Gene Regulatory Network Investigation Using Ordinary Differential Equations

This chapter reviews mathematical models of gene regulation, either as "pure" gene regulatory networks, as signal transduction pathways or as combinations of these. The basic underlying methods are discussed from first principles, relying on rigorous mathematical concepts but with an aim to avoid technical details and focus on the intuitive aspects of this type of mathematical models. After reviewing the principles, some real biological examples are presented to illustrate the practice of modeling, using recent examples from the literature. The proposed examples all arise in the context of plant biology, either at the single cell scale, looking at auxin signaling, or at higher scales, looking at auxin active transport.

A core mechanism for specifying root vascular patterning can replicate the anatomical variation seen in diverse plant species

Pattern formation is typically controlled through the interaction between molecular signals within a given tissue. During early embryonic development, roots of the model plant Arabidopsis thaliana have a radially symmetric pattern, but a heterogeneous input of the hormone auxin from the two cotyledons forces the vascular cylinder to develop a diarch pattern with two xylem poles. Molecular analyses and mathematical approaches have uncovered the regulatory circuit that propagates this initial auxin signal into a stable cellular pattern. The diarch pattern seen in Arabidopsis is relatively uncommon among flowering plants, with most species having between three and eight xylem poles. Here, we have used multiscale mathematical modelling to demonstrate that this regulatory module does not require a heterogeneous auxin input to specify the vascular pattern. Instead, the pattern can emerge dynamically, with its final form dependent upon spatial constraints and growth. The predictions of our simulations compare to experimental observations of xylem pole number across a range of species, as well as in transgenic systems in Arabidopsis in which we manipulate the size of the vascular cylinder. By considering the spatial constraints, our model is able to explain much of the diversity seen in different flowering plant species.

Mobile PEAR transcription factors integrate positional cues to prime cambial growth

Apical growth in plants initiates upon seed germination, whereas radial growth is primed only during early ontogenesis in procambium cells and activated later by the vascular cambium¹. Although it is not known how radial growth is organized and regulated in plants, this system resembles the developmental competence observed in some animal systems, in which pre-existing patterns of developmental potential are established early on^2,3. Here we show that in Arabidopsis the initiation of radial growth occurs around early protophloem-sieve-element cell files of the root procambial tissue. In this domain, cytokinin signalling promotes the expression of a pair of mobile transcription factors-PHLOEM EARLY DOF 1 (PEAR1) and PHLOEM EARLY DOF 2 (PEAR2)-and their four homologues (DOF6, TMO6, OBP2 and HCA2), which we collectively name PEAR proteins. The PEAR proteins form a short-range concentration gradient that peaks at protophloem sieve elements, and activates gene expression that promotes radial growth. The expression and function of PEAR proteins are antagonized by the HD-ZIP III proteins, well-known polarity transcription factors⁴-the expression of which is concentrated in the more-internal domain of radially non-dividing procambial cells by the function of auxin, and mobile miR165 and miR166 microRNAs. The PEAR proteins locally promote transcription of their inhibitory HD-ZIP III genes, and thereby establish a negative-feedback loop that forms a robust boundary that demarks the zone of cell division. Taken together, our data establish that during root procambial development there exists a network in which a module that links PEAR and HD-ZIP III transcription factors integrates spatial information of the hormonal domains and miRNA gradients to provide adjacent zones of dividing and more-quiescent cells, which forms a foundation for further radial growth.

OpenSimRoot: widening the scope and application of root architectural models

OpenSimRoot is an open-source, functional-structural plant model and mathematical description of root growth and function. We describe OpenSimRoot and its functionality to broaden the benefits of root modeling to the plant science community. OpenSimRoot is an extended version of SimRoot, established to simulate root system architecture, nutrient acquisition and plant growth. OpenSimRoot has a plugin, modular infrastructure, coupling single plant and crop stands to soil nutrient and water transport models. It estimates the value of root traits for water and nutrient acquisition in environments and plant species. The flexible OpenSimRoot design allows upscaling from root anatomy to plant community to estimate the following: resource costs of developmental and anatomical traits; trait synergisms; and (interspecies) root competition. OpenSimRoot can model three-dimensional images from magnetic resonance imaging (MRI) and X-ray computed tomography (CT) of roots in soil. New modules include: soil water-dependent water uptake and xylem flow; tiller formation; evapotranspiration; simultaneous simulation of mobile solutes; mesh refinement; and root growth plasticity. OpenSimRoot integrates plant phenotypic data with environmental metadata to support experimental designs and to gain a mechanistic understanding at system scales.

