Elsevier

Mathematical Biosciences

Volume 258, December 2014, Pages 33-43
Mathematical Biosciences

Modeling the dynamics of hypoxia inducible factor-1α (HIF-1α) within single cells and 3D cell culture systems

https://doi.org/10.1016/j.mbs.2014.09.007Get rights and content

Highlights

  • A mathematical model is developed to describe single-cell HIF-1α-hypoxic response.

  • Oxygen measurements are obtained in tumorspheres subject to a hypoxic shock.

  • A mathematical model for the oxygen and HIF-1α dynamics in spheres is developed.

  • High HIF-1α levels are predicted at surface of sphere subject to hypoxic shock.

  • Differential dynamics of transcriptional targets of HIF-1 are predicted.

Abstract

HIF (hypoxia inducible factor) is an oxygen-regulated transcription factor that mediates the intracellular response to hypoxia in human cells. There is increasing evidence that cell signaling pathways encode temporal information, and thus cell fate may be determined by the dynamics of protein levels. We have developed a mathematical model to describe the transient dynamics of the HIF-1α protein measured in single cells subjected to hypoxic shock. The essential characteristics of these data are modeled with a system of differential equations describing the feedback inhibition between HIF-1α and prolyl hydroxylases (PHD) oxygen sensors. Heterogeneity in the single-cell data is accounted through parameter variation in the model. We previously identified the PHD2 isoform as the main PHD sensor responsible for controlling the HIF-1α transient response, and make here testable predictions regarding HIF-1α dynamics subject to repetitive hypoxic pulses. The model is further developed to describe the dynamics of HIF-1α in cells cultured as 3D spheroids, with oxygen dynamics parameterized using experimental measurements of oxygen within spheroids. We show that the dynamics of HIF-1α and transcriptional targets of HIF-1α display a non-monotone response to the oxygen dynamics. Specifically we demonstrate that the dynamic transient behavior of HIF-1α results in differential dynamics in transcriptional targets.

Introduction

Oxygen homeostasis is crucial for the normal function and maintenance of respiring cells. The result of an insufficient supply of oxygen to the cell is hypoxia, a condition that plays a key role in a number of human pathologies. Hypoxia inducible factor (HIF) family members are transcription factors that mediate the intracellular hypoxic response. During hypoxia, HIF induces the transcription of a series of genes involved in diverse adaptive functions such as angiogenesis, glycolysis, cell proliferation and iron metabolism [1]. HIF is a heterodimer, comprised of an oxygen-regulated α subunit and a constitutive β subunit. Dimerization between the two subunits is necessary for DNA binding. In normoxic conditions prolyl hydroxylases (PHDs) catalyze the hydroxylation of HIFα, promoting its proteasomal destruction. Since PHDs require molecular oxygen in order to hydroxylate HIFα, in hypoxic conditions, hydroxylation is decreased. This results in HIFα stabilization and increased transcriptional activity. There is a feedback loop in this system as HIF also induces the transcription of the PHDs. This increase in PHD levels can compensate for the reduction of their activity when oxygen availability drops [2].

Mathematical models of biochemical networks have improved our understanding of biological phenomena. In particular, feedback models have provided insight into robust biological adaptation [3] and into dynamic oscillatory and pulsatile behavior. Examples of dynamic behavior include models of circadian rhythms [4], the cell cycle [5], and the dynamic behavior of key regulatory transcription factors [6], [7]. Cellular signaling pathways are typically very complex, and models must be designed to tackle the scientific questions being addressed subject to the available experimental data. Dynamic feedback models are typically expressed as systems of ordinary or stochastic differential equations, or hybrid combinations thereof. For example stochastic models may be necessary to investigate heterogeneity in single-cell imaging data [8], whereas minimal deterministic models may be sufficient to probe the dynamic properties of biological oscillators [9]. Calibrating or fitting the models to the data is a mathematically complex process, which again depends on the scientific questions being addressed and the experimental data available. Within the context of transcription factor pathways, a system perturbation by a given stimulus that produces an oscillatory or pulsatile response is particularly suitable for feedback modeling. Determining whether experimental data are sufficient to parameterize a model [10], and how to include parameter variability [11] are important concerns when developing dynamic feedback models.

Motivated by novel experimental data describing the dynamics of the HIF-1α protein at the level of single cells, we previously developed a simple mathematical model to capture the HIF-1α-PHD negative feedback [12]. This mathematical model was built based on live imaging experiments of single cells experiencing a single hypoxic transition. Parameters in the model were optimized through a combination of fitting to single-cell dynamic data, and through data collected from additional experiments. While developing this model, it became apparent that it was necessary to explicitly distinguish the PHD isoforms and thus we developed an extended model which included the isoforms PHD1, 2 and 3. Using this extended model we previously identified PHD2 as the main PHD responsible for HIF-1α peak duration [12]. Here, in theory (3.1 Two-component negative feedback model, 3.2 Parameter optimization, 3.3 Extension of model to four components), and results (Section 4.1), we provide more extensive details of model development and calibration. In results (Section 4.2) we then provide additional validation of the model by comparing model predictions with new experimental data (described in Section 2.1) investigating the response of cells to pulsatile hypoxic stimulation.

