1. MS penetrance in the general population–P(MS)

Conclusion: P(MS)≈0.003

Argument: One possible estimate of P(MS) could be the prevalence of MS in a population. However, because the clinical onset of MS occurs largely between the ages of 15 and 45 years (e.g., Fig 1), the measured cross-sectional prevalence of MS (using the entire population as the denominator) will necessarily include individuals with different likelihoods of having already developed MS [35]. For example, using the 2010 United States census data (for the total resident population–see Fig 2) as an approximation, we can divide the general population (Z) into the three mutually exclusive age-bands (A1, A2, and A3), such that:

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 1. The distribution of the age at the clinical-onset of disease in a cohort of 1,463 patients with MS (sd = standard deviation).

Data from Liguori et al. [35].

https://doi.org/10.1371/journal.pone.0246157.g001

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 2. US census data (for each decade from 1980 to 2010) for resident population by age and sex.

Age-band (A1) is highlighted in turquoise; Age-band (A2) is highlighted in yellow; and Age-band (A3) is highlighted in grey.

https://doi.org/10.1371/journal.pone.0246157.g002

Because so few of MS patients have their disease onset prior to the age of 15 years (e.g., Fig 1) it seems a reasonable approximation that:

By contrast, as noted above, the age group (15–45 years) accounts for the large majority of clinical onsets, which have a roughly symmetrical distribution with a mean of 28.3 years (Fig 1). If the distribution were exactly symmetrical and centered on 30 years, the measured prevalence in this age band would be ~50% of the value of P(MS). Therefore, it seems reasonable to estimate:

For the older age band (>45 years) most patients will have already developed the disease (Fig 1). Thus, on the one hand, one might expect that the measured prevalence in this age-band to be equal to P(MS). On the other hand, there is a small but definite excessive mortality in MS such that life expectancy is reduced in MS-patients by about 5–10 years [36–40] although a recent study from Denmark [41] reported that short-term survival has steadily improved for patients beginning in 1950 and continuing through 1999. This excessive mortality will make the estimate too small by some amount although it seems unlikely that this reduction will be more than 25%. Thus, a range of plausible estimates is likely to be: Combining these three different estimates yields the estimate: Defining the measured prevalence in the population as (prev), this estimate translates to:

A second method to estimate P(MS) would be to use a measured prevalence for MS, which is restricted to the age-band of 45–54 years. Thus, within this age-band, almost all patients will have already experienced their clinical-onset and only a few will have experienced their (expected) excessive mortality. Consequently, by this method:

A third method would be to use population-based death data and to consider the percentage of death certificates that mention the diagnosis of MS (not necessarily as, but including, the immediate, underlying, or contributing cause). Thus, by the time of death, any case of clinically evident MS must, by definition, have already declared itself. Consequently, by this method:

In 2001, we took a cursory (unpublished) look at the Kaiser northern California database. At the time, there were 4,352 unique persons in the database with a diagnostic code for MS. With 2.9 million persons enrolled in Kaiser northern California at the time, and if this population is a representative sample, this would translate to an MS-prevalence in northern California of 150 per 100,000 population. Such an estimate is consistent with many other published studies in northern populations, which generally find the prevalence of MS to be 100–200 per 100,000 population [42]. A recent study from the United States, using multiple administrative health claims (AHC) datasets [43] estimated that the prevalence of MS in adults (i.e., age ≥ 18 years)–which represents ~75% of the US population (see Fig 2)–as 288–309 per 100,000 population. For comparison purposes, this translates to a prevalence in the entire population of 216–232 per 100,000 individuals.

Similarly, in a Swedish study by Sundström and co-workers [44], the age-specific prevalence of MS in the 45–54 year age-band was reported to be 304 per 100,000 population. In the AHC study [43], the estimate for this same age-band (considering the entire population) was 314–337 per 100,000 individuals.

And, finally, in a recent population-based multiple-cause-death study from British Columbia [45], a diagnosis of MS was mentioned on 0.28% of the death certificates.

