Basic model

The origin of the platelet hyperserotonemia of autism cannot be understood unless a certain model of the underlying physiological processes is accepted – whether it is an implicit model that is not clearly stated, a model described in words, or a mathematical model. One advantage of mathematical modeling is that it requires a clear description of all relevant interactions among the components of the system. Its greatest disadvantage is that sometimes clear-cut choices have to be made where experimental data may suggest a few possible alternatives. In this section I introduce a model that is based on what is known about the 5-HT circulation outside the CNS and point out two important but unresolved problems.

In search of a factor that can both cause platelet hyperserotonemia and alter normal brain function, many recent studies have focused on the serotonin transporter (SERT) that is expressed in blood platelets and brain neurons [

41]. Despite early promising results [

42], different groups have found little or no linkage [

43] between SERT polymorphisms and autism in various ethnic groups [

40,

44-

47]. I have recently proposed [

48] that the factor that interferes with brain development in autism may also regulate the release of 5-HT from gut enterochromaffin (EC) cells, the main source of blood 5-HT [

36,

49,

50]. First, this hypothesis assumes that EC cells can monitor (directly or by way of gastrointestinal neurons) the 5-HT levels in the surrounding extracellular space and can decrease or increase their 5-HT release accordingly. Similar control mechanisms have long been suspected in the brain, where serotonergic neurons express 5-HT autoreceptors [

51,

52]. Second, the levels of extracellular 5-HT in the gut wall are assumed to be at equilibrium with the levels of free 5-HT in the arterial blood. While the baseline extracellular levels of 5-HT in the gut wall have not been precisely measured, the estimated levels of free 5-HT in the arterial blood appear to be comparable to the extracellular 5-HT levels in the brain [

51,

53], which expresses some of the same 5-HT receptors as the gut [

51,

54-

57].

This hypothesis can be cast in a mathematical form. Suppose that EC cells indirectly monitor the levels of free 5-HT that arrives in the gut with the arterial blood, compare these levels with the expected 5-HT levels, and adjust their 5-HT release to a new value (

*R*_{n+1}), using a pre-set release value (

*R*_{C}) as the reference point. The strength (gain) of this adjustment is controlled by a factor

*α*, which is hypothesized to be different in normal and autistic individuals. After the blood leaves the gut, a large proportion (

*γ*) of the free 5-HT is quickly removed by the liver, lungs and other organs that express SERT and monoamine oxidases (MAOs) [

58-

62]. The numerical value of

*γ *is likely to vary from individual to individual, because the SERT and MAO genes have a number of polymorphic variants distributed in the population [

40,

45,

46,

63-

66]. Therefore,

*γ *is considered to be a random variable with a known probability distribution. The model can then be described by the following system of equations:

*F*_{n + 1 }= (1 - *γ*)*F*_{n }+ *R*_{n + 1}, (2)

Where (1 -

*γ*)

*F*_{n }is the flux of free 5-HT that enters the gut with the arterial blood,

*F*_{C }is the pre-set ("expected") flux, and

*F*_{n + 1 }is the flux of free 5-HT that exits the gut (α ≥ 0, 0 ≤ γ ≤ 1,

*F*_{C }> 0,

*R*_{C }> 0). In the model, the 5-HT release from EC cells does not include the 5-HT that is used for local signaling and is rapidly removed by local gastrointestinal epithelial and neural cells expressing SERT [

54,

67,

68]. This 5-HT could be included in the model, together with the local clearance rate, if estimates of these parameters were available.

It is thought that little free 5-HT is taken up by blood platelets, before most of it is removed by the liver, lungs and other organs [

53,

60]. Also, it has been suggested that platelet 5-HT levels may depend on the levels of free 5-HT in the blood almost linearly [

53]. Then, at the steady state,

*F*_{n + 1 }=

*F*_{n } *F *and

*R*_{n + 1 }=

*R*_{n } *R *for any

*n*, and platelet 5-HT levels are

where *K *> 0 is a constant.

