Model reduction using piecewise-linear approximations preserves dynamic properties of the carbon starvation response in Escherichia coli

The adaptation of the bacterium Escherichia coli to carbon starvation is controlled by a large network of biochemical reactions involving genes, mRNAs, proteins, and signalling molecules. The dynamics of these networks is difficult to analyze, notably due to a lack of quantitative information on parameter values. To overcome these limitations, model reduction approaches based on quasi-steady-state (QSS) and piecewise-linear (PL) approximations have been proposed, resulting in models that are easier to handle mathematically and computationally. These approximations are not supposed to affect the capability of the model to account for essential dynamical properties of the system, but the validity of this assumption has not been systematically tested. In this paper, we carry out such a study by evaluating a large and complex PL model of the carbon starvation response in E. coli using an ensemble approach. The results show that, in comparison with conventional nonlinear models, the PL approximations generally preserve the dynamics of the carbon starvation response network, although with some deviations concerning notably the quantitative precision of the model predictions. This encourages the application of PL models to the qualitative analysis of bacterial regulatory networks, in situations where the reference time scale is that of protein synthesis and degradation.     

Piecewise-linear Models of Genetic Regulatory Networks: Equilibria and their Stability

A formalism based on piecewise-linear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be well-suited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.     

Structural principles for periodic orbits in glass networks

A piecewise-linear differential equation model framework for gene regulatory interactions (Glass networks) has allowed considerable analysis of qualitative dynamics in such systems, including periodicity, an important class of regulatory behaviors. Here, we present new results relating the structure of the network to its dynamics (structural principles). The structure we refer to is the state space of the network, which is a digraph on an n-cube in the case of a single threshold per gene. In particular, we show that for a wide class of cycles in the state space there exist parameter values, consistent with the graph structure, for which a periodic orbit exists in the network. For some classes, we show in addition that stable periodic orbits exist. These results extend greatly earlier work by Glass and Pasternack (J Math Biol 6:207–223, 1978).     

The adenomatous polyposis coli protein unambiguously localizes to microtubule plus ends and is involved in establishing parallel arrays of microtubule bundles in highly polarized epithelial cells

Loss of full-length adenomatous polyposis coli (APC) protein correlates with the development of colon cancers in familial and sporadic cases. In addition to its role in regulating β-catenin levels in the Wnt signaling pathway, the APC protein is implicated in regulating cytoskeletal organization. APC stabilizes microtubules in vivo and in vitro, and this may play a role in cell migration (Näthke, I.S., C.L. Adams, P. Polakis, J.H. Sellin, and W.J. Nelson. 1996. J. Cell Biol. 134:165–179; Mimori-Kiyosue, Y., N. Shiina, and S. Tsukita. 2000. J. Cell Biol. 148:505–517; Zumbrunn, J., K. Inoshita, A.A. Hyman, and I.S. Näthke. 2001. Curr. Biol. 11:44–49) and in the attachment of microtubules to kinetochores during mitosis (Fodde, R., J. Kuipers, C. Rosenberg, R. Smits, M. Kielman, C. Gaspar, J.H. van Es, C. Breukel, J. Wiegant, R.H. Giles, and H. Clevers. 2001. Nat. Cell Biol. 3:433–438; Kaplan, K.B., A. Burds, J.R. Swedlow, S.S. Bekir, P.K. Sorger, and I.S. Näthke. 2001. Nat. Cell Biol. 3:429–432). The localization of endogenous APC protein is complex: actin- and microtubule-dependent pools of APC have been identified in cultured cells (Näthke et al., 1996; Mimori-Kiyosue et al., 2000; Reinacher-Schick, A., and B.M. Gumbiner. 2001. J. Cell Biol. 152:491–502; Rosin-Arbesfeld, R., G. Ihrke, and M. Bienz. 2001. EMBO J. 20:5929–5939). However, the localization of APC in tissues has not been identified at high resolution. Here, we show that in fully polarized epithelial cells from the inner ear, endogenous APC protein associates with the plus ends of microtubules located at the basal plasma membrane. Consistent with a role for APC in supporting the cytoskeletal organization of epithelial cells in vivo, the number of microtubules is significantly reduced in apico-basal arrays of microtubule bundles isolated from mice heterozygous for APC.     

