Abstract
Background
Research in the last fifty years has shown that many autistic individuals have elevated serotonin (5hydroxytryptamine, 5HT) levels in blood platelets. This phenomenon, known as the platelet hyperserotonemia of autism, is considered to be one of the most wellreplicated findings in biological psychiatry. Its replicability suggests that many of the genes involved in autism affect a small number of biological networks. These networks may also play a role in the early development of the autistic brain.
Results
We developed an equation that allows calculation of platelet 5HT concentration as a function of measurable biological parameters. It also provides information about the sensitivity of platelet 5HT levels to each of the parameters and their interactions.
Conclusion
The model yields platelet 5HT concentrations that are consistent with values reported in experimental studies. If the parameters are considered independent, the model predicts that platelet 5HT levels should be sensitive to changes in the platelet 5HT uptake rate constant, the proportion of free 5HT cleared in the liver and lungs, the gut 5HT production rate and its regulation, and the volume of the gut wall. Linear and nonlinear interactions among these and other parameters are specified in the equation, which may facilitate the design and interpretation of experimental studies.
Background
The blood hyperserotonemia of autism is an increase in the serotonin (5hydroxytryptamine, 5HT) levels in the blood platelets of a large subset of autistic individuals. It is usually reported as mean platelet 5HT elevations of 25% to 50% in representative autistic groups [1] that almost invariably contain hyperserotonemic individuals. Since the first report in 1961 [2], this phenomenon has been described in autistic individuals of diverse ethnic backgrounds by many groups of researchers [39]. Despite the fact that the hyperserotonemia of autism is considered to be one of the mostwell replicated findings in biological psychiatry [1], its biological causes remain poorly understood.
Blood platelets themselves do not synthesize 5HT. During their life span of several days, they actively take up 5HT from the blood plasma using a molecular pump, the 5HT transporter (SERT). The plasma 5HT originates in the gut, where most of it is synthesized by enterochromaffin cells (EC) of the gut mucosa [10]. Some of the gut 5HT is used locally as a neurotransmitter of the enteric nervous system and it also can be taken up into gut cells that express SERT and lowaffinity serotonin transporters [11,12]. Some of the gut 5HT diffuses into the general blood circulation, where most of it is rapidly cleared by the liver and the lungs [13,14]. Free 5HT in the blood plasma becomes available to platelets. The circulation of peripheral 5HT is summarized in Figure 1.
Figure 1. The peripheral 5HT circulation. The thick black arrow represents the influx of 5HT from the gut and the red arrows represent the clearance of 5HT. For explanation of the variables, see the text, Table 1, and Appendix 2.
The bloodbrain barrier is virtually impermeable to 5HT and, therefore, free 5HT in the blood plasma is unlikely to reach cerebrospinal fluid or brain parenchyma. However, biological factors that cause the platelet hyperserotonemia may play a role in the early development of the autistic brain, since the brain and peripheral organs express many of the same neurotransmitter receptors and transporters. The consistency of the platelet hyperserotonemia suggests that many of the genes implicated in autism [15,16] may control a small number of functional networks. Since blood platelets are shortlived, the altered processes may remain active in the periphery years after the brain has formed. In contrast, most of the brain developmental processes are over by the time an individual is formally diagnosed with autism. SERT is expressed by brain neurons and blood platelets [17] and its altered function may both affect brain development and lead to abnormal 5HT levels in platelets. To date, most experimental studies have focused on SERT polymorphisms as a likely cause of the platelet hyperserotonemia, but the results have been inconclusive. While SERT polymorphic variants may partially determine platelet 5HT uptake rates [18] or even platelet 5HT levels [19], these polymorphisms, alone, are unlikely to cause the platelet hyperserotonemia of autism [18,20]. Some evidence suggests that the platelet hyperserotonemia may be caused by altered 5HT synthesis or release in the gut [2123] or by interactions among several genes [2426].
To date, most research into the causes of the platelet hyperserotonemia has focused on a specific part of the peripheral 5HT system. However, this system is cyclic by nature and does not allow easy intuitive interpretation. It is not clear what parameters and their interactions platelet 5HT levels are likely to be sensitive to, as well as what parameters should be controlled for when others are varied. For instance, an increase in SERT activity may increase platelet 5HT uptake, but it may also increase 5HT uptake in the gut and lungs and, consequently, may reduce the amount of free 5HT in the blood plasma.