Note that *ser*(*α*, *γ*) is a decreasing function of *γ*. Also, at the steady state,

*R *= *γF*. (4)

It should be emphasized that the mathematical simplicity of equations (1) and (2) in no way implies that the biological regulation of 5-HT release in the gut is simple. The human gut is a remarkably complex organ that uses a wide range of neurotransmitters and that may have at least as many neurons as the spinal cord [

50]. Nevertheless, recent studies suggest that complex biological systems, such as brain neurons, can be "actively linear" [

69], meaning that sophisticated biological mechanisms may act on intrinsically non-linear physical processes to produce quantitative relationships that are mathematically linear.

The dependence of platelet 5-HT levels on

*α *and

*γ *is plotted in Figure , where the numerical values of

*F*_{C }and

*R*_{C }are taken from previously published experimental and theoretical studies [

48,

53,

70], and where the regulation of the 5-HT release from EC cells is assumed to be less than fully functional in autistic individuals (note the low

*α *value). A key feature of this dependence is that, in normal individuals, platelet 5-HT levels remain low with any

*γ*, whereas in autistic individuals these levels may be normal or higher than normal depending on the individual's

*γ*. This dependence captures one of the most puzzling properties of the autistic distribution of platelet 5-HT levels, which always overlaps with the control (normal) distribution, but always includes individuals whose 5-HT levels are higher than normal [

31]. It may also explain why the SERT and MAO genes may appear to be linked with autism but may not actually cause it. As shown in Figure , a low

*γ *is a necessary but not sufficient condition for the platelet hyperserotonemia to occur. Given a low

*γ*, the platelet hyperserotonemia will occur only in those individuals whose regulation of the 5-HT release from EC cells is compromised (i.e., they are autistic and have a low

*α*). It follows then that

*γ *acts only as a modifier of platelet 5-HT levels, and that the statistical distribution of

*γ *may be the same in normal and autistic populations. Assuming an individual's

*γ *value is determined, at least in part, by his/her variants of the SERT and MAO genes expressed in the liver, lungs and other organs, normal and autistic populations may have similar distributions of SERT and MAO polymorphisms. This assumption is supported by recent studies [

40,

45-

47,

63,

64].

Two potentially contentious decisions were made in the model. First, the exact levels of free 5-HT in the blood remain a debated issue. While a number of studies have found "low" but consistently measurable levels of free 5-HT in the human blood [

53,

70,

71], Chen et al. [

72] have suggested that the concentration of free 5-HT in the blood may be negligible, since these researchers have detected virtually no 5-HT in the whole blood of SERT-deficient mice whose blood platelets cannot take up 5-HT. Second, the model assumes that virtually all of the 5-HT stored in blood platelets is taken up by them after the lungs, liver, and other organs have cleared a large proportion of the 5-HT released by the gut. While evidence exists this may be the case [

53,

60], not all researchers agree. One could conceivably take into account both of these views by setting

*ser*(

*α*,

*γ*)

*K*_{1 }*F *+

*K*_{2}(1 -

*γ*)

*F*or, in a more general form,

where

*K*_{1},

*K*_{2 }≥ 0 are constants and

*K*(

*ω*) is a function. However, this would require more detailed information about the dynamics of the 5-HT uptake by platelets, which is not currently available [

31].

Distributions generated by the model

While the model (Fig. ) appears to capture some of the key characteristics of the reported platelet 5-HT levels, it remains unclear whether it would produce similar results if *α *and *γ *took on other numerical values. The regulation of the 5-HT release in EC cells is poorly understood and no experimental estimates for the parameter *α *are available. Is it actually lower in autistic individuals? Likewise, how reasonable is it to suppose that the distribution of *γ *is the same in normal and autistic groups? Importantly, would the model produce consistent numerical values of parameters if different experimental studies were used?

To answer these questions, one may consider the basic framework of the model to be correct, but make no *a priori *assumptions about the values of the parameters (with the exception of those that are experimentally known) or about their differences in normal and autistic individuals. Then the unknown parameters of the model may be allowed to vary in the numerical space until the statistical distributions of 5-HT levels produced by the model closely match those reported in actual clinical studies. In order to be able to do this, one first has to find the theoretical statistical distributions of platelet 5-HT levels produced by the model.