Thus, all three of these methods of estimation are quite consistent with each other. The range of values supported, collectively, by these observations is:

The best support is for the conclusion that, in the northern populations of Europe and the Americas:

However, despite the notable consistency of these three estimates, each of these methods relates only to “diagnosed” MS in the general population (Z). If undiagnosed (i.e., pathological) MS is included in the calculation [46–49], this estimate may increase by as much as 50–100% (see #8 below).

2. Adjusting for the shared environment of MZ-twins–P(MS│IG_MS)

Conclusion: P(MS|IG_MS) = P(MS|G,IG_MS)≈0.134

Argument: Most epidemiological studies in northern populations report the proband-wise concordance rate for MZ-twins to be in the range of 25–30% [27–34, 50]. Using the population-based data from Canada (Table 3; Fig 3), leads to the estimate of:

Suppose that each of the (n) individuals (k = 1,2,…,n) within the general population (Z), has a unique genotype (G_k), in which case:

We can then define the term (IG_MS) such that:

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 3. Summary of epidemiological data regarding MS in Canada circa 2000–2010.

The value for P(H+) was provided by D. Sadovnick, was based on 400 Canadian controls, and the rate was confirmed in a large transplant database (personal communication). The F:M sex-ratio in the general population of Canada was taken from the 2010 Canadian census. Recurrence risks for monozygotic (MZ) twins, dizygotic (DZ) twins, and siblings (S) were taken from the study of Willer et al. [29]. The other summary data was taken from Table 3, and/or from the study of Willer et al. [29]. The F:M sex-ratio among Canadian MS patients at each of the 5-year time-periods (1941–1945 & 1976–1980) was taken from the study of Orton et al. [56].

https://doi.org/10.1371/journal.pone.0246157.g003

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 3. Canadian population data on MZ-twin concordance broken down by (H+)-haplotype and gender-status ^*.

https://doi.org/10.1371/journal.pone.0246157.t003

This is just the expected “adjusted” penetrance for the (MZ_MS) subset. As discussed earlier and, as developed in the S1 File (#1), {P(MS|IG_MS)} can be estimated from the difference in observed concordance rates between siblings and fraternal twins. Using the Canadian population-based data (Table 3; Fig 3) on the recurrence risks in non-twin siblings and DZ-twins (concordance rates for siblings = 2.9%; concordance rates for DZ-twins = 5.4%) to make this adjustment (see above) leads to the estimate of:

3. Proposition #1

Assertions: 1. Upper Solution: x’/2<x≤x’; and Lower Solution: 0<x<x’/2

2. The variance of {X} conforms to the conditions:

3. If the individual values for penetrance within the (G) subset are

distributed in a unimodal manner, then:

4. If the penetrance values in (G) are distributed in a non-unimodal manner, and if: P(MS|G)≥x’/2, then the Upper Solution limits applies and:

5. And finally, for more extreme non-unimodal distributions of (G)–i.e., P(MS|G)<x’/2– then the Lower Solution applies and:

Proof: For notational simplicity, as discussed previously, we use abbreviated terms for several parameters (see Table 2). Among the (n) individuals in the general population (Z), we have already defined the (G) subset, which consists of everyone who has any non-zero life-time probability of developing MS. Thus, each of the (m) individuals in the (G) subset (i = 1,2,…,m) has a unique genotype (G_i), such that:

We define (see Table 2) the parameters, (x_i) and (x), such that:

Thus, (x_i) represents the expected penetrance for MS in the i^th individual of the (G) subset. Even if this penetrance exactly matches that of another person, (x_i) is still unique to the i^th individual. Also, considering the penetrance values for each of the members of the (G) subset, we can define the set {X} such that:

Because the (G) subset forms a partition of the population (Z), each of the (m₂ = n−m) individuals, who are not in the (G) subset, belongs to the mutually exclusive “non-susceptible” subset (G−). Moreover, each of the (m₂) individuals in the (G−) subset (j = 1,2,…,m₂) has a unique genotype (G_j), which has a zero conditional life-time probability of developing MS, so that:

Also, if {Var(X)≠0}, we can partition the subset (G) into two mutually exclusive sub-subsets, (G1) and (G2), suitably defined, such that the sub-subset (G1) has a penetrance greater than that of (G2). Again, for ease of notation, we define the quantities (x’,x₁,x₁’,x₂, & x₂’)– see Table 2 –such that:

In earlier iterations of this analysis [3, 4, 51, 52], we defined the subset (G) differently–i.e., ∀G_i∈G: P(MS|G_i)≥P(MS). We have chosen the current definition because it considerably simplifies the biological interpretation of the findings. Nevertheless, we note that, when circumstances fit the conditions of the Lower Solution (see below), the new sub-subset (G1) is, effectively, identical to the subset (G) defined earlier.

