters that were selected are recorded. This creates one ``virtual person.'' Repeating this process 10,000 times creates a database of 10,000 virtual persons with the phenotypic variation that one expects to find in the cell metabolism in the human population.
A database of virtual individuals and can be used to simulate populations with known genetic polymorphisms. These population models are a new approach to understanding and coping with biological variation. Once this database has been created, the normal tools of statistical analysis can be used to find interesting correlations between variables. However, since the database is based on variation of parameters in a deterministic model, the underlying models can also be used to determine the mechanistic reason for the observed correlations. I have applied these population models to understand the efficacy of selective serotonin reuptake inhibitors (SSRIs) for depression; this application was summarized in our review paper [1].
[1] F. Nijhout, F. Sadre-Marandi, J. Best, M. Reed, Systems biology of phenotypic robustness and plasticity, Integr. Comp. Biol., 57 (2017), pp.171-184. (PDF)
ARSENIC TOXICITY AND IMPLICATIONS FOR PUBLIC HEALTHΨ
Collaboration with Mike Reed and Fred Nijhout (Duke) and Mary Gamble (Columbia)
Large parts of the population in Bangladesh have arsenic poisoning and consequences (melanoma, bladder cancer, etc.) because of arsenic in the water taken from the wells. Collaborator Mary Gamble (Columbia University) conducts clinical trials in Bangladesh to determine an edible treatment to increase the speed of detoxification of arsenic in the general public. In 2007, Gamble showed that folate supplementation for folate deficient individuals lowers blood arsenic. The fundamental question is: Can we find other nutrients or supplements that would also (cheaply) lower blood arsenic?
The reaction diagram for the folate and methionine cycles in OCM is very complicated consisting of loops within loops (shown below). Furthermore, many substrates in the network influence the activity level of enzymes at distant locations in the network (shown in red), through allosteric binding. These long-range regulatory mechanisms are extremely important for stabilizing concentrations and reactions (like DNA methylation and replication) against large changes in amino acid inputs due to meals and changes in the environment. To understand the system properties of the whole network, one needs a mathematical model of OCM, based on the underlying biochemistry and biology, and machine computation.
We have developed a mathematical model for the metabolism of arsenic through the folate and methionine cycles in one carbon metabolism (OCM) based on standard biochemical kinetics. The model consists of a set of ordinary differential equations, one for each metabolite in each compartment, and kinetic equations for metabolism and for transport between compartments. The model can serve as a platform to study, in silico, the efficacy of different treatments for arsenic poisoning. For example, Gamble has a new (unpublished) dataset from clinical trials in Bangladesh where volunteers were given creatine supplements, betaine supplements, and choline supplements over 6-8 weeks while arsenic levels were measured in the blood to test effective treatment supplementation. The model is being used to analyze this new dataset and explain the effects found in Gamble's study. Secondly, using the theory of population models explained above, we are determining the combination of supplements that will work best for each person in Bangladesh based on their individual blood biomarkers.
Collaboration with Mike Reed and Fred Nijhout (Duke) and Mary Gamble (Columbia)
Large parts of the population in Bangladesh have arsenic poisoning and consequences (melanoma, bladder cancer, etc.) because of arsenic in the water taken from the wells. Collaborator Mary Gamble (Columbia University) conducts clinical trials in Bangladesh to determine an edible treatment to increase the speed of detoxification of arsenic in the general public. In 2007, Gamble showed that folate supplementation for folate deficient individuals lowers blood arsenic. The fundamental question is: Can we find other nutrients or supplements that would also (cheaply) lower blood arsenic?
The reaction diagram for the folate and methionine cycles in OCM is very complicated consisting of loops within loops (shown below). Furthermore, many substrates in the network influence the activity level of enzymes at distant locations in the network (shown in red), through allosteric binding. These long-range regulatory mechanisms are extremely important for stabilizing concentrations and reactions (like DNA methylation and replication) against large changes in amino acid inputs due to meals and changes in the environment. To understand the system properties of the whole network, one needs a mathematical model of OCM, based on the underlying biochemistry and biology, and machine computation.
