Chemical Reaction Networks’ Programming for Solving Equations

The computational ability of the chemical reaction networks (CRNs) using DNA as the substrate has been verified previously. To solve more complex computational problems and perform the computational steps as expected, the practical design of the basic modules of calculation and the steps in the reactions have become the basic requirements for biomolecular computing. This paper presents a method for solving nonlinear equations in the CRNs with DNA as the substrate. We used the basic calculation module of the CRNs with a gateless structure to design discrete and analog algorithms and realized the nonlinear equations that could not be solved in the previous work, such as exponential, logarithmic, and simple triangle equations. The solution of the equation uses the transformation method, Taylor expansion, and Newton iteration method, and the simulation verified this through examples. We used and improved the basic calculation module of the CRN++ programming language, optimized the error in the basic module, and analyzed the error’s variation over time.     

A Conformational Switch in a Partially Unwound Helix Selectively Determines the Pathway for Substrate Release from the Carnitine/γ-Butyrobetaine Antiporter CaiT

CaiT is a homotrimeric antiporter that exchanges l-carnitine (CRN) with γ-butyrobetaine (GBB) across the bacterial membrane. Three structures have been resolved to date for CaiT, all in the inward-facing state: CRN-bound (with four CRNs per subunit), GBB-bound (two GBBs per subunit), and apo. One of the reported binding sites is the counterpart of the primary site observed in structurally similar transporters. However, the mechanism and pathway(s) of CRN/GBB unbinding and translocation, or even the ability of the substrates to dislodge from the reported binding sites, are yet to be determined. To shed light on these issues, we performed a total of 1.3 μs of molecular dynamics simulations and examined the dynamics of substrate-bound CaiT structures under different conditions. We find that both CRN and GBB are able to dissociate completely from their primary site into the cytoplasm. Substrate molecules initially located at the secondary sites dissociate even faster (within tens of nanoseconds) into the extra- or intracellular regions. Interestingly, the unbinding pathway from the primary site appears to be dictated by the geometry of the unwound part of the transmembrane (TM) helix 3, mostly around Thr(100) therein. Arg(262) on TM7, which apparently mimics the role of Na(+) in CaiT structural homologues, plays a key role in triggering the dissociation of the substrate away from the primary site and guiding its release to the cytoplasm provided that the unwound part of TM3 switches from a shielding to a yielding pose.     

Variable elimination in post-translational modification reaction networks with mass-action kinetics

We define a subclass of chemical reaction networks called post-translational modification systems. Important biological examples of such systems include MAPK cascades and two-component systems which are well-studied experimentally as well as theoretically. The steady states of such a system are solutions to a system of polynomial equations. Even for small systems the task of finding the solutions is daunting. We develop a mathematical framework based on the notion of a cut (a particular subset of species in the system), which provides a linear elimination procedure to reduce the number of variables in the system to a set of core variables. The steady states are parameterized algebraically by the core variables, and graphical conditions for when steady states with positive core variables imply positivity of all variables are given. Further, minimal cuts are the connected components of the species graph and provide conservation laws. A criterion for when a (maximal) set of independent conservation laws can be derived from cuts is given.     

CLAVATA2 forms a distinct CLE‐binding receptor complex regulating Arabidopsis stem cell specification

CLAVATA1 (CLV1), CLV2, CLV3, CORYNE (CRN), BAM1 and BAM2 are key regulators that function at the shoot apical meristem (SAM) of plants to promote differentiation by limiting the size of the organizing center that maintains stem cell identity in neighboring cells. Previous results have indicated that the extracellular domain of the receptor kinase CLV1 binds to the CLV3‐derived CLE ligand. The biochemical role of the receptor‐like protein CLV2 has remained largely unknown. Although genetic analysis suggested that CLV2, together with the membrane kinase CRN, acts in parallel with CLV1, recent studies using transient expression indicated that CLV2 and CRN from a complex with CLV1. Here, we report detection of distinct CLV2‐CRN heteromultimeric and CLV1‐BAM multimeric complexes in transient expression in tobacco and in Arabidopsis meristems. Weaker interactions between the two complexes were detectable in transient expression. We also find that CLV2 alone generates a membrane‐localized CLE binding activity independent of CLV1. CLV2, CLV1 and the CLV1 homologs BAM1 and BAM2 all bind to the CLV3‐derived CLE peptide with similar kinetics, but BAM receptors show a broader range of interactions with different CLE peptides. Finally, we show that BAM and CLV1 overexpression can compensate for the loss of CLV2 function in vivo. These results suggest two parallel ligand‐binding receptor complexes controlling stem cell specification in Arabidopsis.     

Influence of Complement on the Neutralization of Murine Cytomegalovirus by Rabbit Antibody

Antiserum against murine cytomegalovirus produced in the rabbit contained complement (C')-requiring neutralizing (CRN) antibody. The proportion of CRN was extremely high (up to 98%) during the early portion of an immunization procedure, whereas the antisera produced late had a much lower proportion that required C'. The antiserum produced was specific for MCMV with or without C'.     