We define the term {P(MZMS)} to represent the life-time probability of developing MS for any single individual from an MZ twin-ship (i.e., where the status of their co-twin is unknown). Because identical twinning is considered non-hereditary [53], we expect that:

As noted earlier, we also define the set {X} set to consist of the individual MS-penetrance values for all members of the (G) subset. Thus, the variance of the set {X} can be expressed as:

It follows directly from the definitions of {P(G)} and {P(MS│IG_MS)}–see Methods & #2, above–that:

1. Therefore, the probability {P(MS,G_i|G,IG_MS)} can be re-written as:

2. In turn, the term P(G_i|G,IG_MS) can be re-written as:

Combining these two Equations (i.e., 1 & 2 above) yields:

Notably, this equation can also be rearranged to yield a quadratic in (x) of:

In turn, this quadratic equation can be solved to yield:

which has real, non-negative, solutions only for:

The maximum variance for any distribution [54, 55] on the closed interval [a,b] is: Consequently, the maximum variance for the set {X} is identical to that for the interval [0,x’], which is:

In addition, this maximum variance, (x’/2)², occurs when the distribution of penetrance values in the set {X} is bimodal [54, 55], such that half the (G) subset has a penetrance of (0) and the other half has a penetrance of (x’). From this point of maximum variance, the variance of the {X} subset decreases both when:

x→x’ and: x>x’/2 (the Upper Solution)

and when: x→0 and: x<x’/2 (the Lower Solution)

By definition, any solution requiring {P(MS|G_i) = 0} for any portion of (G) is excluded.

Therefore, the Upper Solution limits become: x’/2<x≤x’

And the Lower Solution limits become: 0<x<x’/2

Moreover because: Therefore, if: ; then: x’ = x

Using abreviated notations (Table 2), notably, there are three other related equivilences: (Eq 1A) (Eq 1B) (Eq 1C)

Because, by definition (x₁>x₂) –see Methods –therefore, applying Eq 1B: also: if: x₂>x’/2; then x>x’/2; and the distribution of {X} will conform

to the Upper Solution (see above).

Also, applying Eq 1C: if: (x₁’>x₂’); then: (x₁’>x’>x₂’)

4. Genetic susceptibility to MS–general considerations

4a. The Upper Solution.

Conclusions: 0.022≤P(G)≤0.045

Argument: From the Upper Solution in Proposition #1 and in conjunction with our estimate from #2 (above) for {P(MS|IG_MS)}, it follows directly, that: We can then apply the relationship developed in the Methods that:

With this we have all the data necessary to establish the limits for the percentage of the population who are members of the (G) subset. Thus, using this range for P(MS│G), together with our estimate for P(MS)–see #1 above–it follows that:

Consequently, by this analysis, only 4.5% or less of the general population (Z) could possibly be genetically susceptible to getting MS and the remainder of the population would have no possibility of getting this condition, regardless of their environmental experiences. Multiple reports from other MS-populations throughout Europe and North America yield very similar Upper Solution estimates for P(G), which seems to be independent of latitude (Table 4).

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 4. MS prevalence, MZ-twin concordance {P(MS│MZ_MS)}, and “genetic susceptibility” {P(G)}–for the upper solution (see Text)–in different geographical locations.

https://doi.org/10.1371/journal.pone.0246157.t004

Notably, we arrived at this estimate for {P(MS|IG_MS)} by adjusting the observed value of {P(MS|MZ_MS)} downward to account for the presumed impact of the shared IU and early post-natal environments of MZ-Twins (see #2 above). To do this, we estimated the magnitude of this impact from the increased recurrence risk in DZ-twins compared to that in non-twin siblings (see Methods; see also #1, in S1 File). Although, the Canadian data suggests a larger discrepancy between {P(MS│DZ_MS) and P(MS│S_MS)} compared to other studies [27–34, 50], it is still possible that our adjustment is too small. Even so, there is a limit to how large any adjustment can be. Thus, from Fig 3, it must be the case that:

Otherwise, there would be no increased risk of MS in persons who have 100% of their genes in common and don’t share their IU and post-natal environments compared to persons who have only 50% of their genes in common and also don’t share their IU and post-natal environments. Importantly, however, even in this case:

Therefore, even using this extreme estimate, the large majority of the population (>79%) would have no chance of getting MS, regardless of their environmental exposures (see Proposition #1).