Modeling amantadine treatment of influenza A virus in vitro

We analyzed the dynamics of an influenza A/Albany/1/98 (H3N2) viral infection, using a set of mathematical models highlighting the differences between in vivo and in vitro infection. For example, we found that including virion loss due to cell entry was critical for the in vitro model but not for the in vivo model. Experiments were performed on influenza virus-infected MDCK cells in vitro inside a hollow-fiber (HF) system, which was used to continuously deliver the drug amantadine. The HF system captures the dynamics of an influenza infection, and is a controlled environment for producing experimental data which lend themselves well to mathematical modeling. The parameter estimates obtained from fitting our mathematical models to the HF experimental data are consistent with those obtained earlier for a primary infection in a human model. We found that influenza A/Albany/1/98 (H3N2) virions under normal experimental conditions at rapidly lose infectivity with a half-life of , and that the lifespan of productively infected MDCK cells is . Finally, using our models we estimated that the maximum efficacy of amantadine in blocking viral infection is ∼74%, and showed that this low maximum efficacy is likely due to the rapid development of drug resistance.     

Allosteric control in Limulus polyphemus hemocyanin: functional relevance of interactions between hexamers

Hemocyanin of the horseshoe crab Limulus polyphemus is composed of 48 oxygen-binding subunits, which are arranged in eight hexameric building blocks. Allosteric interactions in this oligomeric protein have been examined by measurement of high-precision oxygen-equilibrium curves, using an automated Imai cell. Several models were compared in numerical analysis of the data. A number of conclusions can be drawn with confidence. (1) Oxygen binding by Limulus hemocyanin cannot satisfactorily be described by the two-state MWC model [Monod, J., Wyman, J., & Changeux, J.P. (1965) J. Mol. Biol. 12, 88-118] for allosteric transitions with either the hexamer or dodecamer as the allosteric unit. (2) Of the models tested, the data sets can be best described by an extended MWC model that allows for an equilibrium, within the 48-subunit ensemble, between cooperative hexamers and cooperative dodecamers. The model invokes T and R states for both hexamers (T6 and R6) and dodecamers (T12 and R12). Allosteric effectors modulate oxygen affinity and cooperativity by affecting the R to T equilibria within hexamers and dodecamers and by shifting the equilibria between hexamers and dodecamers. (3) The fitted model parameters show that under most conditions the intersubunit contacts within T-state hexamers are more constrained than those within T-state dodecamers. (4) The oxygen affinities of the hexameric and dodecameric R states are the same, but under all conditions examined the conformation of the fully oxygenated molecule is that of the dodecameric R state. (5) Between pH 7.4 and pH 8.5 the dodecameric T state has a higher affinity for oxygen than the hexameric T state, allowing for "T-state cooperativity".(ABSTRACT TRUNCATED AT 250 WORDS)     