Equilibrium points for nonlinear compartmental models.

Equilibrium points for nonlinear autonomous compartmental models with constant input are discussed. Upper and lower bounds for the steady states are derived. Theorems guaranteeing existence and uniqueness of equilibrium points for a large collection of system are proved. New information relating to mean residence times is developed. Asymptotic results and a section on stability are included. A recursive process is discussed that generates iterates that converge to steady states for certain types of models. An interesting range of models are included as examples. An attempt is made to provide general qualitative theory for such nonlinear compartmental systems.     

Operability of Continuous Bioprocesses: Static Behavior of a Large Class of Unstructured Models

The complete static behavior of a large class of unstructured models of continuous bioprocesses is classified using elementary concepts of the singularity theory and continuation techniques. The class consists of models for which the cell growth rate is proportional to the rate of utilization of limiting substrate while the kinetics of cell growth, utilization of limiting substrate and synthesis of the desired non-biomass product are allowed to assume general forms of substrate and product. This class of models was used extensively in the literature to model fermentation processes. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of wash-out conditions with non-trivial steady states and multistability resulting from hysteresis. These general results are applied to a number of experimentally validated models of fermentation processes, and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models. Practical criteria are also derived for the safe operation of the bioprocesses.     

Classification of static behavior of a class of unstructured models of continuous bioprocesses

The stability characteristics of a class of unstructured models of continuous bioreactors are analyzed using elementary concepts of singularity theory and continuation techniques. The class consists of models for which the non-biomass product formation rate is linearly proportional to the utilization rate of limiting substrate. The kinetics expressions of cell growth and product synthesis are allowed to assume general forms of substrate and product. Global analytical conditions are derived that allow the construction of a practical picture in the multidimensional parameter space delineating the different static behavior these models can predict, including unique steady states, coexistence of non-trivial steady states with wash-out conditions, and multistability resulting from hysteresis. These general results are applied to specific examples of bioprocesses and allow the study of the effect of kinetic and operating parameters on the stability characteristics of these models.     

The steady state kinetics of some biological systems: III. Thermodynamic aspects

Some thermodynamic aspects of steady systems are considered. The time rates of changes, “flux”, of various thermodynamic quantities are formulated. In particular the free energy flux in the steady state, the difference between the free energy flux in the steady and time dependent states and the change in free energy flux upon transition between steady states are discussed. Equations are derived which exhibit the formal similarities and differences between the free energy flux and the conventional free energy change. The temperature dependence of the steady state rate is examined and conditions for “mastery” by a single step discussed. A brief discussion of the role ofrate in the coupling of exergonic and endergonic reactions is given.     

Periodicities of open linear systems with positive steady states

Systems of two, three, and four linear non-homogeneous differential equations are examined with a view toward determining whether they can possibly serve as mathematical models to describe periodicities in the concentrations of substances which enhance or inhibit each other's rate of production (or dissipation). The nature of the model demands that the solutions of the differential equations be non-negative at all times, i.e., that all the steady states be positive. Conditions for periodicity and for positive steady states are derived, and it is shown that these conditions are not always compatible with each other. In particular it is shown that certain three- and four-hormone models proposed to account for the periodicities observed in the menstrual cycle cannot satisfy the above conditions for any values of the parameters and hence are inadequate.     

The Irish Potato Famine Pathogen Phytophthora infestans Translocates the CRN8 Kinase into Host Plant Cells

Phytopathogenic oomycetes, such as Phytophthora infestans, secrete an arsenal of effector proteins that modulate plant innate immunity to enable infection. We describe CRN8, a host-translocated effector of P. infestans that has kinase activity in planta. CRN8 is a modular protein of the CRN effector family. The C-terminus of CRN8 localizes to the host nucleus and triggers cell death when the protein is expressed in planta. Cell death induction by CRN8 is dependent on its localization to the plant nucleus, which requires a functional nuclear localization signal (NLS). The C-terminal sequence of CRN8 has similarity to a serine/threonine RD kinase domain. We demonstrated that CRN8 is a functional RD kinase and that its auto-phosphorylation is dependent on an intact catalytic site. Co-immunoprecipitation experiments revealed that CRN8 forms a dimer or multimer. Heterologous expression of CRN8 in planta resulted in enhanced virulence by P. infestans. In contrast, in planta expression of the dominant-negative CRN8^R469A;D470A resulted in reduced P. infestans infection, further implicating CRN8 in virulence. Overall, our results indicate that similar to animal parasites, plant pathogens also translocate biochemically active kinase effectors inside host cells.     