5. Genetic susceptibility in the gender partition–P(F│G) & P(M│G)

Conclusions: 1. The set {X} has, at least, a bimodal distribution

2. 0.145≤P(MS|F,G)≤0.187

3. 0.017≤P(MS|M,G)≤0.034

4. 0.18≤P(F|G)≤0.31

5. 4.3≤P(MS|F,G)/P(MS|M,G)≤8.7

6. 0.041≤P(G)<0.073

Argument: For ease of notation, the Table 2 parameter abbreviations (x,x’,x₁,x₁’,x₂, & x₂’) can be applied to the gender partition by defining both susceptible women and susceptible men such that: {(G1) = (F,G)} and: {(G2) = (M,G)}. The set {X} of penetrance values for members of the (G) subset is, at least, bimodal. Thus, from the data in Fig 3:

Because the sub-subsets (G1) and (G2) have significantly different expected pentrances, we assume that each, considered separately, conforms to the Upper Solution (see Methods). Therefore, from the estimated adjustments for the similar environment of MZ-twins for this partition (see #1.1b, in S1 File), together with the data in Fig 3, it follows that: (Eq 2A) (Eq 2B)

These possible ranges for men and women don’t overlap. Therefore, for this partition, we have defined (above) the sub-subsets (G1) and (G2) correctly because: (x₁>x>x₂) –see Methods. In this case: (a>1>b) and, as a consequence, P(G1|G,MS) must be greater than P(G1|G)– see #2a, in S1 File. The proportion of MS patients who are women from Table 3; Fig 3 is 66%. For the WTCCC data this number is 72%. From the study of Orton and colleagues [56] out of Canada, in the most recent epoch, the percentage of MS patients who are women is 76%. From a recent prevalence estimate for the United States [43], the percentage of women among MS patients is 74%. Using the data from Table 3; Fig 3, one possible upper limit for P(F|G) is: P(F|G)<P(F|G,MS) = 0.66. Nevertheless, any such upper limit is too high. For example, using: P(F) = P(M) = 0.5, together with the other Table 3; Fig 3 data and the above noted ranges for men and women, and from the definition of the (G) subset, we can estimate that:

Because: P(G|F) = P(MS,G|F)/P(MS|F,G)

Therefore: 0.021 = 0.004/0.187≤P(G|F)<0.004/0.093 = 0.043

And similarly: 0.060 = 0.002/0.034≤P(G|M)<0.002/0.017 = 0.118

Therefore, the maximum proportion of women in the (G) subset (using the Table 3; Fig 3 data) must be:

In fact, the gender imbalance may be even greater than this (see #4, in S1 File). Thus, there are four serious concerns about undertaking any calculations that use the limits for (x₁ and: x₂) set forth by Eqs 2A & #2B, above. First, in making the above calculation, we are positing an extreme and tri-modal distribution for the set {X}–i.e., not the unimodal or bimodal distributions under primary consideration. Thus, this calculation, envisions a distribution, in which half of the women have a uniform penetrance of slightly greater than zero and the other half have a uniform penetrance of (x₁’) –i.e., women have the maximum variance possible–and, in which every man has exactly the same penetrance of (x₂’), which is intermediate between these two extreme penetrance groups of women–i.e., men have a zero variance.

Second, such an extreme distribution seems unlikely, especially for circumstances, in which partitioning the (G) subset by a different MS-associated characteristic–i.e., HLA-status (see #6, below)–doesn’t even give a hint of the bimodal nature of {X}.