Quaternary structure of rice nonsymbiotic hemoglobin

Plant nonsymbiotic hemoglobins are hexacoordinate heme proteins found in all plants. Although expression is linked with hypoxic environmental conditions (Taylor, E. R., Nie, X. Z., Alexander, W. M., and Hill, R. D. (1994) Plant Mol. Biol. 24, 853-862), no discrete physiological function has yet been attributed to this family of proteins. The crystal structure of a nonsymbiotic hemoglobin from rice has recently been determined. The crystalline protein is homodimeric and hexacoordinate with two histidine side chains coordinating the heme iron atom. Despite the fact that the amino acids responsible for the subunit interface are relatively conserved among the nonsymbiotic hemoglobins, previous work suggests that this group of proteins might display variability in quaternary structure (Duff, S. M. G., Wittenberg, J. B., and Hill, R. D. (1997) J. Biol. Chem. 272, 16746-16752; Arredondo-Peter, R., Hargrove, M. S., Sarath, G., Moran, J. F., Lohrman, J., Olson, J. S., and Klucas, R. V. (1997) Plant Physiol. 115, 1259-1266). Analytical ultracentrifugation and size exclusion high pressure liquid chromatography were used to investigate the quaternary structure of rice nonsymbiotic hemoglobin at various states of ligation and oxidation. Additionally, site-directed mutagenesis was used to test the role of several interface amino acids in dimer formation and ligand binding. Results were analyzed in light of possible physiological functions and indicate that the plant nonsymbiotic hemoglobins are not oxygen transport proteins but more closely resemble known oxygen sensors.     

From individual behaviour to population models: a case study using swimming algae

Trajectories of swimming algae are analysed, and two random-walk models developed to link the individual-level behaviour of cells to population-level advection-diffusion models for the spatial-temporal distribution of cells. The models are both of the advection-diffusion form but are based on two different sets of assumptions about the underlying random-walk behaviours, a velocity jump behaviour and a velocity diffusion behaviour. The mathematical models were developed to allow for an arbitrary (non-weak) bias in the random walk and a variable swimming speed in order to represent the experimental data. For the algal species considered, Heterosigma akashiwo, the mean upward swimming speed was computed as , and the calculated diffusion constants ranged from 2×10³ to depending on the details of the models. That two widely used modelling approaches yield substantially different population-level predictions when applied to the same empirical data suggests that better theoretical tools are needed for identifying adequate approximations for behavioural characteristics.     

A comparison of muscle thin filament models obtained from electron microscopy reconstructions and low-angle X-ray fibre diagrams from non-overlap muscle

The regulation of striated muscle contraction involves changes in the interactions of troponin and tropomyosin with actin thin filaments. In resting muscle, myosin-binding sites on actin are thought to be blocked by the coiled-coil protein tropomyosin. During muscle activation, Ca2+ binding to troponin alters the tropomyosin position on actin, resulting in cyclic actin-myosin interactions that accompany muscle contraction. Evidence for this steric regulation by troponin-tropomyosin comes from X-ray data [Haselgrove, J.C., 1972. X-ray evidence for a conformational change in the actin-containing filaments of verterbrate striated muscle. Cold Spring Habor Symp. Quant. Biol. 37, 341-352; Huxley, H.E., 1972. Structural changes in actin and myosin-containing filaments during contraction. Cold Spring Habor Symp. Quant. Biol. 37, 361-376; Parry, D.A., Squire, J.M., 1973. Structural role of tropomyosin in muscle regulation: analysis of the X-ray diffraction patterns from relaxed and contracting muscles. J. Mol. Biol. 75, 33-55] and electron microscope (EM) data [Spudich, J.A., Huxley, H.E., Finch, J., 1972. Regulation of skeletal muscle contraction. II. Structural studies of the interaction of the tropomyosin-troponin complex with actin. J. Mol. Biol. 72, 619-632; O'Brien, E.J., Gillis, J.M., Couch, J., 1975. Symmetry and molecular arrangement in paracrystals of reconstituted muscle thin filaments. J. Mol. Biol. 99, 461-475; Lehman, W., Craig, R., Vibert, P., 1994. Ca2+-induced tropomyosin movement in Limulus thin filaments revealed by three-dimensional reconstruction. Nature 368, 65-67] each with its own particular strengths and limitations. Here we bring together some of the latest information from EM analysis of single thin filaments from Pirani et al. [Pirani, A., Xu, C., Hatch, V., Craig, R., Tobacman, L.S., Lehman, W. (2005). Single particle analysis of relaxed and activated muscle thin filaments. J. Mol. Biol. 346, 761-772], with synchrotron X-ray data from non-overlapped muscle fibres to refine the models of the striated muscle thin filament. This was done by incorporating current atomic-resolution structures of actin, tropomyosin, troponin and myosin subfragment-1. Fitting these atomic coordinates to EM reconstructions, we present atomic models of the thin filament that are entirely consistent with a steric regulatory mechanism. Furthermore, fitting the atomic models against diffraction data from skinned muscle fibres, stretched to non-overlap to preclude crossbridge binding, produced very similar results, including a large Ca2+-induced shift in tropomyosin azimuthal location but little change in the actin structure or apparent alteration in troponin position.     