The exact population distribution of

*γ *is unknown, but its mean value is likely to be close to one [

60]. Since SERT gene polymorphisms may occur with comparable frequencies [

73], the statistical distribution of

*γ *in a population can be approximated by a continuous uniform distribution on the interval [

*a*,

*b*] with the probability density function

It can be shown from equations (3) and (5) that the probability density function of platelet 5-HT levels then is

The theoretical population mean

*μ*_{ser}(

*α*,

*a*,

*b*) and variance

(

*α*,

*a*,

*b*) of platelet 5-HT levels follow immediately:

and

where

*U * *F*_{C }-

*R*_{C}*α*.

The standard deviation of platelet 5-HT levels in the population then is

Finding *α *and [*a*, *b*] from clinical data

In order to be able to compare the model's predictions with actual clinical reports, the numerical output of the model has to be scaled to the units of the used experimental studies. This scaling can be done by adjusting the parameter

*K *in equation (3). The studies have reported the following means of the blood 5-HT levels in their normal groups: 3.58 nmol/10

^{9 }platelets [

39], 260 ng/10

^{9 }platelets [

40], and 230 ng/ml [

74]. The last number was obtained by pooling the reported pre-pubertal means of the three ethnic groups. Assuming the flux of free 5-HT to the gut is around 210 ng/min in normal individuals [

48,

53,

70], it follows from equation (3) that

where <...> denotes experimentally obtained means. Now we can calculate the approximate

*K *values for each of the studies by dividing their reported mean 5-HT levels by the approximate flux of free 5-HT to the gut. This yields the following

*K *values for the reports of Mulder et al. [

39], Coutinho et al. [

40] and McBride et al. [

74], respectively: 0.0170 (nmol min ng

^{-1 }10

^{-9 }platelets), 1.2381 (min 10

^{-9 }platelets), and 1.0952 (min ml

^{-1}).

Next, we try to find such numerical values of [*a*, *b*], *α*_{normal}, and *α*_{autistic}, that they minimize the difference between the predicted and observed levels of blood 5-HT. Suppose that the observed levels of blood 5-HT vary from *Min*^{OBS }to *Max*^{OBS }and that the observed mean of blood 5-HT is <*ser*>^{OBS}. The following error function can then be constructed:

where

and *i *= *normal*, *autistic*.

Note that, compared with the mismatch between the predicted and observed ranges of the distributions, the mismatch between the predicted and observed means is penalized "twice as much", because observed means are likely to be more accurate than observed minimal and maximal values.

This error function was numerically minimized by using the standard Nelder-Mead (downhill simplex) and differential evolution methods [

75] implemented in Mathematica's

*NMinimize *function (Wolfram Research, Inc.). Since the values of

*R*_{C }and

*F*_{C }may be approximated from published studies but are not necessarily accurate,

*R*_{C }was centered at 3000 ng/min based on a published estimate [

53] and was allowed to vary ± 33%, whereas the value of

*F*_{C }was centered at 210 ng/min based on published estimates [

48,

53,

70] and was allowed to vary ± 50% (more variation was allowed for

*F*_{C }because less is known about its actual value). No constraints were set for the interval [

*a*,

*b*] (i.e., 0 ≤

*a *<

*b *≤ 1). The variables

*α*_{normal }and

*α*_{autistic }were allowed to vary from 0 to 5 and no

*a priori *assumptions were made about their relative values (i.e., both

*α*_{normal }>

*α*_{autistic }and

*α*_{normal }≤

*α*_{autistic }were allowed). It can be shown that the system (equations (1) and (2)) is stable if

*0≤α<F*_{C}(2 -

*γ*)/[R

_{C}(1 -

*γ*)]. Since the system should be stable for any γ

[

*a*,

*b*] and [

*a*,

*b*] is likely to contain the point

*γ *≈ 0.99 [

60] or

*γ *≈ 0.93 [

48], choosing

*α *between 0 and 5 allows the optimization procedure to use virtually any value of

*α *where the system maintains stability.