Third, it is not possible that the variance of penetrance values for the (F,G) subset to be at its maximum value. Thus, because, (x₁’>x’) –see Table 3; Fig 3 –the maximum variance for the sub-subset (F,G) –(x₁’/2)²–exceeds the maximum total variance possible for the entire (G) subset– (x’/2)². Consequently, the lower limit for the value of (x₁) in Eq 2A –i.e., at its maximum possible variance–must be too low. And fourth, some of the maximum possible variance in the {X} set must be accounted for just by the separation of (x₁) from (x₂) –see #4, in S1 File.

Following the standard development of variance relationships [57], and taking each of these factors into account (see #4, in S1 File), including all solutions (either Upper or Lower), in which the penetrance values of (G1) and (G2) each follow an Upper Solution, leads to the conclusion that:

Importantly, however, if the distribution of {X} follows an Upper Solution, those limits still apply (see #4a, above) although the somewhat different estimates for P(G), in this circumstance, would need to be reconciled. Because the estimate derived from Table 3 for the quantity {P(MS|M,IG_MS)} is based on only two concordant twins, this seems likely to be the least reliable of any in the Table. Thus, if this estimated penetrance were doubled, there would still be an excess of men in the (G) subset such that:

Consequently, an underestimate of {P(MS|M,IG_MS)} would help with any such reconciliation. Similarly, considering only the possibility that the penetrance values of both (G1) and G2) are distributed in a unimodal manner would also help (see #4, in S1 File), as would an underestimate (from Table 3; Fig 3) for the proportion of women among MS patients (see above, this section, see #8, below, & see #3b, in S1 File).

Regardless, however, it seems clear not only that genetic susceptibility is rare in the population, even for Lower Solutions, but also that men are more likely than women to be genetically susceptible to MS. At first pass, it might seem biologically improbable that men would be more likely than women to be in the genetically-susceptible subset (G). Thus, if membership in the (G) subset is envisioned as being due to an individual possessing a sufficient combination of some number of loci in a “susceptible state” [58], it is unclear how men could be more likely than women (or vice versa) to possess certain combinations and not others. This seems especially unlikely for circumstances, where one association study, specifically focused on the X-chromosome, failed to identify any susceptibility loci on this chromosome [7], where another large GWAS found that all but one of the 233 MS-associated loci were located on autosomal chromosomes [14], and where no major gender interaction term has been reported in the literature. Indeed, considering the different “risk” haplotypes in the HLA region identified in the WTCCC, men and women seem equally likely to be carriers [59]. Nevertheless, we can designate (G_ak) to represent each of the (n) autosomal genotypes (k = 1,2,…,n) in the general population (Z) − i.e., omitting any specification of gender. In this circumstance, it is entirely possible that: and, yet, for some specific autosomal genotypes to have the characteristic that:

Indeed, such an explanation for the excess in susceptible men would fit well with the observation that the specific genetic combinations, which underlie susceptibility to MS, seem to be unique to each individual (see #9, below; see also #7, in S1 File). In addition, such a circumstance might also help to rationalize the finding that men seem to have a lower threshold of environmental exposure for developing MS compared to women (see #7, below).

6. Genetic susceptibility in the HLA partition–P(G│H+) & P(G│H−)

Argument: We will designate individuals who possess 1 or 2 copies of the Class II HLA-DRB1*15:01~HLA-DQB1*06:02~a1 haplotype–i.e., the (H+) haplotype–as being members of the (H+) subset and those who possess 0 copies of this haplotype as being members of the (H−) subset. Some epidemiological studies only report HLA-DRB1*15 or HLA~DRB1*15:01 carrier status. Nevertheless, because, in the WTCCC, 93.4% of HLA-DRB1*15-alleles are actually the HLA-DRB1*15:01 allele, and because 99% of HLA-DRB1*15:01 carriers also carry the full haplotype [60], each of these designations will be used interchangeably as (H+).