We have developed a mathematical model for the metabolism of arsenic through the folate and methionine cycles in one carbon metabolism (OCM) based on standard biochemical kinetics. The model consists of a set of ordinary differential equations, one for each metabolite in each compartment, and kinetic equations for metabolism and for transport between compartments. The model can serve as a platform to study, in silico, the efficacy of different treatments for arsenic poisoning. For example, Gamble has a new (unpublished) dataset from clinical trials in Bangladesh where volunteers were given creatine supplements, betaine supplements, and choline supplements over 6-8 weeks while arsenic levels were measured in the blood to test effective treatment supplementation. The model is being used to analyze this new dataset and explain the effects found in Gamble's study. Secondly, using the theory of population models explained above, we are determining the combination of supplements that will work best for each person in Bangladesh based on their individual blood biomarkers.
GENDER DIFFERENCES IN ONE-CARBON METABOLISM
Collaboration with Mike Reed and Fred Nijhout (Duke) and Mary Gamble (Columbia) and Thabat Dahdoul*
There are gender differences in one-carbon metabolism (OCM) and these differences are accentuated throughout pregnancy. Women in the child-bearing years exhibit lower plasma homocysteine (Hcy), higher betaine and choline, and lower S-andenosylmethionine (SAM), shown above. Various enzymes in OCM are upregulated or down regulated in women. For example, phosphatidylethanolamine N-methyltransferase (PEMT) is upregulated by estrogen. Furthermore, insulin and glucose also affect some enzymes of OCM and change during pregnancy. All of these results suggest that a mechanistic understanding of how enzymatic differences in women affect OCM is important for precision medicine.
Using the mathematical model described in the Arsenic Toxicity project, the enzymatic changes in women of child-bearing age are being studied. Since the underlying ODE model is a deterministic model, the causal mechanisms by which the gene expression or enzyme activity changes in women that lead to the metabolite changes can be analyzed. Specifically we are looking to understand how choline and betaine, homocysteine, vitamin deficiencies, and SAM play a role in gender differences. In each case we compare our results to clinical and experimental studies and discuss the causal mechanisms by which the gene expression or enzyme activity changes in women lead to the metabolite changes.
[1] F. Sadre-Marandi, T. Dahdoul*, M. Reed, R. Nijhout, Sex differences in hepatic one-carbon metabolism, Submitted.
*mentored student
Collaboration with Mike Reed and Fred Nijhout (Duke) and Mary Gamble (Columbia) and Thabat Dahdoul*
There are gender differences in one-carbon metabolism (OCM) and these differences are accentuated throughout pregnancy. Women in the child-bearing years exhibit lower plasma homocysteine (Hcy), higher betaine and choline, and lower S-andenosylmethionine (SAM), shown above. Various enzymes in OCM are upregulated or down regulated in women. For example, phosphatidylethanolamine N-methyltransferase (PEMT) is upregulated by estrogen. Furthermore, insulin and glucose also affect some enzymes of OCM and change during pregnancy. All of these results suggest that a mechanistic understanding of how enzymatic differences in women affect OCM is important for precision medicine.
Using the mathematical model described in the Arsenic Toxicity project, the enzymatic changes in women of child-bearing age are being studied. Since the underlying ODE model is a deterministic model, the causal mechanisms by which the gene expression or enzyme activity changes in women that lead to the metabolite changes can be analyzed. Specifically we are looking to understand how choline and betaine, homocysteine, vitamin deficiencies, and SAM play a role in gender differences. In each case we compare our results to clinical and experimental studies and discuss the causal mechanisms by which the gene expression or enzyme activity changes in women lead to the metabolite changes.
[1] F. Sadre-Marandi, T. Dahdoul*, M. Reed, R. Nijhout, Sex differences in hepatic one-carbon metabolism, Submitted.
*mentored student
VIRAL ASSEMBLY
EFFECTS OF RNA ON HIV-1 GAG PROTEIN FLEXIBILITY ΨΨ
Collaboration with Karin Musier-Forysth Lab (Ohio State) and Eric Dykeman (University of York)
Nucleic acid binding and incorporation is an important process in the assembly of an immature HIV-1 particle. For efficient viral assembly, the Gag protein must straighten from its bent shape before the virion is able to bud from the host cell. This is thought to occur after Psi RNA binds with the zinc fingers located on the NC domain of Gag, creating a rigid structure.
Normal mode analysis identifies the natural movements (vibrational modes) of a physical object. The calculation is based on a harmonic approximation around a minimum energy conformation. At this minimum, the potential and kinetic energies can be expanded into quadratic approximations V and T, respectively. The Lagrangian is given by L=T-V, leading to a system of N linear differential equations of motion, where N is the number of atoms in the object. An eigenvalue problem is obtained by assuming a oscillatory solution, where the eigenvectors describe the direction of each particle and the eigenvalues give the corresponding frequency.