The numerical values of the model's parameters (*α*_{normal}, *α*_{autistic}, [*a*, *b*], *F*_{C}, and *R*_{C}) that minimized the error function are given in Table . Note that all three clinical studies yielded similar sets of values. Most importantly, the minimization algorithms yielded the best match between the model and the clinical reports when *α*_{autistic }was virtually zero.

| **Table 1**Estimates of *F*_{C}, *R*_{C}, *a*, *b*, *α*_{normal}, and *α*_{autistic}, obtained by numerical minimization of the error function. |

By plugging these obtained values of the parameters into equations (12), (13), (14) and (9), one can obtain the values of 5-HT levels predicted by the model and compare them with the actual observed levels. As shown in Table , the predicted values closely match the values observed in Mulder et al. [

39] and McBride et al. [

74]. The largest mismatch was between the predicted and observed minimal values. The model predicted slightly higher mean 5-HT levels for Coutinho et al. [

40] than were actually observed; interestingly, Coutinho et al. [

40] have in fact reported unusually low platelet 5-HT levels.

| **Table 2**Predicted and observed ranges, means (<*ser*>), and standard deviations (*SD*) of platelet 5-HT levels, *ser*(*α*, *γ*). The distribution of *γ *was assumed to be continuously uniform; the theoretical *SD *values given in the (more ...) |

Distribution of *γ *can be approximated by beta and normal distributions

One advantage of choosing the uniform distribution to represent

*γ *is that it simplifies calculations and allows finding the exact formulae for means and standard deviations. However, the model tends to overestimate the standard deviations of platelet 5-HT levels (Table ), because in the uniform distribution even extreme

*γ *values occur with same probability as all others. Instead of approximating the distribution of

*γ *as uniform, one may want a distribution of which the probability density function drops off more smoothly near the minimal and maximal values. This can be achieved by replacing the uniform distribution of

*γ *with the beta distribution, the uniform distribution being its special case [

76]. The following deals with mathematical technicalities of this replacement. Non-mathematically inclined readers may skip them and go immediately to Figures and referred to at the end of this section.

Note that if the obtained parameter values (Table ) are plugged into equation (3), the normal and autistic platelet 5-HT levels turn out to depend on

*γ *almost linearly (Fig. ). This allows "warping" the uniform distribution of

*γ *into a symmetric beta distribution on the same interval, with little effect on the theoretical mean values of

*ser*(

*α*,

*γ*). Suppose that

*γ *has a symmetric beta distribution on [

*a*,

*b*], whose shape is determined by the parameters

*m *and

*n*, such that

*m *=

*n *(if

*m *=

*n *= 1, the beta distribution becomes the uniform distribution). We can use a Taylor series to formally linearize

*ser*(

*α*,

*γ*) around

*γ*_{0 }= (

*a *+

*b*)/2 as

*ser*(

*α*,

*γ*) ≈

*ser*(

*α*,

*γ*_{0}) -

*λ *(

*γ *-

*γ*_{0})

*serL*(

*α*,

*γ*),

Then, keeping in mind that *γ *has a beta distribution, the standard deviation of *serL*(*α*, *γ*) becomes

Since the values of

*λ*,

*a*, and

*b *have already been estimated (Table ), it is now possible to obtain the

*m *values that yield such standard deviations of the linearized

*ser*(

*α*,

*γ*) that they precisely match those reported in the clinical studies (Table ). The following

*m *values were obtained for the normal and autistic groups, respectively: 1.2940 and 1.7028 for the data of Mulder et al. [

39]; and 1.8308 and 1.8748 for the data of Coutinho et al. [

40]. Pooled standard variations were unavailable in McBride et al. [

74]. We have earlier assumed that normal and autistic groups have the same

*γ *distribution. Therefore, the actual

*m *values can be approximated by 1.50 for Mulder et al. [

39] and 1.85 for Coutinho et al. [

40].