Here, we develop an equation that yields platelet 5HT levels that are consistent with published experimental data. The equation also provides information about the sensitivity of platelet 5HT levels to a set of biological parameters and their interactions.
Results and Discussion
Platelets take up 5HT at low plasma 5HT concentrations
Suppose blood platelets are produced at a constant rate, their halflife is t_{1/2}, and we are interested in the steady state when the total number of platelets (N_{tot}) remains constant. Then the number of the platelets whose age ranges from x ≥ 0 to x + dx is given by (Appendix 1)
where τ = t_{1/2}/ln 2 ≈ 1.44t_{1/2}.
The 5HT uptake rate of an "average" platelet at time t can be defined as follows:
where u_{i}(t) is the 5HT uptake rate (mol/min) of platelet i at time t.
At any two times t_{1 }and t_{2 }(t_{1 }≠ t_{2}), at least some of the individual platelets in the circulation will be physically different, because platelets are constantly removed from the circulation and replaced by new platelets. Also, at least some individual platelets will be routed by the circulation to different blood vessels, which may have different concentrations of free 5HT in the blood plasma. Since the platelet uptake rate depends on the 5HT concentration in the surrounding plasma, generally, u_{i}(t_{1}) ≠ u_{i}(t_{2}). However, the 5HT uptake and distribution of platelets appear to be little affected by their age or by how much 5HT they have already accumulated [14,27]. Also, the numbers of platelets in blood vessels are very large and can be considered constant. Therefore, should be immune to these replacements and permutations, and the timedependence of can be dropped:
The total amount of 5HT that has been taken up by the subpopulation of platelets whose age ranges from x to x + dx is given by (Appendix 1)
If the total volume of the circulating blood is Ω_{b }and the numerical concentration of platelets is C_{p }= N_{tot}/Ω_{b}, the concentration of platelet 5HT is
It follows that
In normal humans, C_{s}/C_{p }has been experimentally estimated to be around 3.58 · 10^{18 }mol/platelet [7]. The halflife of human platelets is approximately 5 days [28,29], so τ ≈ 1.44t_{1/2 }= 1.04 · 10^{4 }min. Plugging these values into Eq. (6) yields = 3.44 · 10^{22 }mol/min, or an "average" platelet takes up around 3.5 molecules of 5HT every second.
What concentration of free 5HT in the blood plasma corresponds to this uptake rate? Since platelet 5HT uptake obeys MichaelisMenten kinetics [14,18],
where V_{max }is the maximal 5HT uptake rate of one platelet, K_{m }is the MichaelisMenten constant, and c_{i }is the local concentration of free 5HT surrounding platelet i.
If the concentration of free 5HT were the same in all blood vessels (c_{i }≡ C_{f}), we would obtain
and
However, in some blood vessels (such as the ones leaving the gut) the concentration of free 5HT may be considerably higher than in others. We can define
and rely on the evidence that c_{i }≪ K_{m }[14,18,30]. Then
and it follows that
In normal humans, V_{max }≈ 1.26 · 10^{18 }mol/(min · platelet) and K_{m }≈ 0.60 · 10^{6 }mol/L (these values were obtained by weighting the medians of each of the three groups of [18] by the number of subjects in the study). Plugging these values and the obtained into Eq. (12) yields C_{f }≈ = 0.16 · 10^{9 }mol/L = 0.16 nM.
Experimental measurement of free 5HT in the blood plasma poses serious challenges. It is not uncommon to report concentration values of free 5HT that are a few orders of magnitude higher than those obtained in carefully designed studies (for discussion, see [14,30,31]). The theoretically calculated value (0.16 nM) is on the same order as an accurate experimental estimate of free 5HT in the distal venous plasma (0.77 nM) obtained by Beck et al. [30]. These authors note that new experimental methodologies may further reduce their estimate [30]. Taken together, these theoretical and experimental results suggest that virtually all platelets take up 5HT at very low free 5HT concentrations, after most of the 5HT released by the gut has been cleared from the circulation by the liver and the lungs.