Calculation of free-Mg2+ concentration in adenosine 5'-triphosphate containing solutions in vitro and in vivo

We have attempted to resolve the differences between the levels of free Mg2+ in muscle calculated by Wu et al. [Wu, S. T., Pieper, G. M., Salhany, J. M., & Eliot, R. S. (1981) Biochemistry 20, 7399-7403] (2.5 mM in guinea pig heart) and by Gupta and Moore [Gupta, R. K., & Moore, R. D. (1980) J. Biol. Chem. 255, 3987-3993] (0.6 mM in frog skeletal muscle) on the basis of substantially identical measurements by 31P NMR of the phosphate peaks in the spectrum of MgATP2-. The differences depend on the methods of calculation, including which reactions in which multiple equilibria are being considered. Biochemists and physical chemists customarily use different working definitions of the stability constant for MgATP2- in particular. Wu et al. used in their calculations, without reconciliation, methods involving three different operational definitions of the chelation equilibria involved. An algorithm for calculating Mg2+ and total ATP, which can be carried out with a hand calculator, is described here. With it, we calculated Mg2+ levels that agree with those determined by Gupta et al. [Gupta, R. K., Benkovic, J. L., & Rose, Z. B. (1978) J. Biol. Chem. 253, 6165-6171] with their in vitro systems. We therefore agree with the finding of Gupta and Moore that the Mg2+ level in skeletal and cardiac muscle is 0.6 mM.     

Analysis of the high- and low-spin Soret bands of horse-heart metmyoglobin complexes

Interest in the thermodynamics of the iron-binding site in hemoproteins has increased in recent years due to refinements in x-ray crystallographic studies of hemoproteins [see Deathage, J. F., Lee, R. S., Anderson, C. M. & Moffat, K. (1976) J. Mol. Biol. 104 , 687–706; Heidner, E. J., Ladner, R. C. & Perutz, M. F. (1976) J. Mol. Biol. 104 , 707–722; Deathage, J. F., Lee, R. S. & Moffat, K. (1976) J. Mol. Biol. 104 , 723–728; Ladner, R. C., Heidner, E. J. & Perutz, M. F. (1976) J. Mol. Biol. 114 , 385–414; Fermi, G. & Perutz, M. F. (1977) J. Mol. Biol. 114 , 421–431; Takano, T. (1977) J. Mol. Biol. 110 , 537–568 and 569–589], the synthesis and x-ray analysis of model heme compounds [see Scheidt, W. R. (1977) Acc. Chem. Res. 10 , 339–345; Kastner, M. E., Scheidt, W. R., Mashino, T. & Reed, C. A. (1978) J. Am. Chem. Soc. 100 , 666–667; Mashiko, T., Kastner, M. E., Spartalian, K., Scheidt, W. R. & Reed, C. A. (1978) J. Am. Chem. Soc. 100 , 6354–6362; Hill, H. A. O., Skite, P. P., Buchler, J. W., Luchr, H., Tonn, M., Gregson, A. K. & Pellizer, G. (1979) Chem. Commun. 4 , 151–152; and Scheidt, W. R., Cohen, I. A. & Kastner, M. E. (1979) Biochemistry 18 , 3546–3556], and the numerous data on heme–protein interactions that account for the differences observed in ligand binding between the various species of animals. Numerous probes have been used and provide information about the structure and thermodynamics of the binding site, but no single probe can provide the complete picture [see Iizuka, T. & Yonetani, T. (1970) Adv. Biophys. 1 , 157–182; Smith, D. W. & Williams, R. J. P. (1970) Struct. Bond. 7 , 1–45; and Spiro, T. G. (1975) Biochim. Biophys. Acta 416 , 169–189].     