It is clear that (H+) status is considerably enriched in the MS population compared to controls. For example, in WTCCC controls {P(H+) = 0.23}, whereas in cases {P(H+|MS) = 0.50}. This enrichment of (H+) status in MS could occur in two ways (see #5, in S1 File). First, (H+) could make membership in the (G) subset more likely than it is for the (H−)-subset–i.e., it is due to an impact on the ratio of: P(G|H+)/P(G|H−). Second, members of the (G,H+) subset may have a greater penetrance for MS than members of the (G,H−) subset–i.e., it is due to an impact on the ratio of: P(MS|G,H+)/P(MS|G,H−). The available epidemiological data (see #5, in S1 File) suggests that the majority of enrichment is the due to the 1^st of these two possible mechanisms and that:

In addition, the observation (from the Lower Solution) that less than 7.3% of the population is genetically susceptible (see #5; above), together with the WTCCC observation that: P(H+) = 0.23, indicates that fewer than 32% (7.3/23) of (H+)-carriers are even genetically susceptible to MS. Indeed, taken together, the fact that only half of MS-patients are in the (H+) subset and the fact that this estimate for genetic susceptibility represents an upper bound for the Lower Solution, indicates that the actual percentage of (H+) carriers who are genetically susceptible must be far less than this 32% figure. Nevertheless, essentially all of the conserved extended haplotypes (CEHs) that carry (H+) − even those with a single representation in the WTCCC dataset − are associated with MS [60]. Therefore, it seems likely that all (H+)-carrying CEHs can contribute to genetic susceptibility. Despite this contribution, however, the majority of (H+) subset members have no chance whatsoever of developing MS. Therefore, at least with respect to the (H+)-carrying CEHs, genetic susceptibility to MS must result from the combined effect of (H+) together with the effects of certain other (as yet, unidentified) genetic factors (see #7, in S1 File). By itself, however, (H+) membership poses no MS-risk.

7. Environmental factors in MS

1. Conclusions: 1. Environmental factors are critical to MS pathogenesis
2. 2. Susceptible women are more responsive than men to changes in the environmental conditions related to MS pathogenesis.
3. 3. Compared to women, men have a lower threshold of environmental exposure at which they can develop MS
4. 4. Currently, the environmental factors involved in MS pathogenesis are population-wide exposures.
5. 5. Stochastic factors play an important role in MS pathogenesis

Argument: As noted in the Methods, we define (E_T) to be the prevailing environmental conditions (whatever these are) experienced by the population during some time-period (T). We also define (E_i) to be the specific environmental exposure, which is sufficient for MS to develop in the i^th susceptible individual (however many events are involved, whenever these events need to act, and whatever these events might be)–i.e., both the events (E_i and G_i) need to occur jointly in order for MS to develop in the (i^th) individual. Because genetic susceptibility is independent of the environmental conditions, the probability of a sufficient environmental exposure {P(E)} in the (G) subset at time-period (T) can be expressed as:

When {P(E) = 0}, it is not possible for any susceptible person to experience an environment sufficient to cause MS. By contrast, when {P(E) = 1}, every susceptible person experiences an environment sufficient to cause MS. If there are some susceptible individuals, for whom any environmental experience is sufficient to cause MS (i.e., these individuals have “purely genetic” MS), then: 0<P(E)≤1 and thus, {P(E) = 0} cannot be observed. Importantly, those circumstances, in which {P(E) = 0}, only imply that, whatever environmental exposures take place (i.e., E_T), these are insufficient to cause MS in anyone. Regardless, considering the definitions of both P(E) and the (G) subset (see Methods), it is clear that:

Notably, also, the above expression for P(E) explicitly incorporates the possibility that each genotype in (G) may require a unique set of environmental events in order for MS to develop in that individual. Nevertheless, despite this possibility, the existing epidemiological data suggests that many (or most) MS patients are responding to similar environmental events and, thus, any large variability in this regard is probably not a major factor in MS pathogenesis.

For example, despite the fact that every MS patient (except MZ-twins) has a unique combination of “states” at the (>200) susceptibility loci (see #7, in S1 File), the population-based data from Canada indicates that the change in general environmental conditions (whatever these are), which have taken place between the time periods of (1941–1945) and (1976–1980), have produced, at a minimum, a 32% increase in the prevalence of MS (see #6d, in S1 File). Moreover, because this increase has occurred world-wide and predominantly in women [3, 4, 51, 52, 56], the (F:M) sex ratio for MS in Canada has increased during every 5-year increment except one between these two time-periods [56]. Over the entire interval, the ratio has increased from 2.2 in (1941–1945) to 3.2 in (1976–1980). These changes are far too rapid to be genetically based.