Gut 5HT release rate (R)
We denote the gut 5HT release rate R, where R is expressed per unit volume of the gut wall and includes all 5HT released by the gut. Specifically, R includes the 5HT that (i) is taken back up into gut cells, (ii) remains in the extracellular space of the gut wall, and (iii) diffuses into the blood circulation. If the gut 5HT release rate fluctuates but homeostatic mechanisms keep it near some constant value R_{00 }> 0, then we can write
where t is time and λ > 0 is the time constant of the process (the larger is the λ, the slower is the return to R_{00}). We next consider a more general scenario, where the gut 5HT release rate is controlled by the actual state of the peripheral 5HT system.
First, we consider a local mechanism that monitors the extracellular 5HT concentration in the gut wall. The actual sensitivity of the gut 5HT release rate to extracellular 5HT levels is not well understood. In the brain raphe nuclei, 5HT release does not appear to be controlled by 5HT1A autoreceptors unless extracellular 5HT levels become excessive [32]. The gut expresses 5HT1A, 5HT3, and 5HT4 receptors [11], but these receptors may not be activated by the normal levels of endogenous extracellular 5HT in the gut wall [33]. In SERTdeficient mice, 5HT synthesis appears to be increased by around 50%, but the expression and activity of tryptophan hydroxylases 1 and 2 are not altered [34]. In SERTdeficient rats, the expression and activity of tryptophan hydroxylase 2 are also unaltered in the brain, even though the extracellular 5HT levels in the hippocampus are elevated 9fold [35]. From a systemscontrol perspective, the reported insensitivity of 5HT synthesis to extracellular 5HT levels may be due to the inherent ambiguity of the signal. In fact, high extracellular 5HT levels may signal both overproduction of 5HT by tryptophan hydroxylase and an excessive loss of presynaptic 5HT due to its reduced uptake by SERT. If the former is the case, the activity of trypotophan hydroxylase should be decreased; if the latter is the case, it should be increased.
Alternatively, platelet 5HT levels can be regulated by global peripheral mechanisms. Since platelets take up 5HT over their life span, their 5HT levels will change only if an alteration of the peripheral 5HT system is sustained over a considerable period of time. Since platelets act as systemic integrators, we can assume that, formally, the gut 5HT release rate is a function of the platelet 5HT concentration. In essence, we simply assume that the gut 5HT release is controlled by global, systemic changes in the peripheral serotonin system. In biological reality, this relationship would be mediated by latent variables, because platelet 5HT is inaccessible to the gut.
If the gut release rate is controlled by any of the discussed mechanisms,
where G is the extracellular 5HT concentration in the gut wall, P is the platelet 5HT concentration (mol/platelet), and f(., .) is a differentiable function.
Linearization of f(G, P) in the neighborhood of "normal" values of G and P (denoted G_{0 }and P_{0}, respectively) yields
By denoting R_{0 }= R_{00 }+ f (G_{0}, P_{0}) we obtain
Note that Eq. (13) is a special case of Eq. (16) when neither G nor P controls the gut 5HT release rate (i.e., when α = β = 0).
Concentration of extracellular 5HT in the gut wall (G)
The concentration of extracellular 5HT in the gut wall increases due to synthesis and release of 5HT by EC cells and neurons of the gut. It decreases due to two processes: (i) local 5HT uptake by SERT (and perhaps by other, lowaffinity transporters [12,35]) and (ii) 5HT diffusion into gut blood capillaries. Suppose that the blood that has exited the heart through the aorta at time t reaches the gut at time t + s (s > 0). The decrease rate of extracellular 5HT concentration in the gut wall due to the diffusion into blood capillaries is given, according to Fick's First Law, by
where G(t + s) is the concentration of extracellular 5HT in the gut wall at time t + s, D is the 5HT diffusion coefficient across the blood capillary wall, S is the total surface area of the gut blood capillaries, w is the thickness of the capillary wall, Ω_{g }is the effective extracellular volume of the gut wall, Q_{tot }is the total cardiac output, z_{g }is the proportion of the total cardiac output routed to the gut and/or the liver, F(t) is the flow of free 5HT in the aorta at time t, σ is the proportion of blood volume that is not occupied by cells (approximated well by 1  Ht, where Ht is the hematocrit), and d_{g }≡ DS/(wΩ_{g}). Note that z_{g}F (t)/(σz_{g}Q_{tot}) is the concentration of free 5HT in the blood plasma that arrives in the gut at time t + s (Fig. 1).
