A mathematical model for photoreceptor interactions

Mathematical model for rod outer segment dynamics during retinal detachment

Retinal detachment (RD) is the separation of the neural layer from the retinal pigmented epithelium thereby preventing the supply of nutrients to the cells within the neural layer of the retina. In vertebrates, primary photoreceptor cells consisting of rods and cones undergo daily renewal of their outer segment through the addition of disc-like structures and shedding of these discs at their distal end. When the retina detaches, the outer segment of these cells begins to degenerate and, if surgical procedures for reattachment are not done promptly, the cells can die and lead to blindness. The precise effect of RD on the renewal process is not well understood. Additionally, a time frame within which reattachment of the retina can restore proper photoreceptor cell function is not known. Focusing on rod cells, we propose a mathematical model to clarify the influence of retinal detachment on the renewal process. Our model simulation and analysis suggest that RD stops or significantly reduces the formation of new discs and that an alternative removal mechanism is needed to explain the observed degeneration during RD. Sensitivity analysis of our model parameters points to the disc removal rate as the key regulator of the critical time within which retinal reattachment can restore proper photoreceptor cell function.

Spatially-dependent model for rods and cones in the retina

We develop a mathematical model for photoreceptors in the retina. We focus on rod and cone outer segment dynamics and interactions with a nutrient source associated with the retinal pigment epithelium cells. Rod and cone densities (number per unit area of retinal surface) are known to have significant spatial dependence in the retina with cones located primarily near the fovea and the rods located primarily away from the fovea. Our model accounts for this spatial dependence of the rod and cone photoreceptor density as well as for the possibility of nutrient diffusion. We present equilibrium and dynamic solutions, discuss their relation to existing models, and estimate model parameters through comparisons with available experimental measurements of both spatial and temporal photoreceptor characteristics. Our model compares well with existing data on spatially-dependent regrowth of photoreceptor outer segments in the macular region of Rhesus Monkeys. Our predictions are also consistent with existing data on the spatial dependence of photoreceptor outer segment length near the fovea in healthy human subjects. We focus primarily on the healthy eye but our model could be the basis for future efforts designed to explore various retinal pathologies, eye-related injuries, and treatments of these conditions.

A mathematical model of GLUT1 modulation in rods and RPE and its differential impact in cell metabolism

We present a mathematical model of key glucose metabolic pathways in two cells of the human retina: the rods and the retinal pigmented epithelium (RPE). Computational simulations of glucose transporter 1 (GLUT1) inhibition in the model accurately reproduce experimental data from conditional knockout mice and reveal that modification of GLUT1 expression levels of both cells differentially impacts their metabolism. We hypothesize that, under glucose scarcity, the RPE's energy producing pathways are altered in order to preserve its functionality, impacting the photoreceptors' outer segment renewal. On the other hand, when glucose is limited in the rods, aerobic glycolysis is preserved, which maintains the lactate contribution to the RPE.

Inverse Problem Reveals Conditions for Characteristic Retinal Degeneration Patterns in Retinitis Pigmentosa Under the Trophic Factor Hypothesis

Retinitis pigmentosa (RP) is the most common inherited retinal dystrophy with a prevalence of about 1 in 4,000, affecting approximately 1.5 million people worldwide. Patients with RP experience progressive visual field loss as the retina degenerates, destroying light-sensitive photoreceptor cells (rods and cones), with rods affected earlier and more severely than cones. Spatio-temporal patterns of retinal degeneration in human RP have been well characterised; however, the mechanism(s) giving rise to these patterns have not been conclusively determined. One such mechanism, which has received a wealth of experimental support, is described by the trophic factor hypothesis. This hypothesis suggests that rods produce a trophic factor necessary for cone survival; the loss of rods depletes this factor, leading to cone degeneration. In this article, we formulate a partial differential equation mathematical model of RP in one spatial dimension, spanning the region between the retinal centre (fovea) and the retinal edge (ora serrata). Using this model we derive and solve an inverse problem, revealing for the first time experimentally testable conditions under which the trophic factor mechanism will qualitatively recapitulate the spatio-temporal patterns of retinal regeneration observed in human RP.