Theoretical approaches to understanding root vascular patterning: a consensus between recent models

The root vascular tissues provide an excellent system for studying organ patterning, as the specification of these tissues signals a transition from radial symmetry to bisymmetric patterns. The patterning process is controlled by the combined action of hormonal signaling/transport pathways, transcription factors, and miRNA that operate through a series of non-linear pathways to drive pattern formation collectively. With the discovery of multiple components and feedback loops controlling patterning, it has become increasingly difficult to understand how these interactions act in unison to determine pattern formation in multicellular tissues. Three independent mathematical models of root vascular patterning have been formulated in the last few years, providing an excellent example of how theoretical approaches can complement experimental studies to provide new insights into complex systems. In many aspects these models support each other; however, each study also provides its own novel findings and unique viewpoints. Here we reconcile these models by identifying the commonalities and exploring the differences between them by testing how transferable findings are between models. New simulations herein support the hypothesis that an asymmetry in auxin input can direct the formation of vascular pattern. We show that the xylem axis can act as a sole source of cytokinin and specify the correct pattern, but also that broader patterns of cytokinin production are also able to pattern the root. By comparing the three modeling approaches, we gain further insight into vascular patterning and identify several key areas for experimental investigation.

Dynamic regulation of auxin oxidase and conjugating enzymes AtDAO1 and GH3 modulates auxin homeostasis

The hormone auxin is a key regulator of plant growth and development, and great progress has been made understanding auxin transport and signaling. Here, we show that auxin metabolism and homeostasis are also regulated in a complex manner. The principal auxin degradation pathways in Arabidopsis include oxidation by Arabidopsis thaliana gene DIOXYGENASE FOR AUXIN OXIDATION 1/2 (AtDAO1/2) and conjugation by Gretchen Hagen3s (GH3s). Metabolic profiling of dao1-1 root tissues revealed a 50% decrease in the oxidation product 2-oxoindole-3-acetic acid (oxIAA) and increases in the conjugated forms indole-3-acetic acid aspartic acid (IAA-Asp) and indole-3-acetic acid glutamic acid (IAA-Glu) of 438- and 240-fold, respectively, whereas auxin remains close to the WT. By fitting parameter values to a mathematical model of these metabolic pathways, we show that, in addition to reduced oxidation, both auxin biosynthesis and conjugation are increased in dao1-1 Transcripts of AtDAO1 and GH3 genes increase in response to auxin over different timescales and concentration ranges. Including this regulation of AtDAO1 and GH3 in an extended model reveals that auxin oxidation is more important for auxin homoeostasis at lower hormone concentrations, whereas auxin conjugation is most significant at high auxin levels. Finally, embedding our homeostasis model in a multicellular simulation to assess the spatial effect of the dao1-1 mutant shows that auxin increases in outer root tissues in agreement with the dao1-1 mutant root hair phenotype. We conclude that auxin homeostasis is dependent on AtDAO1, acting in concert with GH3, to maintain auxin at optimal levels for plant growth and development.

GH3-Mediated Auxin Conjugation Can Result in Either Transient or Oscillatory Transcriptional Auxin Responses

The conjugation of the phytohormone auxin to amino acids via members of the gene family GH3 is an important component in the auxin-degradation pathway in the model plant species Arabidopsis thaliana, as well as many other plant species. Since the GH3 genes are themselves up-regulated in response to auxin, providing a negative feedback on intracellular auxin levels, it is hypothesised that the GH3s have a role in auxin homoeostasis. To investigate this, we develop a mathematical model of auxin signalling and response that includes the auxin-inducible negative feedback from GH3 on the rate of auxin degradation. In addition, we include a positive feedback on the rate of auxin input via the auxin influx transporter LAX3, shown previously to be expressed in response to auxin and to have an important role during lateral root emergence. In the absence of the LAX3 positive feedback, we show that the GH3 negative feedback suffices to generate a transient transcriptional response to auxin in the shape of damped oscillations of the model system. When LAX3 positive feedback is present, sustained oscillations of the system are possible. Using steady-state analyses, we identify and discuss key parameters affecting the oscillatory behaviour of the model. The transient peak of auxin and subsequent transcriptional response caused by the up-regulation of GH3 represents a possible protective homoeostasis mechanism that may be used by plant cells in response to excess auxin.

Sequential induction of auxin efflux and influx carriers regulates lateral root emergence

Emergence of a new lateral root primordium through the outer layers of the parental root requires the sequential auxin-mediated induction of two auxin transporters. This positive feedback regulatory loop coordinates patterned gene expression in outer tissues.

The emergence of lateral roots through several tissues requires the precise regulation of gene expression in overlaying cells to trigger cell separation.

Auxin derived from new lateral root primordia induces a positive feedback loop in the outer tissues by promoting the expression of the auxin influx transporter LAX3.

A mathematical model based on realistic 3D geometries predicted the involvement of an auxin efflux carrier that was later identified to be PIN3.

The model also revealed that PIN3 must be expressed before LAX3 to ensure a ‘robust' pattern of LAX3 induction in just two overlaying cortical cell files, thereby delimiting cell separation.