Calculations of the mean square fluctuations (crystallographic B-factors of the amino-acid residues are commonly used for characterizing the normal modes, where the B-factor is a measure of flexibility or rigidity of each atom. By analyzing the vibrational modes of all-atom data for NC-RNA constructs, we are able to match experimental results showing that while NC does not distinguish between the RNA stem loops SL2 and SL3, the addition of the SP1 peptide confirms the high affinity binding of SL3. Modeling results also indicate that Psi RNA binding to NC may promote the helical conformational switch of the SP1 region.
[1] F. Sadre-Marandi, S. Liu, W. Cantara, E. Dykeman, K. Musier-Forsyth, Normal mode analysis identifies SP1 domain of HIV-1 Gag as a key modulator of protein-RNA complex stability, Submitted.
Collaboration with Karin Musier-Forysth Lab (Ohio State) and Eric Dykeman (University of York)
Nucleic acid binding and incorporation is an important process in the assembly of an immature HIV-1 particle. For efficient viral assembly, the Gag protein must straighten from its bent shape before the virion is able to bud from the host cell. This is thought to occur after Psi RNA binds with the zinc fingers located on the NC domain of Gag, creating a rigid structure.
Normal mode analysis identifies the natural movements (vibrational modes) of a physical object. The calculation is based on a harmonic approximation around a minimum energy conformation. At this minimum, the potential and kinetic energies can be expanded into quadratic approximations V and T, respectively. The Lagrangian is given by L=T-V, leading to a system of N linear differential equations of motion, where N is the number of atoms in the object. An eigenvalue problem is obtained by assuming a oscillatory solution, where the eigenvectors describe the direction of each particle and the eigenvalues give the corresponding frequency.
Calculations of the mean square fluctuations (crystallographic B-factors of the amino-acid residues are commonly used for characterizing the normal modes, where the B-factor is a measure of flexibility or rigidity of each atom. By analyzing the vibrational modes of all-atom data for NC-RNA constructs, we are able to match experimental results showing that while NC does not distinguish between the RNA stem loops SL2 and SL3, the addition of the SP1 peptide confirms the high affinity binding of SL3. Modeling results also indicate that Psi RNA binding to NC may promote the helical conformational switch of the SP1 region.
[1] F. Sadre-Marandi, S. Liu, W. Cantara, E. Dykeman, K. Musier-Forsyth, Normal mode analysis identifies SP1 domain of HIV-1 Gag as a key modulator of protein-RNA complex stability, Submitted.
VIRAL CAPSID NUCLEATION
Collaboration with James Liu, Simon Tavener (Colorado State), and Yuewu Liu, Xiufen Zou (Wuhan University)
Funded by NSF EAPSI Fellowship, IIA-1415117, PI; "Modeling Virus Nucleation Using Dynamical Systems." and the generous support of CSTEC.
The major goal of this research project is to develop models for viral capsid assembly. Existing work has modeled viral capsid assembly using one large-size dynamical system, combining the two sub-stages: nucleation and elongation.
Our approach focuses on nuclei growth (nucleation), relatively independent of capsid elongation. Investigating the nucleation stage first gives this model a unique advantage for characterizing conditions required to start capsid formation and producing the building blocks for the mature capsid. It also allows us to examine the favorable and unfavorable conditions for nucleation.
Since some biological parameters in these models are difficult to measure in experiments, mathematical analysis enables us to characterize the most important or sensitive parameters. A 6-species cascaded dynamical system model is created, parameters are estimated to fit biological data, and sensitivity analysis of model parameters is performed. The sensitivity analysis confirms the biological experiments that the dimer intermediate is vital for capsid protein self assembly.
[1] F. Sadre-Marandi, Y. Liu, J. Liu, S. Tavener, X. Zou, Modeling HIV-1 viral capsid nucleation by dynamical systems, Math. Biosci., 270 (2015), pp.95-105. (PDF)
Collaboration with James Liu, Simon Tavener (Colorado State), and Yuewu Liu, Xiufen Zou (Wuhan University)
Funded by NSF EAPSI Fellowship, IIA-1415117, PI; "Modeling Virus Nucleation Using Dynamical Systems." and the generous support of CSTEC.
The major goal of this research project is to develop models for viral capsid assembly. Existing work has modeled viral capsid assembly using one large-size dynamical system, combining the two sub-stages: nucleation and elongation.