Insights into pathological mechanisms and interventions revealed by analyzing a mathematical model for cone metabolism

This work analyzes a mathematical model for the metabolic dynamics of a cone photoreceptor, which is the first model to account for energy generation from fatty acids oxidation of shed photoreceptor outer segments (POS). Multiple parameter bifurcation analysis shows that joint variations in external glucose, the efficiency of glucose transporter 1 (GLUT1), lipid utilization for POS renewal, and oxidation of fatty acids affect the cone’s metabolic vitality and its capability to adapt under glucose-deficient conditions. The analysis further reveals that when glucose is scarce, cone viability cannot be sustained by only fueling energy production in the mitochondria, but it also requires supporting anabolic processes to create lipids necessary for cell maintenance and repair. In silico experiments are used to investigate how the duration of glucose deprivation impacts the cell without and with a potential GLUT1 or oxidation of fatty acids intervention as well as a dual intervention. The results show that for prolonged duration of glucose deprivation, the cone metabolic system does not recover with higher oxidation of fatty acids and requires greater effectiveness of GLUT1 to recover. Finally, time-varying global sensitivity analysis (GSA) is applied to assess the sensitivity of the model outputs of interest to changes and uncertainty in the parameters at specific times. The results reveal a critical temporal window where there would be more flexibility for interventions to rescue a cone cell from the detrimental consequences of glucose shortage.

Mathematical Models of Retinitis Pigmentosa: The Trophic Factor Hypothesis

Retinitis pigmentosa (RP) is the term used to denote a group of inherited retinal-degenerative conditions that cause progressive sight loss. Individuals with this condition lose their light-sensitive photoreceptor cells, known as rods and cones, over a period of years to decades; degeneration starting in the retinal periphery, and spreading peripherally and centrally over time. RP is a rod-cone dystrophy, meaning that rod health and function are affected earlier and more severely than that of cones. Rods degenerate due to an underlying mutation, whereas the reasons for cone degeneration are unknown. A number of mechanisms have been proposed to explain secondary cone loss and the spatio-temporal patterns of retinal degeneration in RP. One of the most promising is the trophic factor hypothesis, which suggests that rods produce a factor necessary for cone survival, such that, when rods degenerate, cone degeneration follows. In this paper we formulate and analyse mathematical models of human RP under the trophic factor hypothesis. These models are constructed as systems of reaction-diffusion partial differential equations in one spatial dimension, and are solved and analysed using a combination of numerical and analytical methods. We predict the conditions under which cones will degenerate following the loss of a patch of rods from the retina, the critical trophic factor treatment rate required to prevent cone degeneration following rod loss and the spatio-temporal patterns of cone loss that would result if the trophic factor mechanism alone were responsible for retinal degeneration.

The role of RdCVFL in a mathematical model of photoreceptor interactions

Recent experimental and mathematical work has shown the interdependence of the rod and cone photoreceptors with the retinal pigment epithelium in maintaining sight. Accelerated intake of glucose into the cones via the theoredoxin-like rod-derived cone viability factor (RdCVF) is needed as aerobic glycolysis is the primary source of energy production. Reactive oxidative species (ROS) result from the rod and cone metabolism and recent experimental work has shown that the long form of RdCVF (RdCVFL) helps mitigate the negative effects of ROS. In this work we investigate the role of RdCVFL in maintaining the health of the photoreceptors. The results of our mathematical model show the necessity of RdCVFL and also demonstrate additional stable modes that are present in this system. The sensitivity analysis shows the importance of glucose uptake, nutrient levels, and ROS mitigation in maintaining rod and cone health in light-damaged mouse models. Together, these suggests areas on which to focus treatment in order to prolong the photoreceptors, especially in situations where ROS is a contributing factor to their death such as retinitis pigmentosa.

Optimal control with MANF treatment of photoreceptor degeneration

People afflicted with diseases such as retinitis pigmentosa and age-related macular degeneration experience a decline in vision due to photoreceptor degeneration, which is currently unstoppable and irreversible. Currently there is no cure for diseases linked to photoreceptor degeneration. Recent experimental work showed that mesencephalic astrocyte-derived neurotrophic factor (MANF) can reduce neuron death and, in particular, photoreceptor death by reducing the number of cells that undergo apoptosis. In this work, we build on an existing system of ordinary differential equations that represent photoreceptor interactions and incorporate MANF treatment for three experimental mouse models having undergone varying degrees of photoreceptor degeneration. Using MANF treatment levels as controls, we investigate optimal control results in the three mouse models. In addition, our numerical solutions match the experimentally observed surviving percentage of photoreceptors and our uncertainty and sensitivity analysis identifies significant parameters in the math model both with and without MANF treatment.