If all three discussed processes are taken into consideration,
where k_{g }is the 5HT uptake rate constant in the gut wall. This constant is likely to be a function of SERT activity (γ), i.e., k_{g }≡ k_{g}(γ). Importantly, k_{g}(0) is not necessarily zero, since 5HT uptake in the gut may be mediated by lowaffinity 5HT transporters, at least in the absence of SERT [12,35].
Flow of free 5HT in the aorta (F)
We next consider the flow (mol/min) of free 5HT in the blood circulation from the time blood exits the heart through the aorta (at time t) to the time it returns to the aorta after one circulation cycle (at time t + T; Fig. 1). Since blood transit times from organ to organ are relatively short (seconds), we will ignore 5HT diffusion parallel to the flow. After the blood leaves the heart, a proportion (z_{g}) of the total cardiac output is routed to the gut and/or the liver. On arrival in the gut at time t + s (0 <s <T), the blood is replenished with new 5HT synthesized in the gut wall. According to Fick's First Law, this flow of 5HT into the blood is
where all parameters and G(t + s) are defined as in Eq. (17), F(t) is the flow of free 5HT in the aorta, and d_{b }≡ DS/w (note that d_{b}/d_{g }= Ω_{g}).
After the 5HT flow leaves the gut, it passes through the liver that removes a large proportion (1  θ_{h}) of free 5HT [13,14]. After exiting the liver, the 5HT flow is joined by the 5HT flow that did not enter the gut and/or the liver and the merged flow passes through the lungs that remove another large proportion (1  θ_{p}) of free 5HT [13,14]. Experimental results suggest that θ_{h }≈ 0.25 and θ_{p }≈ 0.08 [13]. Since the lungs express SERT [36], θ_{p }may be considered to be a function of SERT activity, i.e., θ_{p }≡ θ_{p}(γ). It is likely that θ_{p}(0)≠ 0, since no obvious toxic 5HT effects are seen in mice that lack SERT [12].
Platelet 5HT uptake is a slow process compared with the blood circulation through the gut, liver, and lungs. Therefore, in this circulation, platelet uptake should have a negligible effect on free 5HT levels in the blood plasma [13,14]. However, platelets spend a considerable proportion of the circulation cycle in the vascular beds of other organs (the "nongut" system of Fig. 1), where platelet 5HT uptake may have an impact on the already low levels of free 5HT.
Taking all these considerations together, the 5HT flow that leaves the heart after one full circulation cycle is
where 1  θ_{v }is the proportion of free 5HT cleared by the platelets in the "nongut" system (Fig. 1) and z_{ng }= 1  z_{g}.
Platelet 5HT concentration at the steady state ()
Denote the steadystate flow of free 5HT in the aorta. The system is in its steady state if the following is true: dR/dt = 0, dG/dt = 0, F(t) = F(t  T) = , and if F(t  s) ≈ F(t  s  x) = for all x > 0 for which N_{tot }exp(x/τ) ≫ 1, where 0 <s <T (for the last condition, see Eqs. (36) and (47) in Appendix 2).
At the steady state, the platelet 5HT concentration is (Appendix 2)
where k_{p }≡ k_{p}(γ) is the 5HT uptake rate constant of one platelet. In mice lacking SERT, the amount of 5HT stored in blood platelets in virtually zero [12], suggesting that k_{p}(0) = 0.
Solving Eqs. (16), (18), (20), and (21) at the steady state yields
where
and
where for brevity we defined
and
In the derivation, we used the relationship d_{g }= d_{b}/Ω_{g}.
The values of the parameters can be approximated based on published experimental results (Table 1). Since little is known about the regulation of 5HT release from the gut, we can initially assume that α = β = 0 (in this case, platelet 5HT concentration is independent of G_{0 }and P_{0}). Plugging the parameter values into Eq. (22) yields = 2.40 · 10^{18 }mol/platelet, or 4.23 · 10^{16}g/platelet. Since the platelet concentration in the blood has been estimated to be 4.28 · 10^{8 }platelets/mL [7], the obtained value is equivalent to the wholeblood 5HT concentration of 1.02 μM or 0.18 μg/mL. These values are well within the range of normal 5HT concentrations obtained in experimental studies (Fig. 2). Platelet 5HT concentrations when α > 0 are plotted in Fig. 2.