Phosphorylation of myosin-II regulatory light chain by cyclin-p34cdc2: a mechanism for the timing of cytokinesis

To understand how cytokinesis is regulated during mitosis, we tested cyclin-p34cdc2 for myosin-II kinase activity, and investigated the mitotic-specific phosphorylation of myosin-II in lysates of Xenopus eggs. Purified cyclin-p34cdc2 phosphorylated the regulatory light chain of cytoplasmic and smooth muscle myosin-II in vitro on serine-1 or serine-2 and threonine-9, sites known to inhibit the actin-activated myosin ATPase activity of smooth muscle and nonmuscle myosin (Nishikawa, M., J. R. Sellers, R. S. Adelstein, and H. Hidaka. 1984. J. Biol. Chem. 259:8808-8814; Bengur, A. R., A. E. Robinson, E. Appella, and J. R. Sellers. 1987. J. Biol. Chem. 262:7613-7617; Ikebe, M., and S. Reardon. 1990. Biochemistry. 29:2713-2720). Serine-1 or -2 of the regulatory light chain of Xenopus cytoplasmic myosin-II was also phosphorylated in Xenopus egg lysates stabilized in metaphase, but not in interphase. Inhibition of myosin-II by cyclin-p34cdc2 during prophase and metaphase could delay cytokinesis until chromosome segregation is initiated and thus determine the timing of cytokinesis relative to earlier events in mitosis.     

A fitness based analysis of Daisyworld

The Gaia hypothesis [Lovelock, J., Margulis, L., 1974. Atmospheric homeostasis: the Gaia hypothesis. Tellus 26, 1], that the earth functions as a self-regulating system, has never sat particularly comfortably with ideas in mainstream biology [Anon, 2002. In pursuit of arrogant simplicities. Nature 416, 247]. A lack of any clear role for evolution in the model has led to claims of teleology-that self-regulation emerges because it is pre-ordained to do so [Doolittle, W.F., 1981. Is nature really motherly? CoEvol. Q. 58-63; Dawkins, R., 1979. The Extended Phenotype. Oxford University Press, Oxford]. The Daisyworld parable [Watson, A.J., Lovelock, J.E., 1983. Biological homeostasis of the global environment--the parable of Daisyworld. Tellus B 35, 284], a simple mathematical illustration of Gaia, went some way to addressing these critiques but, despite recent success in incorporating natural selection [Stocker, S.,1995. Regarding mutations in Daisyworld models. J. Theor. Biol. 175, 495; Lenton, T.M., 1998. Gaia and natural selection. Nature 394, 439; Lenton, T.M., Lovelock, J.E., 2001. Daisyworld revisited: quantifying biological effects on planetary self-regulation. Tellus B 53, 288; Wood, A.J., Ackland, G.J., Lenton, T.M., 2006. Mutation of albedo and growth response leads to oscillations in a spatial Daisyworld. J. Theor. Biol. 242, 188], it remains a widely held view that the ideas are inconsistent with biological principles. We show that standard methodology from quantitative genetics can be used to predict the stationary states and dynamic behaviour of Daisyworlds. The system regulates its temperature due to the low-level evolutionary dynamics of competition between the thermally coupled daisies, no higher level principle is invoked. A reconciliation of Gaia with evolutionary theory may allow further development of evolutionary arguments for the existence of global self-regulatory systems.     

Phosphatidylinositol 4-kinase from Saccharomyces cerevisiae. Kinetic analysis using Triton X-100/phosphatidylinositol-mixed micelles.