A Mathematical Analysis of Aerobic Glycolysis Triggered by Glucose Uptake in Cones

Patients affected by retinitis pigmentosa, an inherited retinal disease, experience a decline in vision due to photoreceptor degeneration leading to irreversible blindness. Rod-derived cone viability factor (RdCVF) is the most promising mutation-independent treatment today. To identify pathologic processes leading to secondary cone photoreceptor dysfunction triggering central vision loss of these patients, we model the stimulation by RdCVF of glucose uptake in cones and glucose metabolism by aerobic glycolysis. We develop a nonlinear system of enzymatic functions and differential equations to mathematically model molecular and cellular interactions in a cone. We use uncertainty and sensitivity analysis to identify processes that have the largest effect on the system and their timeframes. We consider the case of a healthy cone, a cone with low levels of glucose, and a cone with low and no RdCVF. The three key processes identified are metabolism of fructose-1,6-bisphosphate, production of glycerol-3-phosphate and competition that rods exert on cone resources. The first two processes are proportional to the partition of the carbon flux between glycolysis and the pentose phosphate pathway or the Kennedy pathway, respectively. The last process is the rods' competition for glucose, which may explain why rods also provide the RdCVF signal to compensate.

A Rare Form of Retinal Dystrophy Caused by Hypomorphic Nonsense Mutations in CEP290

PURPOSE

To identify the gene defect and to study the clinical characteristics and natural course of disease in a family originally diagnosed with oligocone trichromacy (OT), a rare congenital cone dysfunction syndrome.

METHODS

Extensive clinical and ophthalmologic assessment was performed on two siblings with OT and long-term follow up data were analyzed. Subsequently, whole exome sequencing (WES) and Sanger sequence analysis of CEP290 was performed in the two siblings. Additionally, the identified CEP290 mutations were analyzed in persons with achromatopsia (ACHM) (n = 23) and autosomal recessive or isolated cone dystrophy (CD; n = 145).

RESULTS

In the first decade of life, the siblings were diagnosed with OT based on low visual acuity, photophobia, nystagmus, and absent cone response on electroretinography , but with normal color discrimination. Over time, the phenotype of OT evolved to a progressive degenerative disease without any CEP290-associated non-ocular features. In both siblings, two nonsense mutations (c.451C>T; p.(Arg151*) and c.4723A>T; p.(Lys1575*)) in CEP290 were found. Previously, p.(Arg151*) was demonstrated to induce nonsense-mediated alternative splicing events leading to intact open reading frames of the resulting mRNA products (p.(Leu148_Glu165del) and p.(Leu148_Lys172del)). mRNA analysis for p.(Lys1575*) confirmed a suspected hypomorphic character, as exon 36 skipping was observed in a small fraction of CEP290 mRNA, resulting in a 36 aa in-frame deletion (p.(Glu1569_Trp1604del)). No additional cases carrying these variants were identified in the ACHM and CD cohorts.

CONCLUSIONS

Compound heterozygous hypomorphic mutations in CEP290 may lead to a rare form of cone-dominated retinal dystrophy, a novel phenotype belonging to the CEP290-associated spectrum of ciliopathies. These findings provide insight into the effect of CEP290 mutations on the clinical phenotype.

Quantifying the metabolic contribution to photoreceptor death in retinitis pigmentosa via a mathematical model

Retinitis pigmentosa (RP) is a family of inherited retinal degenerative diseases that leads to blindness. In many cases the disease-causing allele encodes for a gene exclusively expressed in the night active rod photoreceptors. However, because rod death always leads to cone death affected individuals eventually lose their sight. Many theories have been proposed to explain the secondary loss of cones in RP; however, most fail to fully explain the different pathological transition stages seen in humans. Incorporating experimental data of rod and cone death kinetics from two mouse models of RP, we use a mathematical model to investigate the interplay and role of energy consumption and uptake of the photoreceptors as well as nutrient availability supplied through the retinal pigment epithelium (RPE) throughout the progression of RP. Our data driven mathematical model predicts that the system requires a total reduction of approximately 27-31% in nutrients available to result in the complete demise of all cones. Simulations utilizing retinal degeneration 1 (rd1) mouse cell count data in which cone death was delayed by altering cell metabolism in cones show that preventing a 1-2% decrease in nutrients available can permanently halt cone death even when 90% have already died. Our results also indicate that the ratio of energy consumption to uptake of cones, Dc, is mainly disrupted during the death wave of the rods with negligible changes thereafter and that the subsequent nutrient decrease is mainly responsible for the demise of the cones. The change in this ratio Dc highlights the compensation that the cones must undergo during rod death to meet the high metabolic demands of the entire photoreceptor population. Global sensitivity analysis confirms the results and suggests areas of focus for halting RP, even at later stages of the disease, through feasible therapeutic interventions.