Likewise, *γ *can be assumed to have a normal distribution with mean (*a *+ *b*)/2 and standard deviation *σ*. Then the standard deviation of *serL*(*α*, *γ*) becomes

*σ*_{serL}(*α*, *a*, *b*, *σ*) = *λσ*, (17)

where

*λ *is the same as in equation (15), and we obtain the following

*σ *values for the normal and autistic groups, respectively: 0.0410 and 0.0370 for the data of Mulder et al. [

39]; and 0.0630 and 0.0624 for the data of Coutinho et al. [

40]. Therefore the actual

*σ *values can be approximated by 0.04 for Mulder et al. [

39] and 0.06 for Coutinho et al. [

40].

The model now easily generates "normal" and "autistic" samples of platelet 5-HT levels that closely match the actual reported data (Fig. ). Most importantly, the switch from the normal distribution to the autistic distribution requires changing only one parameter, *α*.

It is not known what normal and autistic distributions would look like if one could sample a very large number of subjects. The model can predict the shape of these distributions by simulating such large sampling (Fig. ).

Is the 5-HT synthesis rate altered in autism?

One of the most important questions in autism research is whether the rate of 5-HT synthesis is altered in the brain and gut of autistic individuals. If 5-HT synthesis is altered in the autistic brain, as some studies have suggested [

77-

79], this potentially may have a great impact on brain development [

80,

81] (but caution should be exercised in predicting the extent of these alterations [

82]).

The brain 5-HT and the gut 5-HT are synthesized by two different tryptophan hydroxylases [

49] that, at least in humans, have different properties and are regulated differently [

83]. While the biological factor underlying the parameter

*α *of the model is hypothesized to play a role in the developing brain (Fig. ), the model makes no assumptions about its exact function in the brain. In the brain, it may not regulate 5-HT release from serotonergic neurons and may have a different function (see, for example, Figure 4 of [

48]). Therefore, this section focuses only on the 5-HT synthesis and release in the gut.

It is important to note that the model says nothing about the rate of 5-HT synthesis in the gut and rather deals with the rate of 5-HT release from the gut. However, most clinical and experimental studies make no such distinction and, therefore, their relevance to the model is discussed assuming higher 5-HT synthesis rates do lead to higher 5-HT release rates.

It follows from equations (3) and (4) that, at the steady state,

and that this relationship is independent of *γ*. This means that if one were to sample any group of individuals and could measure their platelet 5-HT levels and gut 5-HT release rates precisely, the correlation coefficient between these two variables would always be minus one, irrespective of the distribution of *γ*. In other words, equation (18) predicts that individuals with higher platelet 5-HT levels should have lower 5-HT release rates.

How can lower 5-HT release rates lead to higher platelet 5-HT levels? Note that, in the model, both the platelet 5-HT levels and the 5-HT release rate are dynamically linked through the 5-HT clearance rate, *γ*. As *γ *grows lower, less 5-HT is removed from the system and more of 5-HT is accumulated in blood platelets. At the same time, these higher 5-HT levels drive down the 5-HT release rate in the gut, as required by equation (1).

Still, it appears that the results of clinical studies are inconsistent with equation (18). Three important findings should be noted:

(i) Minderaa et al. [

36] have found no significant correlation between whole blood 5-HT levels and 5-HT synthesis in the gut, measured as the production of urinary 5-HIAA [

36]. Similar results have been obtained by Launay et al. [

84] and other groups (reviewed in [

31]).

(ii) Croonenberghs et al. [

85] have shown that the 5-HT synthesis in the gut of autistic individuals may be higher than that in normal individuals, at least when subjects are administered 5-hydroxytryptophan (5-HTP), an immediate precursor of 5-HT.

(iii) Carcinoid tumors, derived from gut EC cells, may result in excessive synthesis and release of 5-HT, which in turn may lead to elevated platelet 5-HT levels [

86].

A more careful analysis reveals that these findings are not only consistent with the model, but that the model can reconcile some of the apparent contradictions among them:

(i) It follows from the model that the measured correlation between platelet 5-HT levels and 5-HT release rates should be close to zero in autistic groups, even though equation (18) holds.