Table 1. Parameter values
Figure 2. Platelet 5HT levels. Normal platelet 5HT concentrations reported in published reports (a [6], b [19], c [7], d [8], e [9]; the circles are the means and the bars indicate the range), compared with the theoretical values obtained with α = 0 (square) and α > 0. The values of the other parameters are given in Table 1 and β = 0. The theoretical platelet 5HT concentrations reach a limit when α is large (inset).
Sensitivity of platelet 5HT to parameters
Equation (22) represents the minimal set of relationships that have to be taken into account in experimental studies. It provides information about the sensitivity of platelet 5HT levels to biological parameters and their interactions, some of which have not been considered or controlled for in experimental approaches. Here, we limit sensitivity analysis to the simplest case when parameters in Eq. (22) can be considered independent.
First, we calculate the local rate of change in with respect to each of the parameters, i.e., we evaluate the partial derivatives of with respect to each of the parameters at the parameter values given in Table 1 (see Appendix 3 for details). We express this rate of change as the percentagewise change in if a parameter increases by 10% with respect to its normal value (assuming the relationship can be approximated as linear). The results of these calculations are given in Table 2.
Table 2. Sensitivity of platelet 5HT concentration to changes in parameters
Second, we inverse the problem and calculate the percentagewise change in each of the parameters needed to reach a 25% or 50% increase in platelet 5HT concentration. These increases represent the typical range of elevation in platelet 5HT levels in autism [1]. The required changes of the parameters are calculated using Eq. (22) without linearization. The results of these calculations are given in Table 3.
Table 3. Parameter changes causing 25% and 50% increases in platelet 5HT concentration
Tables 2, 3 indicate that platelet 5HT concentration is highly sensitive to the platelet 5HT uptake rate constant (k_{p}), the baseline gut 5HT release rate (R_{0}), the proportion of 5HT cleared in the liver and lungs (θ_{h}, θ_{p}), and the volume of the gut wall (Ω_{g}). Some experimental evidence suggests that k_{p }is altered in autistic individuals [37,38]. The analysis also suggests that the hyperserotonemia of autism may be caused by altered extracellular 5HTdependent regulation of the gut release rate (α). We have recently shown that mice lacking the 5HT1A receptor, expressed in the gut [39], develop an autisticlike blood hyperserotonemia [23], which may be caused by altered regulation of the gut 5HT release rate. Another potentially important 5HT receptor is the 5HT4 receptor that is expressed throughout the gastrointestinal tract in humans [40]. The analysis also shows that the 5HT uptake rate constant in the gut wall (k_{g}) and the rate constant of 5HT diffusion into the blood (d_{b}) should have little effect on platelet 5HT levels. A recent study has found no link between platelet hyperserotonemia and increased intestinal permeability in children with pervasive developmental disorders [41].
In the analysis we assumed that each parameter can be manipulated independently of the other parameters. In particular, this assumes that Ω_{g }can be changed independently of d_{b}, which is a function of S, the capillary surface area of the gut. However, increasing Ω_{g }is likely to increase S. To make Ω_{g }and d_{b }truly independent, it is sufficient to make an assumption that a unit volume of the gut wall contains a constant surface area of blood capillaries, i.e., S/Ω_{g }≡ ρ = const. Since d_{b }= DS/w, this yields
After this correction, the sensitivity of platelet 5HT concentration to the gut volume remains virtually unchanged (Tables 2, 3).
Care should be exercised in manipulating the parameters k_{p}, k_{g}, θ_{p}, and θ_{v}, which may not be independent. All of them may be determined, at least in part, by SERT activity (γ). Given the lack of experimental data regarding their actual relationships, two extreme scenarios can be considered. As assumed in the sensitivity analysis, these parameters can be considered to be virtually independent, since each of them is likely to be determined (in addition to SERT) by other factors in the platelet, gut, and lungs. Alternatively, all four parameters may be functions of only one variable, γ. In this case, platelet 5HT levels may increase or decrease with different γ values, even if each of the functions were linear. This behavior of as a function of γ is important to consider in SERT polymorphism studies. The ambiguity could be resolved if an experimentallyobtained covariance matrix for k_{p}, k_{g}, θ_{p}, and θ_{v }were available. Equation (22) also suggests that platelet 5HT levels may be highly sensitive to interactions among the platelet uptake rate, the proportion of 5HT cleared in the liver and lungs, the gut 5HT release rate, and the volume of the gut wall. The length of the human gut is known to be remarkably variable [42], which may underlie some variability in platelet 5HT levels. This possibility has not been investigated experimentally or theoretically. It is worth noting that 5HT itself plays important roles in gastrulation [43] and morphogenesis [44], and that changes in gut length may have had a major impact on the evolution of the human brain [45].