Our approach focuses on nuclei growth (nucleation), relatively independent of capsid elongation. Investigating the nucleation stage first gives this model a unique advantage for characterizing conditions required to start capsid formation and producing the building blocks for the mature capsid. It also allows us to examine the favorable and unfavorable conditions for nucleation.
Since some biological parameters in these models are difficult to measure in experiments, mathematical analysis enables us to characterize the most important or sensitive parameters. A 6-species cascaded dynamical system model is created, parameters are estimated to fit biological data, and sensitivity analysis of model parameters is performed. The sensitivity analysis confirms the biological experiments that the dimer intermediate is vital for capsid protein self assembly.
[1] F. Sadre-Marandi, Y. Liu, J. Liu, S. Tavener, X. Zou, Modeling HIV-1 viral capsid nucleation by dynamical systems, Math. Biosci., 270 (2015), pp.95-105. (PDF)
VIRAL PROTEIN TRAFFICKING AND BINDING
Collaboration with Yuanbin Wang, Xiufen Zou (Wuhan University), and Jingyin Tan (Huazhong Agricultural University), and James Liu (Colorado State)
Funded by NSF, DMS-1419077, "Developing Novel Numerical Methods for Flow and Transport in Porous Media"
Quantitative results for intracellular trafficking and assembly of gag proteins have critical importance for gaining insights into the processes of virus replication and for developing novel control strategies. Our recent work has established a model for integrating the simultaneous treatments of gag monomers and trimers in the dynamical process of transport and binding. The model characterizes the dynamics of virus trafficking and the transformation between monomeric and trimeric states by coupling different types of differential equations.
Numerical simulation results show that the gag protein trimers will accumulate at the membrane of the cell. Numerical results on the time when the first new virions appear near the cell membrane (Ta) are in very good agreement with published experiment data. Sensitivity analysis of Ta to the model parameters indicates that the diffusion and transport process affects the time of initial appearance of HIV-1 virions on the cell membrane.
[1] Y. Wang, J. Tang, F. Sadre-Marandi, J. Liu, X., Zou, Mathematical modeling for intracellular transport and binding of HIV-1 gag proteins, Math. Biosci. 262 (2015), pp. 198-205. (PDF)
Collaboration with Yuanbin Wang, Xiufen Zou (Wuhan University), and Jingyin Tan (Huazhong Agricultural University), and James Liu (Colorado State)
Funded by NSF, DMS-1419077, "Developing Novel Numerical Methods for Flow and Transport in Porous Media"
Quantitative results for intracellular trafficking and assembly of gag proteins have critical importance for gaining insights into the processes of virus replication and for developing novel control strategies. Our recent work has established a model for integrating the simultaneous treatments of gag monomers and trimers in the dynamical process of transport and binding. The model characterizes the dynamics of virus trafficking and the transformation between monomeric and trimeric states by coupling different types of differential equations.
Numerical simulation results show that the gag protein trimers will accumulate at the membrane of the cell. Numerical results on the time when the first new virions appear near the cell membrane (Ta) are in very good agreement with published experiment data. Sensitivity analysis of Ta to the model parameters indicates that the diffusion and transport process affects the time of initial appearance of HIV-1 virions on the cell membrane.
[1] Y. Wang, J. Tang, F. Sadre-Marandi, J. Liu, X., Zou, Mathematical modeling for intracellular transport and binding of HIV-1 gag proteins, Math. Biosci. 262 (2015), pp. 198-205. (PDF)
VIRAL CAPSID
MODELING THE LATTICE STRUCTURES OF VIRAL CAPSIDS
Collaboration with James Liu, Simon Tavener, Chaoping Chen (Colorado State), and Praachi Das* (Ohio State)
Partially funded by Ohio State Undergraduate Research Scholar Award (Praachi Das*)
Virus capsids are best described by fullerene-like structures, modeled by a simple 3-valent, n-vertex polyhedron. This creates a caged polyhedral arrangement, consisting of capsid proteins with group into clusters of six (hexamers) or five (pentamers). It is known that viral capsids could be categorized into three major types: icosahedron, tube, and cone. Mathematical models for the lattice structure help understand the underlying biological mechanisms in the formation of viral capsids. While the models for icosahedral capsids are established and well-received, tubular and conical capsids are not yet fully understood.
Our work establishes a unified approach for the three common capsid shapes by extending the Caspar and Klug (CK) Theory and overcomes the flaw of incomplete closure when existing models are inappropriately applied. In particular, one generating vector is needed to build an icosahedron, while two and three generating vectors are used to characterize respectively the lattice structures of tubular and conical capsids.