It is conceivable that this observed sex-ratio change might be artifactual. For example, if women were more likely than men to have minimally symptomatic MS, then, with such patients now being diagnosed by our improved imaging and laboratory methods, women might represent a disproportionate number of these newly diagnosed cases. Alternatively, in earlier eras, vague symptoms of MS in women may have been written off as “non-organic” more often than they were in men. Nevertheless, four lines of evidence argue strongly against this change being an artifact. First, this increase in the sex ratio began before, and continued up to, the advent of modern imaging and laboratory methods [56]. Second, among asymptomatic individuals, incidentally, found to have MS by MRI, the (F:M) ratio is approximately the same as current estimates for symptomatic MS and 80% of the those with spinal cord lesions are women − i.e., those lesions having, by far, the greatest odds for progression to “clinical” MS [61]. Third, if (as seems likely), women have a higher threshold for developing MS than men, this would require the difference in exposure between the genders to be one of degree not one of kind (see below, this section; see also #6e, in S1 File). Finally, and most persuasively, the greater penetrance of MS in women is confirmed independently by the MZ-twin data (see #5 above). Consequently, the increase observed in the (F:M) sex ratio of Canada [56] almost certainly has an environmental basis.

In addition, a prior Epstein Barr viral (EBV) infection seems to be a prerequisite for most (or all) genotypes in (G) to develop MS [3, 4, 51, 52, 62–64]. Indeed, if (as suggested by these studies) a prior EBV infection occurs in 100% of MS cases, this would indicate that EBV exposure can be designated as a ‘necessary factor’ and, as such, must be part of the causal pathway leading to MS [51]. In addition, the likelihood that members of the (G) subset will develop MS seems to be influenced greatly by vitamin D deficiency, latitude, migration, and the IU environment [3, 4, 51, 52, 62–64]. Each of these additional observations also indicates that similar environmental changes can affect a large proportion of genetically susceptible individuals in a similar manner (i.e., contribute to MS pathogenesis).

Using the standard methods of survival analysis [65], we can define the cumulative survival {S(u)} and failure {F(u)} functions as well as the hazard-rate functions {h(u)} and {g(u)} for developing MS at different environmental exposures in “susceptible” men and women (respectively). These hazard-rate functions are assumed (initially) to be proportional. The implications of non-proportionality are considered in the in S1 File (#6e) and in the legend of Fig 4. However, assuming proportionality, then:

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Fig 4. Response curves for the likelihood of developing MS in genetically susceptible men and women with an increasing probability of a sufficient environmental exposure {P(E)}, assuming proportional hazards (see #7).

Response curves are derived from the change in the (F:M) sex-ratio over time in Canada [56] and using the estimates for P(MS)₂ & P(G) estimated in the Text. The probability of getting MS in a genetically-susceptible individual–i.e., P(MS,E|G) –is shown on the y-axis. The exposure level {P(E)} for the population is shown on the x-axis using transformed “exposure units” (a)–see #7, Text. Labels for points Zw = P(MS,E|G,F) and Zm = P(MS,E|G, M) are provided at time-points (1) and (2). One “exposure unit” is defined arbitrarily as: (a₂−a₁) for men and (a₂^app−a₁^app) for women (see #7). Solid-line plots have been constructed using the known values of (Zw₂) and (Zm₂), estimating that: P(F|MS)₂ = 0.66 (Table 3; Fig 3), together with the estimates of: {C = 0.6}, {R = 1}, {P(G) = 0.044} & {P(F|G) = 0.25}– the latter two parameters being valued in the middle of their predicted ranges (see, Text). The limiting values for (Zm) and (Zw) are: (c = 0.035) and (d = 0.228), respectively. Increasing the estimate of P(F|G) will reduce the separation of the response curves by lowering the plateau for women and raising it for men; increasing the estimate of P(F|MS)₂ will increase the separation of between the curves in men from women for the opposite reason; increasing the estimate of P(G) will reduce the plateaus of both response curves. Response curves for women under conditions (R = 0.67) and: (R = 1.5) are also depicted and are shown in grey lines (dashed and dotted, respectively). Changes to the value of (C) will slightly alter the units of the y-axis. As seen in the Figure, men have a lower threshold for developing MS compared to women (see #7, Text), and changes to the value of (R) alter how quickly the curves reach their plateau (limit). If the hazard is not proportional, for women, each of the points (Zw₁, Zw₂, Zm₁, and Zm₂) would be the same as depicted for (R = 1), although the scale of the x-axis for the two exponential curves would be transformed non-linearly and, thus, the response-curve in men could not be plotted on the same graph as women. Moreover, the x-intercept for the curve in women would be at (a^app = λw = 0). Nevertheless, the limiting values (c) and (d) would be unchanged and, under any circumstances, women (relative to men) would still be seen to exhibit a greater responsiveness to those changes in environmental exposure, which have taken place between the two time-periods. If some cases of MS were “purely genetic” (i.e., P(E|M) = 0, or P(E|F) = 0 or both were not possible), this could elevate the zero-point on y-axis for “environmental” MS to the intersection of the curves for men and women (see Text) and this would make the threshold difference then disappear (i.e., λ = 0) –see #7, Text.