It should be noted that Eq. (22) remains valid if some or all of the parameters are expressed as functions of new, independent parameters. In this case, the original parameters may no longer be independent and changing one of the new parameters may alter more than one of the original parameters. For instance, serotonin uptake in blood platelets has been recently shown to be dependent on interaction between SERT and integrin αIIbβ3 [46]. Denoting the activity of the integrin y, we can write k_{p }= k_{p}(γ, y). It is possible that some other parameters in Eq. (22) can also be expressed as functions of integrin αIIbβ3 activity. All of these functions can be plugged into Eq. (22), which remains to be correct and now allows calculation of platelet concentration as a function of integrin αIIbβ3 activity, i.e., = (y). Generally, further theoretical progress will largely depend on understanding the relationships among the current set parameters. Whether they can be expressed as functions of a smaller set of parameters is not known.
Assumptions and caveats
Many of the assumptions in the model are "natural" in the sense that they are commonly used to explain experimental results (even though they may not be explicitly stated). In essence, the model simply formalizes the idea that peripheral 5HT is produced in the gut, from which it can diffuse into the systemic blood circulation, where it can be transported into blood platelets. The strength of the model is in its "bird'seye" view of the entire system. In particular, the model does not allow focusing on one parameter without explicitly stating what assumptions are made regarding the other parameters (some of which may be equally important in determining platelet 5HT levels). For example, studies on SERT polymorphisms often focus on 5HT uptake in platelets but do not explain how the same polymorphisms may affect 5HT release from the gut (which also expresses SERT). The model also indicates which parameters and their interactions platelet 5HT concentration is likely to be sensitive to, thus limiting one's freedom in choosing which factors can fall "outside the scope" of a study. By its very nature, the platelet hyperserotonemia of autism is a systems problem.
Some of the model assumptions are not critical, such as the assumption that the gut 5HT release rate can be controlled by extracellular 5HT in the gut wall or by platelet 5HT levels. In the model, the absence of control is simply a special case of this more general scenario, since we can always set α = β = 0. If control is present, the assumption of its linearity (Eq. (16)) is necessary to obtain Eq. (22). While the Taylor series, used in Eq. (15), guarantees nearlinear behavior of the control mechanisms in the neighborhood of G_{0 }and P_{0}, nothing is said about how far one can move away from G_{0 }and P_{0 }before nonlinearities can no longer be ignored.
The assumption of the independence of the parameters in Eq. (22) is not necessary and is used here only to simplify the numerical sensitivity analysis. Some or all of the parameters may be tightly linked, which does not change Eq. (22) (but it may change the results obtained in the sensitivity analysis). Interdependent parameters can be expressed as functions of other, independent parameters (or "parameterized" in the mathematical sense), and these functions can be substituted for the parameters in Eq. (22). In this case, becomes a function of the new parameters, as already discussed with regard to integrin αIIbβ3.
The model assumes that the gut 5HT release rate is constant at the steadystate. Strictly speaking, this assumption is incorrect, since gut activity exhibits circadian and other rhythmic behavior. Likewise, platelet counts exhibit normal fluctuations due to a number of factors, such as exercise, digestion, exposure to ultraviolet light, and others [47]. However, platelets accumulate 5HT over days; therefore, R and N_{tot }can be thought of as "baseline" values.
A potentially important assumption is made regarding the nature of the 5HT diffusion from the gut into the blood circulation. Passive diffusion is assumed, and the value of the diffusion coefficient (D) is considered to be comparable to typical values observed in liquids. Virtually no experimental data are available on the exact nature of the 5HT diffusion (which may be facilitated), and its D value remains to be determined.
