Model Status
This CellML model cannot be opened in COR and it cannot be run in PCEnv due to the use of vectors - something which may be included in the CellML 1.2 specification. The way this model is written it is not ideally suited for expression in CellML. However, we have attempted to describe the model in CellML with the hope that in the future it may be further validated and curated.
Model Structure
ABSTRACT: In this paper, concepts from network automata are adapted and extended to model complex biological systems. Specifically, systems of nephrons, the operational units of the kidney, are modelled and the dynamics of such systems are explored. Nephron behaviour can fluctuate widely and, under certain conditions, become chaotic. However, the behaviour of the whole kidney remains remarkably stable and blood solute levels are maintained under a wide range of conditions even when many nephrons are damaged or lost. A network model is used to investigate the stability of systems of nephrons and interactions between nephrons. More sophisticated dynamics are explored including the observed oscillations in single nephron filtration rates and the development of stable ionic and osmotic gradients in the inner medulla which contribute to the countercurrent exchange mechanism. We have used the model to explore the effects of changes in input parameters including hydrostatic and osmotic pressures and concentrations of ions, such as sodium and chloride. The intrinsic nephron control, tubuloglomerular feedback, is included and the effects of coupling between nephrons are explored in two-, eight- and 72-nephron models.
The original paper reference is cited below:
A computational model for emergent dynamics in the kidney, Moss R, Kazmierczak E, Kirley M, and Harris P, 2009,
Philosophical Transactions of the Royal Society A
, 367, 2125-40.
PubMed ID: 19414449
Schematic diagram representing the model for the single-nephron tubule as a network automata, showing the edges that capture fluid flow, solute transport and tubulo-glomerular feedback.
$t$
$t=\mathrm{time}$
$\mathrm{delta\_t}=1.0$
$\mathrm{R\_A}_{t+\mathrm{delta\_t}}=\mathrm{epsilon}_{t}e^{\mathrm{gamma}\mathrm{H\_A}}$
$\mathrm{P\_G}_{t+\mathrm{delta\_t}}=\frac{\mathrm{P\_V}+\mathrm{P\_A}\frac{\mathrm{R\_E}_{t}}{\mathrm{R\_A}_{t}}\frac{\mathrm{C\_A}_{t}}{\mathrm{C\_E}_{t}}}{(1.0+\frac{\mathrm{R\_E}_{t}}{\mathrm{R\_A}_{t}})\frac{\mathrm{C\_A}_{t}}{\mathrm{C\_E}_{t}}}$
$\mathrm{phi}_{t+\mathrm{delta\_t}}=\mathrm{epsilon\_max}-\frac{\mathrm{psi}}{1.0+e^{\mathrm{kappa}\mathrm{Na\_M}_{t-\mathrm{D\_TGF}}-\mathrm{Na\_half}}}$
$\mathrm{epsilon}_{t+\mathrm{delta\_t}}=\frac{\mathrm{phi}_{t}+\mathrm{epsilon}_{t}(\frac{2.0\mathrm{zeta}}{\mathrm{omega}\mathrm{delta}}+\frac{2.0}{(\mathrm{omega}\mathrm{delta})^{2.0}})-\frac{\mathrm{epsilon}_{t-\mathrm{delta}}}{(\mathrm{omega}\mathrm{delta})^{2.0}}}{1.0+\frac{2.0\mathrm{zeta}}{\mathrm{omega}\mathrm{delta}}+\frac{1.0}{(\mathrm{omega}\mathrm{delta})^{2.0}}}$
$\mathrm{pi\_G}=\mathrm{pi}\frac{\mathrm{C\_A}+\mathrm{C\_E}}{2.0}$