Mathematical and computational models of the retina in health, development and disease

The retina confers upon us the gift of vision, enabling us to perceive the world in a manner unparalleled by any other tissue. Experimental and clinical studies have provided great insight into the physiology and biochemistry of the retina; however, there are questions which cannot be answered using these methods alone. Mathematical and computational techniques can provide complementary insight into this inherently complex and nonlinear system. They allow us to characterise and predict the behaviour of the retina, as well as to test hypotheses which are experimentally intractable. In this review, we survey some of the key theoretical models of the retina in the healthy, developmental and diseased states. The main insights derived from each of these modelling studies are highlighted, as are model predictions which have yet to be tested, and data which need to be gathered to inform future modelling work. Possible directions for future research are also discussed. Whilst the present modelling studies have achieved great success in unravelling the workings of the retina, they have yet to achieve their full potential. For this to happen, greater involvement with the modelling community is required, and stronger collaborations forged between experimentalists, clinicians and theoreticians. It is hoped that, in addition to bringing the fruits of current modelling studies to the attention of the ophthalmological community, this review will encourage many such future collaborations.

CNGA3 deficiency affects cone synaptic terminal structure and function and leads to secondary rod dysfunction and degeneration

PURPOSE

To investigate rod function and survival after cone dysfunction and degeneration in a mouse model of cone cyclic nucleotide-gated (CNG) channel deficiency.

METHODS

Rod function and survival in mice with cone CNG channel subunit CNGA3 deficiency (CNGA3-/- mice) were evaluated by electroretinographic (ERG), morphometric, and Western blot analyses. The arrangement, integrity, and ultrastructure of photoreceptor terminals were investigated by immunohistochemistry and electron microscopy.

RESULTS

The authors found loss of cone function and cone death accompanied by impairment of rods and rod-driven signaling in CNGA3-/- mice. Scotopic ERG b-wave amplitudes were reduced by 15% at 1 month, 30% at 6 months, and 40% at 9 months and older, while scotopic a-wave amplitudes were decreased by 20% at 9 months, compared with ERGs of age-matched wild-type mice. Outer nuclear layer thickness in CNGA3-/- retina was reduced by 15% at 12 months compared with age-matched wild-type controls. This was accompanied by a 30%-40% reduction in expression of rod-specific proteins, including rhodopsin, rod transducin α-subunit, and glutamic acid-rich protein (GARP). Cone terminals in the CNGA3-/- retina showed a progressive loss of neurochemical and ultrastructural integrity. Abnormalities were observed as early as 1 month. Disorganized rod terminal ultrastructure was noted by 12 months.

CONCLUSIONS

These findings demonstrate secondary rod impairment and degeneration after cone degeneration in mice with cone CNG channel deficiency. Loss of cone phototransduction accompanies the compromised integrity of cone terminals. With time, rod synaptic structure, function, and viability also become compromised.

℮-conome: an automated tissue counting platform of cone photoreceptors for rodent models of retinitis pigmentosa

Background

Retinitis pigmentosa is characterized by the sequential loss of rod and cone photoreceptors. The preservation of cones would prevent blindness due to their essential role in human vision. Rod-derived Cone Viability Factor is a thioredoxin-like protein that is secreted by rods and is involved in cone survival. To validate the activity of Rod-derived Cone Viability Factors (RdCVFs) as therapeutic agents for treating retinitis Pigmentosa, we have developed e-conome, an automated cell counting platform for retinal flat mounts of rodent models of cone degeneration. This automated quantification method allows for faster data analysis thereby accelerating translational research.

Methods

An inverted fluorescent microscope, motorized and coupled to a CCD camera records images of cones labeled with fluorescent peanut agglutinin lectin on flat-mounted retinas. In an average of 300 fields per retina, nine Z-planes at magnification X40 are acquired after two-stage autofocus individually for each field. The projection of the stack of 9 images is subject to a threshold, filtered to exclude aberrant images based on preset variables. The cones are identified by treating the resulting image using 13 variables empirically determined. The cone density is calculated over the 300 fields.

Results

The method was validated by comparison to the conventional stereological counting. The decrease in cone density in rd1 mouse was found to be equivalent to the decrease determined by stereological counting. We also studied the spatiotemporal pattern of the degeneration of cones in the rd1 mouse and show that while the reduction in cone density starts in the central part of the retina, cone degeneration progresses at the same speed over the whole retinal surface. We finally show that for mice with an inactivation of the Nucleoredoxin-like genes Nxnl1 or Nxnl2 encoding RdCVFs, the loss of cones is more pronounced in the ventral retina.

Conclusion

The automated platform ℮-conome used here for retinal disease is a tool that can broadly accelerate translational research for neurodegenerative diseases.