A set of critical assumptions limits relationships between the parameters (given in Table 1), which are assumed to be constant in an individual, and the four dynamic variables (R, G, F, and P), which can evolve in time. While any parameter can be a function of any other parameters, original or new, none of the parameters (original or new) can be a function of any of the dynamic variables. If this condition is not met, the steadystate platelet 5HT concentration will have a form different from Eq. (22). Suppose extracellular 5HT in the gut wall controls SERT expression, or free 5HT in the blood plasma controls the proportion of internalized SERT in blood platelets [24]. In these cases, the model may fail because the uptake rate constants will depend on the dynamic variables, i.e., k_{g }= k_{g}(G(t)) and k_{p }= k_{p}(F (t)). Likewise, Eqs. (16), (18), (20), and (21) are assumed to exhaust all relationships between the four dynamic variables. If, for instance, Eq. (16) were changed to
where F_{0 }and β' are constants and β' ≠ 0, the solution in Eq. (22) would no longer be correct.
These critical assumptions define the limits within which the model should perform reasonably well. New experimental data will be needed to further refine it.
Conclusion
We developed an equation that allows calculation of platelet 5HT levels as a function of biological parameters. While the main goal is to understand the origin of the hyperserotonemia of autism, the equation can also be used to calculate platelet 5HT levels in normal individuals and in individuals whose peripheral 5HT system may be altered due to conditions unrelated to autism. In the simplest case when each parameter is manipulated independently, theoretical analysis predicts that platelet 5HT concentration should be sensitive to changes in the platelet 5HT uptake rate constant, the proportion of free 5HT cleared in the liver and lungs, the gut 5HT production rate and its regulation, and the volume of the gut wall. The equation also specifies linear and nonlinear interactions among these and other parameters, some of which may also play a role in the developing autistic brain.
Methods
Authors' contributions
SJ developed the model and wrote the manuscript.
Appendix
1. Distribution of blood platelets by age
To derive Eqs. (1) and (4), consider the platelets whose age is between x = kΔx and x + Δx, where Δx > 0 is small and k = 0, 1, 2... If the platelet production rate is denoted r, the total number of platelets produced in the interval x is rΔx. With each time step Δx, this number decreases by a factor of q, where q = e^{Δx/τ }(this follows directly from the fact that the decay of platelet numbers can be described by a constant halflife). The number of the remaining platelets after k time steps is given by
The total number of platelets currently circulating in the blood then is
It follows from Eqs. (29) and (30) that
Since Δx ≪ τ and
we obtain
Then the platelets whose age is between x and x + Δx have taken up the following amount of 5HT:
where is the 5HT uptake rate of an "average" platelet, defined in Eqs. (2) and (3). If Δx is allowed to tend to zero, Eqs. (33) and (34) become Eqs. (1) and (4).
2. Platelet 5HT concentration
Consider the circulation of peripheral 5HT (Fig. 1). We start by dividing the peripheral 5HT system into the "gut" system (Gsystem) and the "nongut" system (NGsystem). In the Gsystem, arterial blood exits the heart through the aorta, perfuses the gut and/or the liver, joins the venous blood flow to the heart, passes through the lungs, and returns to the heart with the oxygenated blood. In the NGsystem, arterial blood exits the heart through the aorta, perfuses various peripheral organs, and joins the venous blood flow. In further considerations, the blood flow rate (measured in m^{3}/min) is clearly distinguished from the 5HT flow rate (measured in mol/min). In fact, if a blood vessel carrying 5HTenriched blood is joined by another blood vessel with virtually no 5HT in its blood, the blood flow rate of the merged vessel increases, but its 5HT flow rate remains the same. We intentionally avoid the term "flux", which often denotes flow rate per unit area.
Denote Q_{tot }the total cardiac output, z_{ng }the proportion of the cardiac output that does not pass through the gut and/or the liver, and N_{tot }the total number of blood platelets in the circulation. Then the blood flow rate of the NGsystem is Q_{ng }= z_{ng}Q_{tot }and at any time the NGsystem contains N_{ng }= ηN_{tot }uniformly distributed platelets (0 ≤ z_{ng}, η ≤ 1). If every platelet passing through the NGsystem travels an approximate linear distance L, we can subdivide L into K (not necessarily equal) segments, each of which contains the same number of platelets ΔN_{ng }= ηN_{tot}/K (Fig. 1). Assuming these groups of platelets advance in discrete time steps, each of them will spend the same constant time, Δt, in each of the linear segments:
where C_{p }is the concentration of platelets in the blood (the number of platelets per unit volume of blood). Denote F(t) the flow of free 5HT that exits the heart through the aorta at time t. Next, consider ΔN_{ng }platelets that exit the heart through the aorta at time t and enter the NGsystem at time t + s (s > 0). The flow of free 5HT that enters the NGsystem with these platelets at time t + s is z_{ng}F (t). The concentration of free 5HT around these platelets at time t + s then is
where σ is the proportion of blood volume not occupied by cells.