While our previous work focused on single cells experiencing a single hypoxic transition, in tissues cells respond to continuous temporal changes in oxygen as well as spatial gradients. For example, the vascularization and oxygenation status of solid tumors vary over time due to the dynamic process whereby new blood vessels are formed and sub-functional vessels collapse [13]. The transient cycles of hypoxia-reoxygenation (intermittent hypoxia) that are known to occur in solid tumors is a poorly appreciated therapeutic problem and it is associated with resistance to radiation therapy and impaired delivery of chemotherapeutic agents [14]. Moreover, cells will experience different levels of oxygenation, depending on their proximity to blood vessels. These can also vary over time due to cell migration within a tissue. Hence, due to a combination of spatial and temporal factors, cells will experience constant and complex changes in their oxygenation.

The results presented in this paper explore the implications of HIF-1α temporal dynamics within a spatial setting. In order to achieve this, we used experimental data which measured the oxygen concentration within a 3D system of cancerous neuroblastoma cells forming tumorspheres (described in Section 2.2). These tumorspheres were subjected to different degrees of hypoxia and oxygen concentration distributions within the spheres were measured using phosphorescent Pt-porphyrin based oxygen nanosensors [15], [16]. These data were used to parameterize a spatial reaction–diffusion model for oxygen concentration (3.4 Model for oxygen concentration, 4.3 The oxygen dynamics in a tumorsphere can be captured by a diffusion model), which was then coupled to the intracellular model of the HIF-1α-PHD feedback loop (Section 3.5). By explicitly incorporating the dynamic behavior of HIF-1α, instead of assuming that the HIF-1α concentration takes an equilibrium value dependent on oxygen concentration, we uncovered unexpected HIF-1α and PHD spatio-temporal dynamics. Specifically our model predicts that in a tumorsphere, HIF-1α levels may display overshoot dynamics near the surface even though the absolute oxygen levels may be higher at the surface than in the center (Section 4.4). We also used the model to predict how the proteins PHD2 and PHD3, which are transcriptional targets of HIF-1, display different temporal dynamics at different spatial locations in the tumorsphere (Section 4.5).

Section snippets

Single-cell imaging of HIF-1α and ODD

Cell culture and hypoxic incubation: HeLa cells were grown in Dulbecco's Modified Eagle's Medium (DMEM) supplemented with 10% fetal calf serum (FCS) (v/v) and 1% non essential amino acids (v/v), at 37°C, 5% CO2. Cells (between passages 8–20) were plated at 1 × 105 cells⋅ml−1. For imaging experiments, cells were plated in 35 mm glass bottom dishes (Greiner bio-one, UK). Hypoxic incubation was performed directly on the microscope stage equipped with a PeCon incubator with an O2 controller. The

Two-component negative feedback model

We consider the following minimal feedback-model: HIF-1α (x) induces the transcription of PHD (y) at rate k and HIF-1α (x) is degraded via PHD (y) dependent hydroxylation at a maximal rate h. To prevent elimination of HIF-1α (x) and unbounded growth of PHD (y) we further suppose HIF-1α (x) is produced through basal synthesis at rate S and PHD (y) is degraded at rate d. The sensitivity to oxygen in this model is represented by the function h(C), where C is oxygen concentration, which is the rate

A minimal two-component model for HIF-1α-PHD negative feedback can capture dynamic single-cell data

Numerical simulations of model (1) were compared with experimental data [12]. By varying parameters in the model, we previously obtained a good fit to the experimental bell-shaped data (Fig. 1, adapted from [12]). Furthermore, in most cases, the optimal parameters were independent of initial estimates for the parameters, thus suggesting that we have found a unique global optimal set. Specifically, Fig. 2shows the results of the parameter optimization, in which 50 initial sets of parameter were

Conclusions

In vivo, cells do not experience hypoxia as a binary on/off switch; instead cells respond to a range of spatial and temporal gradients in oxygen, mediated through a dynamic signaling pathway. A key novelty was to demonstrate that the protein HIF-1α does not simply track oxygen levels, but displays transient overshoots in response to a hypoxic shock. By developing a model based on this single-cell experimental data, we have been able to predict how the dynamics of HIF-1α are affected by a range

Acknowledgments

J.L. was a recipient of a University of Liverpool studentship. V.S. was a recipient of a BBSRC David Phillips fellowship (BB/C520471/1). J.B. was a recipient of a BBSRC DTG studentship. A.H. has been funded by the Neuroblastoma Society and by the Alder Hey Oncology Fund ref. no. 8098.

References (35)

  • GoldbeterA.