$\mathrm{pi}_{C}=aC+bC^{2.0}$
$\mathrm{SNGFR}=\mathrm{Kf}(\mathrm{P\_G}-\mathrm{P\_BC}+\mathrm{pi\_G})$
$\mathrm{Na\_I}_{t}=\frac{\mathrm{Na\_R}_{t}}{0.67}$
$\mathrm{H2O\_I}_{t}=\frac{\mathrm{H2O\_R}_{t}}{0.67}$
$\mathrm{H2O\_D\_M}_{t}=\mathrm{H2O\_D\_M}_{t+\mathrm{delta\_t}}$
$\mathrm{Na\_D\_M}_{t}=\mathrm{Na\_D\_M}_{t+\mathrm{delta\_t}}$
$\mathrm{Na\_D}_{t}=\frac{\mathrm{Na\_M}_{t}}{\mathrm{H2O\_M}_{t+\mathrm{delta\_t}}}\mathrm{H2O\_D}_{t+\mathrm{delta\_t}}$
$\mathrm{Na\_M}_{t+\mathrm{delta\_t}}=\mathrm{H2O\_M}_{t}(\frac{\mathrm{Na\_A}_{t+\mathrm{delta\_t}}}{\mathrm{H2O\_A}_{t}}+\mathrm{Na\_G})$
$\mathrm{H2O\_D}_{t+\mathrm{delta\_t}}=\frac{\mathrm{Na\_D}_{t}}{\mathrm{Na\_D\_M}_{t}}\mathrm{H2O\_D\_M}_{t}$
$\mathrm{H2O\_M}_{t+\mathrm{delta\_t}}=\frac{\mathrm{Na\_M}_{t}}{\mathrm{Na\_D\_M}_{t}}\mathrm{H2O\_D\_M}_{t}$
$\mathrm{Na\_A}_{t+\mathrm{delta\_t}}=\frac{\mathrm{H2O\_A}_{t}}{\mathrm{H2O\_M\_A}_{t+\mathrm{delta\_t}}}(\mathrm{Na\_M\_A}_{t}-\mathrm{Na\_G}_{t}\mathrm{H2O\_M}_{t})$
$\mathrm{Na\_G}=\mathrm{Na\_M}-\mathrm{Na\_A}$
$\mathrm{Na\_M\_A}=\mathrm{Na\_M}+\mathrm{Na\_A}$
$\mathrm{H2O\_M\_A}=\mathrm{H2O\_M}+\mathrm{H2O\_A}$
$\mathrm{R\_ADH}=\frac{\lg (\mathrm{ADH\_I}\times 1E12)}{2.0}$
$\mathrm{R\_ALD}=\frac{\lg (\mathrm{ALD\_I}\times 1E11)}{3.0}$
$\mathrm{H2O\_R}=\mathrm{R\_ADH}(\mathrm{H2O\_I}-\mathrm{Na\_I}\frac{\mathrm{H2O\_I\_IF}}{\mathrm{Na\_I\_IF}})$
$\mathrm{Na\_R}=\mathrm{R\_ALD}\mathrm{Na\_max}\mathrm{Na\_I}$
Species
Gene
sodium concentration gradient between the interstitial fluid and the ascending limb of the loop of Henle
water filtrate in the proximal tubule
water in the ascending limb of the loop of Henle
venous pressure in the glomerulus
damping coefficient in the TGF signal
sodium concentration in the interstitial fluid of the loop of Henle
sodium concentration in the interstitial fluid of the loop of Henle
concentration of sodium filtrate in the proximal tubule
Robert Moss
ED Kazmierczak
Michel Kirley
Peter Harris
water filtrate in the proximal tubule
sodium concentration in the interstitial fluid of the loop of Henle
water in the interstitial fluid of the loop of Henle
concentration of aldosterone in the late distal tubule
water in the interstitial fluid of the loop of Henle
resistance of the afferent arteriole
Moss network model
tubuloglomerular feedback signal
pressure in the distal tubule ("at the start of the tubule" is mentioned in the paper, I assume distal tubule!)
sodium concentration in the interstitial fluid of the loop of Henle
emergent dynamics
nephron models
viscous resistance coefficient in the glomerulus
sodium concentration in the interstitial fluid and the ascending limb of the loop of Henle
water conservation (quantity) in the descending limb and interstitial fluid of the loop of Henle
water conservation (quantity) in the descending limb and interstitial fluid of the loop of Henle
sodium concentration in the interstitial fluid of the loop of Henle
concentration of protein (blood plasma) in the afferent arteriole (?? "plasma membrane protein" from FMA??)