Phosphatidylinositol 4-kinase (ATP:phosphatidylinositol 4-phosphotransferase, EC 2.7.1.67) was purified from Saccharomyces cerevisiae by an improved procedure over that previously reported (Belunis, C.J., Bae-Lee, M., Kelley, M.J., and Carman, G.M. (1988) J. Biol. Chem. 263, 18897-18903) for the enzyme. The molecular mass of the enzyme was 45 kDa. The 35-kDa protein previously identified as PI 4-kinase was a proteolysis product of the 45-kDa protein. A detailed kinetic analysis of the purified enzyme was performed with Triton X-100/phosphatidylinositol-mixed micelles according to the "surface dilution" (Deems, R.A., Eaton, B.R., and Dennis, E.A. (1975) J. Biol. Chem. 250, 9013-9020) and "dual phospholipid" (Hendrickson, H.S., and Dennis, E.A. (1984) J. Biol. Chem. 259, 5734-5739) kinetic models. Phosphatidylinositol 4-kinase activity followed saturation kinetics with respect to the bulk and surface concentrations of phosphatidylinositol at concentrations of phosphatidylinositol below 0.1 mM. Above 0.1 mM activity was only dependent on the surface concentration of phosphatidylinositol. The enzyme more closely followed the dual phospholipid model where the enzyme associated with Triton X-100 micelles when phosphatidylinositol was present. The interfacial Michaelis constant (KmB) for phosphatidylinositol was 0.0036 mol fraction and the dissociation constant (KsA) for phosphatidylinositol in the micelle surface was 0.26 mM. The results of glycerol gradient centrifugation studies showed that the enzyme was physically associated with Triton X-100/phosphatidylinositol micelles.     

The spread of infectious diseases in spatially structured populations: an invasory pair approximation

The invasion of new species and the spread of emergent infectious diseases in spatially structured populations has stimulated the study of explicit spatial models such as cellular automata, network models and lattice models. However, the analytic intractability of these models calls for the development of tractable mathematical approximations that can capture the dynamics of discrete, spatially-structured populations. Here we explore moment closure approximations for the invasion of an SIS epidemic on a regular lattice. We use moment closure methods to derive an expression for the basic reproductive number, R(0), in a lattice population. On lattices, R(0) should be bounded above by the number of neighbors per individual. However, we show that conventional pair approximations actually predict unbounded growth in R(0) with increasing transmission rates. To correct this problem, we propose an 'invasory' pair approximation which yields a relatively simple expression for R(0) that remains bounded above, and also predicts R(0) values from lattice model simulations more accurately than conventional pair and triple approximations. The invasory pair approximation is applicable to any spatial model, since it takes into account characteristics of invasions that are common to all spatially structured populations.     

The complete classification for dynamics in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population was proposed by Li et al. in (J Math Biol 10:1037–1052, 1986). The sufficient and necessary conditions for coexistence and the sufficient conditions for stability of the boundary equilibria were provided. This paper will present a thorough classification of dynamics for this model in terms of the first and second so called reproductive numbers of infection in strains I and J. This classification not only solves a conjecture proposed in (Li et al., J Math Biol 10:1037–1052, 1986) but also gives the sufficient and necessary conditions for the competitive exclusion. Supported by the NSF of China grants 10531030 and 10671143.     

Direct interaction of Gas11 with microtubules: implications for the dynein regulatory complex