    A model for circadian oscillations in the Drosophila period protein (Per)

    Proc. Roy. Soc. B

    (1995)
  • GerardC. et al.

    Temporal self-organization of the cyclin/Cdk network driving the mammalian cell cycle

    Proc. Natl. Acad. Sci. USA

    (2009)
  • NelsonD.E.

    Oscillations in NF-kappa B signaling control the dynamics of gene expression

    Science

    (2004)
  • Geva-ZatorskyN.

    Oscillations and variability in the p53 system

    Mol. Syst. Biol.

    (2006)
  • AshallL.

    Pulsatile stimulation determines timing and specificity of NF-kappa B-dependent transcription

    Science

    (2009)
  • GerardC. et al.

    A skeleton model for the network of cyclin-dependent kinases driving the mammalian cell cycle

    Interf. Focus

    (2011)
  • Balsa-CantoE. et al.

    An iterative identification procedure for dynamic modeling of biochemical networks

    BMC Syst. Biol.

    (2010)
  • Cited by (23)

    • Modeling hypoxia-related inflammation scenarios

      2023, Mathematical Biosciences
      Citation Excerpt :

      Then, we will include HIF2 and PHD3 to conclude with the dynamics of the full problem discussing the possible scenarios linking hypoxia and inflammation. Focusing on Eqs. (10) and (11), we will now use experimental data (i) to check the fitting between experimental data and model output on the kinetics of HIF1 and PHD2 and (ii) to better estimate the production rates of the two chemicals, which in the literature have been so far evaluated through preliminary simulations only (see [4,12,18]). Then, being aware of the variability of response from tissue to tissue and even from cell to cell within the same cell type, we will analyze the dependence of the response on the parameters.

    • In silico-guided optimisation of oxygen gradients in hepatic spheroids

      2019, Computational Toxicology
      Citation Excerpt :

      The importance of understanding the local environment and mechanisms within tumour spheroids to develop medical applications has led to considerable mathematical modelling efforts in this area. Some work has focused on reproducing growth and development processes [8,20], while others investigate the effects of various oxygen consumption kinetics in different geometries and hypoxic environments [19,18,31]. Acknowledgement of the importance of spatial dynamics in spheroids has also led to mathematical models that simulate 3D pharmacological processes such as drug delivery and metabolism [46,37].

    • Attenuation of doxorubicin-induced cardiotoxicity in a human in vitro cardiac model by the induction of the NRF-2 pathway

      2019, Biomedicine and Pharmacotherapy
      Citation Excerpt :

      A Michaelis-Menten oxygen uptake term was assumed with maximal oxygen consumption rates (Vmax) for the three cell types taken from a study by Sekine et al. [63] which estimated oxygen consumption rates of 1.97 × 10−16, 5.28 × 10-17, 6.11 × 10-17 mol/s/cell, respectively, for human iPSC-CM, cardiac fibroblasts (FB) and human cardiac microvascular endothelial cells (EC). The Michaelis-Menten half maximal parameter (Km) and the rate of oxygen diffusion within the spheroid were taken to be 6.24 × 10-3 mol/m3 [64] and 6.33 × 10-10 m2/sec [65], respectively. Cell densities were calculated using the average spheroid radius of 88 μm which comprises of a total of 498 cells in a mixed ratio of 4:2:1 for CM:FB:EC (285:142:71 cells) – giving 9.89 × 1013, 4.97 × 1013, 2.487 × 1013 cells/m3 for CM, FB and EC cells, respectively.

    • Phosphorescence based O<inf>2</inf> sensors – Essential tools for monitoring cell and tissue oxygenation and its impact on metabolism

      2016, Free Radical Biology and Medicine
      Citation Excerpt :

      Oxygenation at the cell monolayer or within a complex 3D tissue model usually differs significantly from atmospheric or macroscopic O2 levels, because cells consume dissolved O2 continuously, acting as an O2 sink and creating O2 gradients [17,18]. Thus, a static non-perfused monolayer cell culture (2D) with a layer of medium and headspace as O2 reservoir (Fig. 4A) can be described in simple diffusion-uptake terms [18,19]. The steady-state O2 gradient, which propagates from bottom to top of the sample, is determined by a number of parameters, which include: atmospheric [O2], thickness of the medium layer and cell/tissue layer, cell respiration activity, rate of O2 diffusion in media, temperature, etc.

    • Three-dimensional simulations of the cell growth and cytokinesis using the immersed boundary method

      2016, Mathematical Biosciences
      Citation Excerpt :

      Eventually, the cell divides into two daughter cells. Until recently, many studies have focused on cell motion and cell cytokinesis [3–24]. Among these studies, determining the site of cell division is an active problem in cell biology.

    View all citing articles on Scopus
    1

    Present address: Faculty of Life Sciences, University of Manchester, Manchester M13 9PT, UK.

    2

    Present address: B. Braun Melsungen AG, Graz, Austria.

    View full text