Coupled nephrons

Fig. 12.17 Tubular pressure variations for a pair of coupled nephrons in a normotensive rat. The pressure variations remain nearly in phase forthe entire observation time (or 25 periods of oscillation).

In order to examine the synchronization phenomena that can arise in larger ensembles of nephrons, we recently developed a model of a vascular-coupled nephron tree [35], focusing on the effect of the hemodynamic coupling. As explained above, the idea is here that, as one nephron reduces its arterioler diameter to lower the incoming blood flow, more blood is distributed to the other nephrons in accordance with the flow resistances in the network. An interesting aspect of this particular coupling is that the nephrons interact both via the blood flow that controls their tendency to oscillate and via the oscillations in this blood flow that control their tendency to synchronize. We refer to such a structure as a resource distribution chain, and we have shown that phenomena similar to those that we describe here... 

Fig. 12.19 Left sketch of a vascular-coupled nephron tree including the interlobular artery, the afferent arterioles and the glomeruli. Right oscillation amplitudes as function of the arterial pressure and the position of the branching point along the vascular tree.

The mathematical model of our vascular-coupled nephron tree thus consists of (i) 12 sets of coupled ODEs describing individual nephrons, (ii) a set of linear algebraic equations that determines the blood pressure drop from one branching point to another, and (iii) algebraic relations for the vascular interaction. 

In the aldosterone-sensitive distal nephron (ASDN), ENaC-mediated Na+ absorption under the control of aldosterone is critical to balance urinary Na+ excretion with the daily intake. ENaC allows Na+ entry from the tubule lumen at the apical membrane, and the Na+/K.+ ATPase extrudes Na+ at the basolateral side. The Na+ absorption in the distal nephron is coupled to K+ secretion via K+ channels (ROMK2) located at the apical membrane. 

The above discussion completes our description of the single-nephron model. In total we have six coupled ordinary differential equations, each representing an essential physiological relation. Because of the need to numerically evaluate Ce in each integration step, the model cannot be brought onto an explicit form. The applied parameters in the single-nephron model are specified in Table 12.1. They have all been adopted from the experimental literature, and their specific origin is discussed in Jensen et al. [13]. 

The parameters y and e relate to the model of nephron-nephron interaction to be discussed in Section 12.5. In order to represent the hemodynamic coupling it is necessary to also introduce a parameter Cgj0 that describes the elastic response of... 

As demonstrated by the power spectra in Figs. 12.2a and 12.3b, regulation of the blood flow to the individual nephron involves several oscillatory modes. The two dominating time scales are associated with the period Tsiow 30—40 s of the slow TGF-mediated oscillations and the somewhat shorter time scale Tjast 5—10 s defined by the myogenic oscillations of the afferent arteriolar diameter. The two modes interact because they both involve activation of smooth muscle cells in the arteriolar wall. Our model describes these mechanisms and the coupling between the two modes, and it also reproduces the observed multi-mode dynamics. We can, therefore, use the model to examine some of the phenomena that can be expected to arise from the interaction between the two modes. 

Arrows point to couples of nephrons that share a piece of common arteriole. We suppose that hemodynamic coupling can be important for such nephrons. For other nephrons, the vascularly propagated coupling is likely to dominate. 

To implement the hemodynamic coupling in our model, a piece of common afferent arteriole is included into the system, and the total length of the incoming blood vessel is hereafter divided into a fraction s < that is common to the two interacting nephrons, a fraction 1 — that is affected by the TGF signal, and a remaining fraction — e for which the flow resistance is considered to remain constant As compared with the equilibrium resistance of the separate arterioles, the piece of shared arteriole is assumed to have half the flow resistance per unit length. 

Here, Rao denotes the total flow resistance for each of the two nephrons in equilibrium. ri and ri are the normalized radii of the active parts of the afferent arterioles for nephron 1 and nephron 2, respectively, and Pg and Pg2 are the corresponding glomerular pressures. As a base value of the hemodynamic coupling parameter we shall use s = 0.2. This parameter measures the fraction of the arteriolar length that is shared between the two nephrons. 

Because of the implicit manner in which the glomerular pressure is related to the efferent colloid osmotic pressure and the filtration rate [Eqs. (5)—(8)], direct solution of the set of coupled algebraic equations for the two-nephron system becomes... 

Fig. 12.15 (a) Phase plot for one of the nephrons and (b) temporal variation of the tubular pressures for both nephrons in a pair of coupled chaotically oscillating units, a = 32, T = 16 s, and e = y = 0.2. The figure illustrates the phenomenon of chaotic phase synchronization. By virtue of their mutual coupling the two chaotic oscillators adjust their (average) periods to be identical. The amplitudes, however, vary incoherently and in a chaotic manner [27],... 

Let us consider the case of a = 30 corresponding to a weakly developed chaotic attractor in the individual nephron. The coupling strength y = 0.06 and the delay time T2 in the second nephron is considered as a parameter. Three different chaotic states can be identified in Fig. 12.16. For the asynchronous behavior both of the rotation numbers ns and n f differ from 1 and change continuously with T2. In the synchronization region, the rotation numbers are precisely equal to 1. Here, two cases can be distinguished. To the left, the rotation numbers ns and n/ are both equal to unity and both the slow and the fast oscillations are synchronized. To the right (T2 > 14.2 s), while the slow mode of the chaotic oscillations remain locked, the fast mode drifts randomly. In this case the synchronization condition is fulfilled only for one of oscillatory modes, and we speak of partial synchronization. A detailed analysis of the experimental data series reveals precisely the same phenomena [31]. 

Direct coupling of adenosine triphosphate (ATP) hydrolysis is an example of an active transport process. The most important of these in the nephron is Nafi K -ATPase, which is located on the basolateral membranes of the tubulo-epithelial cells (Figure 45-6). This enzymatic transporter accounts for much of renal oxygen consumption and drives more than 99% of renal sodium reabsorption. Other examples of primary active transport mechanisms are a Ca " -ATPase, an H -ATPase, and an H, K -ATPase. These enzymes establish ionic gradients, polarizing cell membranes and thus driving secondary transport processes. 

The major effect is on the distal tubules of nephrons, where aldosterone promotes sodium retention and potassium excretion. Under the influence of aldosterone, sodium ions are actively transported out of the distal tubular cell into blood, and this transport is coupled to passive potassium flux in the opposite direction. Consequently, intracellular [Na" ] is diminished and intracellular [K+] is elevated. This intracellular diminution of [Na+] promotes the diffusion of sodium from the filtrate into the cell, and potassium diffuses into the filtrate. Aldosterone also stimulates sodium reabsorption from salivary fluid in the salivary gland and from luminal fluid in the intestines, but these sodium-conserving actions are of minor importance. 

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