Modifiable chemically crosslinked poli(κ-carrageenan) particles

Micron size κ-carrageenan hydrogel particles, p(CRN) from linear κ-carrageenan, were prepared via microemulsion polymerization using divinyl sulfone (DVS) as chemical crosslinker in a sodium bis(2-ethylhexyl) sulfosuccinate (AOT) reverse miceller system. Magnetic field responsive (m-p(CRN)) composite particles were also synthesized by encapsulating magnetic ferrite (Fe₃O₄) nanoparticles together with linear κ-carrageenan within the AOT reverse micelle before the crosslinking reaction. The synthesized bare p(CRN) particles were further modified to produce positive charges on the particles (q-p(CRN)) by a quaternization reaction with an 3-chloro-2-hydroxypropyl trimethyl ammonium chloride aqueous solution. Scanning electron microscopy (SEM), dynamic light scattering (DLS), zeta potential measurements and FT-IR analysis confirmed that particle sizes and charges were altered by chemical modification. Furthermore, a model drug, phenylephrine HCl was used for in vitro drug delivery studies to compare the effectiveness of modification of p(CRN) microgels by comparing bare p(CRN), m-p(CRN) and q-p(CRN) particles drug release capabilities in phosphate buffer solution (PBS) at pH 7.4.     

Thermodynamics of information transfer between subunits in oligomeric enzymes and kinetic cooperativity. 1. Thermodynamics of subunit interactions, partition functions and enzyme reaction rate

The strength of quaternary constraints between two subunits of a polymeric enzyme depends upon the number of neighboring subunits and upon whether these subunits are liganded or not. These quaternary constraints between two subunits of a complex polymeric enzyme may be expressed, however, in terms of quaternary constraints that exist within ideal dimers. The influence of quaternary constraints on the reaction rate of a complex polymeric enzyme may thus be expressed in terms of the intersubunit strain that exists within dimers. This conclusion, that was far from evident, appears to be the consequence of the postulates of structural kinetics, and derive as well from usual thermodynamic principles. The structural steady-state equations may be expressed in terms of partition and sub-partition functions. As applied to structural kinetic models, a partition function expresses how, during the steady state, the energy of a population of enzyme molecules is distributed over n states. Similarly a sub-partition function describes how, during the steady state, the energy of these enzyme molecules is partitioned among only n-k of these states. Although the concept of partition function was initially formulated for equilibrium processes, it may be extended without any loss of generality to non-equilibrium processes. Moreover it is reminiscent of the concept of binding polynomial presented some years ago by Wyman for the equilibrium binding of a ligand to a protein. With this formalism derived from statistical mechanics, a structural rate equation may be derived from the ratio of a sub-partition function of degree n-1 and of a partition function of degree n. Again these properties are the consequence of the postulates of structural kinetics associated with simple ideas derived from statistical thermodynamics.     

The receptor kinase CORYNE of Arabidopsis transmits the stem cell-limiting signal CLAVATA3 independently of CLAVATA1

Stem cells in shoot and floral meristems of Arabidopsis thaliana secrete the signaling peptide CLAVATA3 (CLV3) that restricts stem cell proliferation and promotes differentiation. The CLV3 signaling pathway is proposed to comprise the receptor kinase CLV1 and the receptor-like protein CLV2. We show here that the novel receptor kinase CORYNE (CRN) and CLV2 act together, and in parallel with CLV1, to perceive the CLV3 signal. Mutations in CRN cause stem cell proliferation, similar to clv1, clv2, and clv3 mutants. CRN has additional functions during plant development, including floral organ development, that are shared with CLV2. The CRN protein lacks a distinct extracellular domain, and we propose that CRN and CLV2 interact via their transmembrane domains to establish a functional receptor.     

Identification and Characterisation CRN Effectors in Phytophthora capsici Shows Modularity and Functional Diversity

Phytophthora species secrete a large array of effectors during infection of their host plants. The Crinkler (CRN) gene family encodes a ubiquitous but understudied class of effectors with possible but as of yet unknown roles in infection. To appreciate CRN effector function in Phytophthora, we devised a simple Crn gene identification and annotation pipeline to improve effector prediction rates. We predicted 84 full-length CRN coding genes and assessed CRN effector domain diversity in sequenced Oomycete genomes. These analyses revealed evidence of CRN domain innovation in Phytophthora and expansion in the Peronosporales. We performed gene expression analyses to validate and define two classes of CRN effectors, each possibly contributing to infection at different stages. CRN localisation studies revealed that P. capsici CRN effector domains target the nucleus and accumulate in specific sub-nuclear compartments. Phenotypic analyses showed that few CRN domains induce necrosis when expressed in planta and that one cell death inducing effector, enhances P. capsici virulence on Nicotiana benthamiana. These results suggest that the CRN protein family form an important class of intracellular effectors that target the host nucleus during infection. These results combined with domain expansion in hemi-biotrophic and necrotrophic pathogens, suggests specific contributions to pathogen lifestyles. This work will bolster CRN identification efforts in other sequenced oomycete species and set the stage for future functional studies towards understanding CRN effector functions.