In Arabidopsis, lateral roots originate from pericycle cells deep within the primary root. New lateral root primordia (LRP) have to emerge through several overlaying tissues. Here, we report that auxin produced in new LRP is transported towards the outer tissues where it triggers cell separation by inducing both the auxin influx carrier LAX3 and cell-wall enzymes. LAX3 is expressed in just two cell files overlaying new LRP. To understand how this striking pattern of LAX3 expression is regulated, we developed a mathematical model that captures the network regulating its expression and auxin transport within realistic three-dimensional cell and tissue geometries. Our model revealed that, for the LAX3 spatial expression to be robust to natural variations in root tissue geometry, an efflux carrier is required—later identified to be PIN3. To prevent LAX3 from being transiently expressed in multiple cell files, PIN3 and LAX3 must be induced consecutively, which we later demonstrated to be the case. Our study exemplifies how mathematical models can be used to direct experiments to elucidate complex developmental processes.

Reduction of off-flavor generation in soybean homogenates: a mathematical model

The generation of off-flavors in soybean homogenates such as n-hexanal via the lipoxygenase (LOX) pathway can be a problem in the processed food industry. Previous studies have examined the effect of using soybean varieties missing one or more of the 3 LOX isozymes on n-hexanal generation. A dynamic mathematical model of the soybean LOX pathway using ordinary differential equations was constructed using parameters estimated from existing data with the aim of predicting how n-hexanal generation could be reduced. Time-course simulations of LOX-null beans were run and compared with experimental results. Model L(2), L(3), and L(12) beans were within the range relative to the wild type found experimentally, with L(13) and L(23) beans close to the experimental range. Model L(1) beans produced much more n-hexanal relative to the wild type than those in experiments. Sensitivity analysis indicates that reducing the estimated K(m) parameter for LOX isozyme 3 (L-3) would improve the fit between model predictions and experimental results found in the literature. The model also predicts that increasing L-3 or reducing L-2 levels within beans may reduce n-hexanal generation.

PRACTICAL APPLICATION

This work describes the use of mathematics to attempt to quantify the enzyme-catalyzed conversions of compounds in soybean homogenates into undesirable flavors, primarily from the compound n-hexanal. The effect of different soybean genotypes and enzyme kinetic constants was also studied, leading to recommendations on which combinations might minimize off-flavor levels and what further work might be carried out to substantiate these conclusions.

Controls on soil carbon sequestration and dynamics: lessons from land-use change

Soil carbon (C) dynamics and sequestration are controlled by interactions of chemical, physical and biological factors. These factors include biomass quantity and quality, physical environment and the biota. Management can alter these factors in ways that alter C dynamics. We have focused on a range of managed sites with documented land use change from agriculture or grassland to forest. Our results suggest that interactions of soil type, plant and environment impact soil C sequestration. Above and below ground C storage varied widely across sites. Results were related to plant type and calcium on sandy soils in our Northern sites. Predictors of sequestration were more difficult to detect over the temperature range of 12.4°C in the present study. Accrual of litter under pines in the moist Mississippi site limited C storage in a similar manner to our dry Nebraska site. Pre-planting heterogeneity of agricultural fields such as found in Illinois influences C contents. Manipulation of controls on C sequestration such as species planted or amelioration of soil quality before planting within managed sites could increase soil C to provide gains in terrestrial C storage. Cost effective management would also improve soil C pools positively affecting soil fertility and site productivity.

Randomized clinical trial of preoperative intravenous iron sucrose to reduce blood transfusion in anaemic patients after colorectal cancer surgery

BACKGROUND

The transfusion rate following colorectal cancer resection is between 10 and 30 per cent. Receipt of allogeneic blood is not without risk or cost. A preoperative adjunct that reduced the need for transfusion would mitigate these risks. This study was designed to determine whether iron sucrose reduces the likelihood of postoperative blood transfusion in patients undergoing elective colorectal cancer resection.

METHODS

In this randomized prospective blinded placebo-controlled trial of patients undergoing resectional surgery with a preoperative diagnosis of colorectal cancer, 600 mg iron sucrose or placebo was given intravenously in two divided doses, at least 24 h apart, 14 days before surgery. The primary outcome measures were serum haemoglobin concentration, recorded at recruitment, immediately before surgery and at discharge, and perioperative blood transfusions.

RESULTS

No difference was demonstrated between treatment groups (iron sucrose, 34 patients; placebo, 26) for any of the primary outcome measures, for either the whole study population or a subgroup of anaemic patients.

CONCLUSION

This pilot study provided no support for the use of intravenous iron sucrose as a preoperative adjunct to increase preoperative haemoglobin levels and thereby reduce the likelihood of allogeneic blood transfusion for patients undergoing resectional surgery for colorectal cancer.