We previously described the Trypanin family of cytoskeleton-associated proteins that have been implicated in dynein regulation [Hill et al., J Biol Chem2000; 275(50):39369-39378; Hutchings et al., J Cell Biol2002;156(5):867-877; Rupp and Porter, J Cell Biol2003;162(1):47-57]. Trypanin from T. brucei is part of an evolutionarily conserved dynein regulatory system that is required for regulation of flagellar beat. In C. reinhardtii, the trypanin homologue (PF2) is part of an axonemal 'dynein regulatory complex' (DRC) that functions as a reversible inhibitor of axonemal dynein [Piperno et al., J Cell Biol1992;118(6):1455-1463; Gardner et al., J Cell Biol1994;127(5):1311-1325]. The DRC consists of an estimated seven polypeptides that are tightly associated with axonemal microtubules. Association with the axoneme is critical for DRC function, but the mechanism by which it attaches to the microtubule lattice is completely unknown. We demonstrate that Gas11, the mammalian trypanin/PF2 homologue, associates with microtubules in vitro and in vivo. Deletion analyses identified a novel microtubule-binding domain (GMAD) and a distinct region (IMAD) that attenuates Gas11-microtubule interactions. Using single-particle binding assays, we demonstrate that Gas11 directly binds microtubules and that the IMAD attenuates the interaction between GMAD and the microtubule. IMAD is able to function in either a cis- or trans-orientation with GMAD. The discovery that Gas11 provides a direct linkage to microtubules provides new mechanistic insight into the structural features of the dynein-regulatory complex.     

Reverse-engineering of gene networks for regulating early blood development from single-cell measurements

Background

Recent advances in omics technologies have raised great opportunities to study large-scale regulatory networks inside the cell. In addition, single-cell experiments have measured the gene and protein activities in a large number of cells under the same experimental conditions. However, a significant challenge in computational biology and bioinformatics is how to derive quantitative information from the single-cell observations and how to develop sophisticated mathematical models to describe the dynamic properties of regulatory networks using the derived quantitative information.

Methods

This work designs an integrated approach to reverse-engineer gene networks for regulating early blood development based on singel-cell experimental observations. The wanderlust algorithm is initially used to develop the pseudo-trajectory for the activities of a number of genes. Since the gene expression data in the developed pseudo-trajectory show large fluctuations, we then use Gaussian process regression methods to smooth the gene express data in order to obtain pseudo-trajectories with much less fluctuations. The proposed integrated framework consists of both bioinformatics algorithms to reconstruct the regulatory network and mathematical models using differential equations to describe the dynamics of gene expression.

Results

The developed approach is applied to study the network regulating early blood cell development. A graphic model is constructed for a regulatory network with forty genes and a dynamic model using differential equations is developed for a network of nine genes. Numerical results suggests that the proposed model is able to match experimental data very well. We also examine the networks with more regulatory relations and numerical results show that more regulations may exist. We test the possibility of auto-regulation but numerical simulations do not support the positive auto-regulation. In addition, robustness is used as an importantly additional criterion to select candidate networks.

Conclusion

The research results in this work shows that the developed approach is an efficient and effective method to reverse-engineer gene networks using single-cell experimental observations.

     

Microsecond time scale dynamics in the RXR DNA-binding domain from a combination of spin-echo and off-resonance rotating frame relaxation measurements

Slow protein dynamics can be studied by 15N spin-echo (CPMG) and off-resonance rotating frame relaxation through the effective field dependence of the exchange-mediated relaxation contribution. It is shown that, by a combination of these complementary techniques, a more extended sampling of the microsecond time scale processes is achieved than by either method alone. 15N R2 and improved off-resonance R1 experiments [Mulder et al. (1998) J. Magn. Reson., 131, 351–357] were applied to the 9- cis-retinoic acid receptor DNA-binding domain and allowed the identification of 14 residues exhibiting microsecond time scale dynamics. Assuming exchange between two conformational substates, average lifetimes ranging from 37 to 416 s, and chemical shift differences of up to 3 ppm were obtained. The largest perturbation of tertiary structure was observed for the second zinc finger region, which was found to be disordered in the solution structure [Holmbeck et al. (1998) J. Mol. Biol., 281, 271–284]. Since this zinc-coordinating domain comprises the principal dimerization interface for RXR in a wide repertoire of complexes with different hormone receptors to their cognate response elements, this finding has important implications for our understanding of nuclear receptor assembly on DNA direct repeats. The flexibility observed for the dimerization domain may explain how RXR, through the ability to adaptively interact with a wide variety of highly homologous partner molecules, demonstrates such a versatile DNA-binding repertoire.