In fact, we can rewrite equation (18) as

Now consider two random variables, *η *and *ξ*, that are linearly dependent such that

*η *= *wξ *+ *q*, (20)

where *w *and *q *are constants. It follows from equation (20) that the correlation between them is either -1 or 1, depending on the sign of *w*.

Denote the means of these variables *μ*_{η }and *μ*_{ξ}, respectively, and their standard deviations *σ*_{η }and *σ*_{ξ}, respectively. Suppose next that the errors of measurement of *η *and *ξ *are independent random variables *ε*_{η }and *ε*_{ξ}, such that their expected values are zero and standard deviations are *δ*_{η }and *δ*_{ξ}, respectively. Note that experimentally we can measure only *η** = *η *+ *ε*_{η }and *ξ** = *ξ *+ *ε*_{ξ}. The expected values of *η** and *ξ** are the same as those of *η *and *ξ*. However, the theoretical correlation coefficient between *η** and *ξ** now becomes

If the standard deviations of the errors of measurement are small, we obtain *ρ*(*η**, *ξ**) ≈ ± 1, as expected from equation (20).

Now we return to equation (19). Any experimental measurement of *R *(5-HT release) and *ser*(*α*, *γ*) (platelet 5-HT levels) will contain a measurement error. Denoting these measured values *ser**(*α*, *γ*) and *R**, one obtains from equations (19), (20), and (21) that the correlation coefficient between *R** and *ser**(*α*, *γ*) is

where

*w *= -(*αR*_{C})/(*KF*_{C}), (23)

*σ*_{ser }> 0 is the standard deviation of *ser*(*α*, *γ*), and δ_{R} > 0 and δ_{ser} > 0 are the standard deviations of the errors of measurement of *R *and *ser*(*α*, *γ*), respectively. The estimated values of *K*, *F*_{C}, *R*_{C}, and *α *can be obtained from Table and the values of *σ*_{ser }from Table or from the original published data.

Consider now an autistic group whose

*α *→ 0 (Table ). Then, from equation (23),

*w *→ 0, and it follows from equation (22) that

.

(ii) Croonenberghs et al. [

85] have recently shown that oral administration of 5-hydroxytryptophan (5-HTP) leads to higher platelet 5-HT levels in autistic patients, and the authors have suggested that the 5-HT synthesis rate may be higher in the gut of autistic subjects compared with normal subjects.

Suppose that the administered 5-HTP is converted to 5-HT at the same rate in both normal and autistic groups. It is likely that the exogenous influx of 5-HTP results in a comparable exogenous influx of 5-HT, because the rate-limiting step in the synthesis of 5-HT is not the 5-HTP conversion to 5-HT, but rather the tryptophan conversion to 5-HTP [

87].

Notice that the system is not in its steady state during the experiment and, therefore, we have to use equations (1) and (2), which now should contain the exogenous source of 5-HT. It is straightforward to see that the system then becomes

*F*_{n + 1 }= (1 - *γ*)*F*_{n }+ *R*_{n + 1 }+ *R*_{EX}, (25)

where *R*_{EX }is the exogenous flux of 5-HT.

Solving equations (24) and (25) step-by-step essentially replicates the major finding of Croonenberghs et al. [

85] (Fig. ). However, the model predicts that the higher blood 5-HT levels in autistic subjects are not due to a higher 5-HT synthesis rate, but rather to the failure of their gut to decrease the release of endogenous 5-HT in response to the high concentration of 5-HT caused by the administration of 5-HTP.

(iii) In the case of carcinoid tumors, abnormally large amounts of 5-HT may be released into the blood. It is likely that the normal mechanisms regulating 5-HT release are compromised or absent in carcinoid tumors. Then instead of equations (1) and (2) one can consider only one equation (2), which can be rewritten as

*F*_{n + 1 }= (1 - *γ*)*F*_{n }+ *R*_{CARCINOID}, (26)

where *R*_{CARCINOID }is large and relatively constant. Then, at the steady state,

*F *= *R*_{CARCINOID}/*γ*

and

It is obvious that in this abnormal case higher 5-HT release rates will lead to higher platelet 5-HT levels, as reported by Kema et al. [

86].