REGISTRATION NUMBER

2005-003608-13UK (Medicines and Healthcare products Regulatory Agency).

Pre-operative oral iron supplementation reduces blood transfusion in colorectal surgery - a prospective, randomised, controlled trial

INTRODUCTION

Allogeneic blood transfusion confers a risk to the recipient. Recent trials in colorectal surgery have shown that the most significant factors predicting blood transfusion are pre-operative haemoglobin, operative blood loss and presence of a transfusion protocol. We report a randomised, controlled trial of oral ferrous sulphate 200 mg TDS for 2 weeks' pre-operatively versus no iron therapy.

PATIENTS AND METHODS

Patients diagnosed with colorectal cancer were recruited from out-patient clinic and haematological parameters assessed. Randomisation was co-ordinated via a telephone randomisation centre.

RESULTS

Of the 49 patients recruited, 45 underwent colorectal resection. There were no differences between those patients not receiving iron (n = 23) and the iron-supplemented group (n = 22) for haemoglobin at recruitment, operative blood loss, operation duration or length of hospital stay. At admission to hospital, the iron-supplemented group had a higher haemoglobin than the non-iron treated group (mean haemoglobin concentration 13.1 g/dl [range, 9.6-17 g/dl] versus 11.8 g/dl [range, 7.8-14.7 g/dl]; P = 0.040; 95% CI 0.26-0.97) and were less likely to require operative blood transfusion (mean 0 U [range, 0-4 U] versus 2 U [range, 0-11 U] transfused; P = 0.031; 95% CI 0.13-2.59). This represented a cost reduction of 66% (47 U of blood = pound4700 versus oral FeSO(4) at pound30 + 15 U blood at pound1500). At admission, ferritin in the iron-treated group had risen significantly from 40 microg/l (range, 15-222 microg/l) to 73 microg/l (range, 27-386 microg/l; P = 0.0036; 95% CI 46.53-10.57).

CONCLUSIONS

Oral ferrous sulphate given pre-operatively in patients undergoing colorectal surgery offers a simple, inexpensive method of reducing blood transfusions.

Prospective randomized controlled trial of acute normovolaemic haemodilution in major gastrointestinal surgery

BACKGROUND

The efficacy of acute normovolaemic haemodilution (ANH) remains uncertain because of a lack of well-designed prospective randomized controlled trials. The aim of this study was to assess the effects of ANH on allogeneic transfusion, postoperative complications, and duration of stay.

METHODS

Consecutive patients undergoing major gastrointestinal surgery were randomized to a planned 3-unit ANH, or no ANH. Both groups underwent identical management including adherence to a transfusion protocol after surgery. Outcome measures included the number of patients receiving allogeneic blood, complications, and duration of stay.

RESULTS

380 patients were screened of which 160 were included in the study, median age was 62 yr (range 23-90), 'ANH' n=78, 'no ANH' n=82. There was no significant difference between groups in the number of patients receiving allogeneic blood 22/78 (28%) vs 25/82 (30%), the total number of allogeneic units transfused (90 vs 93), complication rate, or duration of stay. Haemodilution significantly increased anaesthetic time, median 55 (range 15-90) vs 40 min (range 17-80) (P<0.001). Significantly fewer patients in the ANH group experienced oliguria in the immediate postoperative period 37/78 (47%) vs 55/82 (67%) (P=0.012). The most significant factors affecting transfusion were blood loss, starting haemoglobin, and age. When compared with ASA-matched historical controls, the introduction of a transfusion protocol reduced the transfusion rate in colorectal patients from 136/333 (41%) to 37/138 (27%), P=0.004.

CONCLUSIONS

In this large pragmatic study, ANH did not affect allogeneic transfusion rate in major gastrointestinal surgery. Preoperative haemoglobin, blood loss, and transfusion protocol are the key factors influencing allogeneic transfusion.

Characteristics of murine histidinaemia and its potential for genetic manipulation

BACKGROUND

Histidinaemia is an autosomal recessive disorder affecting the hepatic enzyme histidine ammonia lyase (histidase) resulting in elevated plasma and urinary histidine and is prototypic of a series of hepatic cytosolic enzyme defects.

AIMS

To characterise the physiology of murine histidinaemia with respect to histidine excretion and catabolism, and explore the potential for manipulating cellular and whole body histidase metabolism by gene transfer.

MATERIALS AND METHODS

We studied his/his mice which have a G to A substitution in the gene encoding histidase, using both in vitro transduction of isolated hepatocytes by lipofection with wild-type histidase cDNA, and in vivo transduction of whole liver using a retroviral construct.

RESULTS AND CONCLUSION

Histidase cDNA expression restored histidase activity in vivo and in vitro towards normal levels, demonstrated both at the cellular level and by whole body metabolic studies, establishing the potential of this model for the development of new gene therapeutic approaches.