concentration of sodium in the descending limb and interstitial fluid of the loop of Henle
sodium concentration in the interstitial fluid and the ascending limb of the loop of Henle
sodium concentration in the ascending limb of the loop of Henle
concentration of antidiuretic hormone (ADH) in the distal tubule (ADH is also known as vasopressin)
concentration of antidiuretic hormone (ADH) in the distal tubule (ADH is also known as vasopressin)
sodium concentration in the interstitial fluid of the loop of Henle
natural frequency (angular frequency) of the TGF mechanism
concentration of protein (blood plasma) in the afferent arteriole (?? "plasma membrane protein" from FMA??)
water in the ascending limb of the loop of Henle
concentration of protein (blood plasma) in the afferent arteriole (?? "plasma membrane protein" from FMA??)
water conservation (quantity) in the descending limb and interstitial fluid of the loop of Henle
water in the descending limb of the loop of Henle
tubuloglomerular feedback signal
concentration of sodium in the descending limb and interstitial fluid of the loop of Henle
response of the afferent arteriole to the tubuloglomerular feedback signal
concentration of aldosterone in the late distal tubule
delay in sending the TGF signal
concentration of sodium delivery in the glomerulus for which the TGF response is half maximal
water in the ascending limb of the loop of Henle
hydrostatic pressure in the glomerulus
In this paper, concepts from network automata are adapted and extended to model complex biological systems. Specifically, systems of nephrons, the operational units of the kidney, are modelled and the dynamics of such systems are explored. Nephron behaviour can fluctuate widely and, under certain conditions, become chaotic. However, the behaviour of the whole kidney remains remarkably stable and blood solute levels are maintained under a wide range of conditions even when many nephrons are damaged or lost. A network model is used to investigate the stability of systems of nephrons and interactions between nephrons. More sophisticated dynamics are explored including the observed oscillations in single nephron filtration rates and the development of stable ionic and osmotic gradients in the inner medulla which contribute to the countercurrent exchange mechanism. We have used the model to explore the effects of changes in input parameters including hydrostatic and osmotic pressures and concentrations of ions, such as sodium and chloride. The intrinsic nephron control, tubuloglomerular feedback, is included and the effects of coupling between nephrons are explored in two-, eight- and 72-nephron models.
arterial pressure in the glomerulus
water in the interstitial fluid of the loop of Henle
range constant for the TGF signal
A computational model for emergent dynamics in the kidney
water in the interstitial fluid of the loop of Henle
sodium concentration in the descending limb of the loop of Henle
resistance of the efferent arteriole
sodium concentration in the ascending limb of the loop of Henle
resistance of the afferent arteriole
sodium concentration in the ascending limb of the loop of Henle
oncotic pressure in the glomerulus
sodium concentration in the descending limb of the loop of Henle
sodium concentration in the ascending limb of the loop of Henle
sodium concentration gradient between the interstitial fluid and the ascending limb of the loop of Henle
water in the interstitial fluid and the ascending limb of the loop of Henle
concentration of sodium filtrate in the proximal tubule
concentration of protein (bloood pasma) in the efferent arteriole (?? "plasma membrane protein" from FMA??)
water in the descending limb of the loop of Henle
concentration of maximal amount of sodium reabsorption that can be achieved by the late distal tubule
concentration of sodium filtrate in the proximal tubule
concentration of sodium in the descending limb and interstitial fluid of the loop of Henle
water in the interstitial fluid of the loop of Henle
hydrostatic pressure in the glomerulus
sodium concentration gradient between the interstitial fluid and the ascending limb of the loop of Henle
compartment2
compartment3
concentration of protein (bloood pasma) in the efferent arteriole (?? "plasma membrane protein" from FMA??)
water in the ascending limb of the loop of Henle
oncotic pressure in the glomerulus
permeability constant for the glomerulus from blood in lumen of glomerulus to ultrafiltrate in lumen of bowman's capsule
single nephron glomerular filtration rate (SNGFR)
average afferent haematocrit (the ratio of the volume of red blood cells to the total volume of blood) for the glomerulus
concentration of protein (bloood pasma) in the efferent arteriole (?? "plasma membrane protein" from FMA??)
water in the interstitial fluid and the ascending limb of the loop of Henle