For icosahedral capsids, the number of hexamers per capsid varies depending on the capsid size and CK Theory dictates there are exactly twelve pentamers needed to form a closed capsid. However, for a significant number of viruses, including viruses of the Papovaviridae family, the theory doesn't apply. The CK Theory correctly predicts the location of the capsid units, but predicts 12 pentamers and 60 hexamers rather than the known 72 pentamers. The anomaly of the CK Theory raised a new question: "For which Caspar and Klug models can each hexamer be replaced with a pentamer (called a pentagonal polyhedra) while still following icosahedral symmetry?" My analysis with undergraduate student Praachi Das (Ohio State University) in [2] shows the existence of pentagonal polyhedra for a subclass of the triangulation number.
[1] F. Sadre-Marandi, J. Liu, S. Tavener, C. Chen, Generating vectors for the lattice structures of tubular and conical viral capsids, Mol. Based Math. Biol., 2 (2014), pp.128–140. (PDF)
(Selected as journal cover)
[2] F. Sadre-Marandi, P. Das*, Extension of Caspar-Klug theory to higher order pentagonal polyhedra, Comp. and Math. Biophys., 6 (2018), pp.1-13. (PDF)
*mentored student
Collaboration with James Liu, Simon Tavener, Chaoping Chen (Colorado State), and Praachi Das* (Ohio State)
Partially funded by Ohio State Undergraduate Research Scholar Award (Praachi Das*)
Virus capsids are best described by fullerene-like structures, modeled by a simple 3-valent, n-vertex polyhedron. This creates a caged polyhedral arrangement, consisting of capsid proteins with group into clusters of six (hexamers) or five (pentamers). It is known that viral capsids could be categorized into three major types: icosahedron, tube, and cone. Mathematical models for the lattice structure help understand the underlying biological mechanisms in the formation of viral capsids. While the models for icosahedral capsids are established and well-received, tubular and conical capsids are not yet fully understood.
Our work establishes a unified approach for the three common capsid shapes by extending the Caspar and Klug (CK) Theory and overcomes the flaw of incomplete closure when existing models are inappropriately applied. In particular, one generating vector is needed to build an icosahedron, while two and three generating vectors are used to characterize respectively the lattice structures of tubular and conical capsids.
For icosahedral capsids, the number of hexamers per capsid varies depending on the capsid size and CK Theory dictates there are exactly twelve pentamers needed to form a closed capsid. However, for a significant number of viruses, including viruses of the Papovaviridae family, the theory doesn't apply. The CK Theory correctly predicts the location of the capsid units, but predicts 12 pentamers and 60 hexamers rather than the known 72 pentamers. The anomaly of the CK Theory raised a new question: "For which Caspar and Klug models can each hexamer be replaced with a pentamer (called a pentagonal polyhedra) while still following icosahedral symmetry?" My analysis with undergraduate student Praachi Das (Ohio State University) in [2] shows the existence of pentagonal polyhedra for a subclass of the triangulation number.
[1] F. Sadre-Marandi, J. Liu, S. Tavener, C. Chen, Generating vectors for the lattice structures of tubular and conical viral capsids, Mol. Based Math. Biol., 2 (2014), pp.128–140. (PDF)
(Selected as journal cover)
[2] F. Sadre-Marandi, P. Das*, Extension of Caspar-Klug theory to higher order pentagonal polyhedra, Comp. and Math. Biophys., 6 (2018), pp.1-13. (PDF)
*mentored student
CURVATURE CONCENTRATION ON HIV CONICAL CORES
Collaboration with James Liu, Simon Tavener, and Chaoping Chen (Colorado State)
Viral capsids follow the fullerene-like structure with exactly 12 pentamers by the Euler's theorem. Difference distributions of the pentamers result in various shaped capsids. These pentamers introduce declination and curvature on the capsids. Our project intends to provide an explicit and quantitative characterization of curvature on virus capsids.
The discrete setting of the Gauss-Bonnet Theorem is applied to viral capsids for calculating the angle defect at each hexamer and pentamer. For the HIV (5,7)-cone, it is shown in [1] that the curvature concentration at the narrow end is about five times higher than that at the broad end. This leads to a conclusion that the narrow end is the weakest part on the HIV-1 capsid and a conjecture
that the narrow end closes last during maturation and opens first during entry into a host cell.