If the 5HT uptake rate constant of one platelet is k_{p}, the total 5HT amount taken up by the NGsystem platelets from time t_{1 }to t_{1 }+ Δt is
Next, consider a time period from t_{1 }to t_{2 }= t_{1 }+ MΔt (M = 2, 3...). During this time, the total amount of 5HT taken up by the NGsystem platelets is
If M ≥ K,
where
and
Since blood platelets accumulate 5HT over a period of time that is a few orders of magnitude longer than one blood circulation cycle, we are interested in the situation when M ≫ K. Then, if F(t) satisfies mild constraints (e.g., does not fluctuate rapidly), δ_{1 }and δ_{2 }can be dropped and Eq. (39) becomes
If we allow Δt to tend to zero,
Thus far, we have ignored the fact that blood platelets are destroyed and replaced by new platelets. However, the halflife of platelets is only approximately 5 days [28,29]. Consider a past time t_{0 }(t_{0 }<t, where t is present time). Among the presently circulating platelets, the proportion of the platelets that are older than t  t_{0}, according to Eq. (1), is
Only these platelets were taking up 5HT when the concentration of free 5HT in the blood entering the NGsystem at time t_{0 }was c_{ng}(t_{0}). Therefore, the total 5HT amount accumulated by the NGsystem platelets at time t can be found by weighting the past concentrations of free 5HT by the proportion of the presently circulating platelets that were taking up 5HT at these past times:
where U_{ng}(t) ≡ U_{ng}(∞, t).
After changing the dummy variable under the integral sign, we obtain
At a steady state,
for all x for which N_{tot }exp(x/τ) ≫ 1 (i.e., more than one currently circulating platelet was produced before time t x). Then the steadystate amount of 5HT accumulated by the platelets of the NGsystem is
By analogy, the 5HT accumulated by the platelets of the Gsystem at the steady state is expected to be
where is the steadystate concentration of free plasma 5HT in the ith compartment of the Gsystem, μ_{i}> 0, and Σ_{i}μ_{i }= 1. Then
where is the mean of the steadystate concentrations of free 5HT in the compartments of the Gsystem:
We have already shown that virtually all platelets take up 5HT at very low free 5HT concentrations. This is not surprising, since the blood that has left the gut reaches the liver within seconds [13,14], and the liver removes more than 70% of free 5HT [13]. Assuming , we obtain the total amount of 5HT accumulated by all blood platelets of the peripheral 5HT system at the steady state:
The concentration of platelet 5HT at the steady state then is
Since, according to Eq. (36),
where is the steadystate flow of free 5HT in the aorta, we obtain
For the convenience of notation, we will further consider Eq. (55) to be exact.
3. Sensitivity of platelet 5HT levels to changes in parameters
We investigate the sensitivity of to changes in the parameters, which for the purpose of this analysis are considered to be independent. For the convenience of notation, we denote the set of parameters in Eq. (22) X = (X_{1}, X_{2}, X_{3}, X_{4}, ...) ≡ (α, β, k_{g}, k_{p}, ...), (X) ≡ . Two approaches are used.
In the first approach, for each parameter X_{i }we calculate the normalized differential
where are the values of the parameters given in Table 1 and . The obtained values represent the percentagewise increase in if a parameter increases by 10%, assuming the relationship can be approximated as linear for this small change. The results are given in Table 2. In the second approach, we assign all, or all but one of the parameters the values from Table 1:
or
respectively, and then numerically solve the equations
for each X_{i}, where q = 1.25 or q = 1.5. The results are given in Table 3.
Acknowledgements
This study was supported, in part, by the Santa Barbara Cottage Hospital – UCSB Special Research Award. I thank the anonymous reviewers for their constructive comments and Vaiva for her support.
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