The modeling results should be helpful for better understanding the HIV-1 capsid structure and the underlying biology. Curvature formalism is novel to the structural virology field and can be used to rank the stability of (related) capsids.
[1] J. Liu, F. Sadre-Marandi, S. Tavener, C. Chen, Curvature concentrations on the HIV-1 capsid, Mol. Based Math. Biol., 3 (2015), pp.43-53. (PDF)
Collaboration with James Liu, Simon Tavener, and Chaoping Chen (Colorado State)
Viral capsids follow the fullerene-like structure with exactly 12 pentamers by the Euler's theorem. Difference distributions of the pentamers result in various shaped capsids. These pentamers introduce declination and curvature on the capsids. Our project intends to provide an explicit and quantitative characterization of curvature on virus capsids.
The discrete setting of the Gauss-Bonnet Theorem is applied to viral capsids for calculating the angle defect at each hexamer and pentamer. For the HIV (5,7)-cone, it is shown in [1] that the curvature concentration at the narrow end is about five times higher than that at the broad end. This leads to a conclusion that the narrow end is the weakest part on the HIV-1 capsid and a conjecture
that the narrow end closes last during maturation and opens first during entry into a host cell.
The modeling results should be helpful for better understanding the HIV-1 capsid structure and the underlying biology. Curvature formalism is novel to the structural virology field and can be used to rank the stability of (related) capsids.
[1] J. Liu, F. Sadre-Marandi, S. Tavener, C. Chen, Curvature concentrations on the HIV-1 capsid, Mol. Based Math. Biol., 3 (2015), pp.43-53. (PDF)
EFFICIENT NUMERICAL METHODS FOR FLOW IN POROUS MEDIA
Collaboration with James Liu, Zhuoran Wang (Colorado State), and Guang Lin (Purdue)
Funded by NSF, DMS-1419077, "Developing Novel Numerical Methods for Flow and Transport in Porous Media"
The cytoplasm can be treated as a porous medium which includes micro-tubules, actin filaments, and intracellular organelles. Therefore, viral protein trafficking can be modeled as transport in a porous medium. Efficient numerical methods for flow and transport problems in porous media are indispensable mathematical tools for simulating intracellular transport processes.
Existing finite element methods, e.g., the classical mixed methods and the widely used discontinuous Galerkin methods are examined against the novel weak Galerkin finite element methods (WGFEMs) in [2]. The comparison reveal that the WGFEMs hold advantages over the existing methods in terms of efficiency and ease of implementation, in addition to preserving physical properties, e.g., local mass conservation. A Matlab toolbox has being developed to put the WGFEMs into practical use (see SOFTWARE page), including simulations of problems in math biology.
[1] J. Liu, F. Sadre-Marandi, Z. Wang, DarcyLite: A Matlab toolbox for Darcy flow computation, Procedia Computer Science, 80 (2016), pp.1301-1312. (PDF)
[2] G. Lin, J. Liu, F. Sadre-Marandi, A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods, Journal of Computational and Applied Mathematics, 273 (2015), pp. 346-362. (PDF)
Collaboration with James Liu, Zhuoran Wang (Colorado State), and Guang Lin (Purdue)
Funded by NSF, DMS-1419077, "Developing Novel Numerical Methods for Flow and Transport in Porous Media"
The cytoplasm can be treated as a porous medium which includes micro-tubules, actin filaments, and intracellular organelles. Therefore, viral protein trafficking can be modeled as transport in a porous medium. Efficient numerical methods for flow and transport problems in porous media are indispensable mathematical tools for simulating intracellular transport processes.
Existing finite element methods, e.g., the classical mixed methods and the widely used discontinuous Galerkin methods are examined against the novel weak Galerkin finite element methods (WGFEMs) in [2]. The comparison reveal that the WGFEMs hold advantages over the existing methods in terms of efficiency and ease of implementation, in addition to preserving physical properties, e.g., local mass conservation. A Matlab toolbox has being developed to put the WGFEMs into practical use (see SOFTWARE page), including simulations of problems in math biology.
[1] J. Liu, F. Sadre-Marandi, Z. Wang, DarcyLite: A Matlab toolbox for Darcy flow computation, Procedia Computer Science, 80 (2016), pp.1301-1312. (PDF)
[2] G. Lin, J. Liu, F. Sadre-Marandi, A comparative study on the weak Galerkin, discontinuous Galerkin, and mixed finite element methods, Journal of Computational and Applied Mathematics, 273 (2015), pp. 346-362. (PDF)