https://doi.org/10.1371/journal.pone.0246157.g004

For men, we can transform exposure from (u) units into (a) units, first by defining {H(u)}to be the definite integral of the hazard-function {h(u)} from a (u) level of exposure to a (0) level of exposure and, second, by defining the (a) units to be:

Because these (a) units are arbitrary, we can assign “1 unit” of environmental exposure in men to be the difference in exposure level between any two time points (e.g., a₁ and a₂) such that:

For women, we can similarly transform exposure into a different scale of so-called “apparent” exposure units (a^app) such that: and where we now define “1 unit” of environmental exposure (on this scale) as: The choice of which gender (men or women) to assign to which scale is completely arbitrary.

A standard derivation from survival analysis methods [65], demonstrates that the survival curves are exponential with respect to their hazard functions.

Thus, for men:

and, for women:

So that, for men:

and, for women:

In considering the probability of failure (i.e., of developing MS), we will use subscripts (1) and (2) to denote the failure probabilities and the values of other parameters at the 1^st and 2^nd time-periods respectively. Importantly, unlike true survival (where everyone fails given a sufficient amount of time), the probability of developing MS may not become 100% as the probability of a sufficient environmental exposure increases to {P(E) = 1}. Moreover, the limiting value for the cumulative probability of developing MS in men (c) need not be the same as that in women (d). However, because the new definition of the subset (G) differs from earlier iterations of our analysis [3, 4, 51, 52], the environmental exposure at which the development of MS becomes possible (i.e., the threshold) must occur at {P(E) = 0} for, at least, one of these two sub-subsets–provided that this exposure level is possible for either one or both of these 2 gender subgroups (see Fig 4, and above).

From these definitions, the failure probability for susceptible women (Zw) and men (Zm) at the 1^st time period is:

By our definitions of “1 exposure unit”, these equations, at the 2^nd time point, become:

Because the observations at time-periods (1) and (2) represent two points on the exponential response curves for both women and men, and because any two points on an exponential curve defines the curve (both uniquely and completely), we can use the observations regarding the (F:M) sex-ratio change over time in Canada [56], to derive and construct these two response curves.

Thus, from the definition of {P(E)} and using: {P(G,F) = P(F|G)*P(G) = 0.25*0.044 = 0.011 & P(F|MS)₂ = 0.66 − i.e., P(F|G) and P(G) are in the middle of their estimated ranges (see #5 above; see also #4, in S1 File) and P(F|MS)₂ is taken from Table 3 − we can estimate the values of (Zw₂) and (Zm₂) as: and: Moreover, as demonstrated in the in S1 File (#6a & #6d), we define the term (C) such that: and, thereby, re-express (Zw₁) and (Zm₁) in terms of (Zw₂) and (Zm₂) such that:

Consequently, based on the population data from Canada, the prevalence of MS must have increased by more than 32% between these two time periods.

Finally (see #6b, in S1 File), we can estimate the value of both (c) and (d) as:

Thus, using the observed change in the (F:M) sex-ratio over time in Canada, together with our estimates for P(G) and P(F│G), we have all the data needed to construct the complete response curves for the probability of developing MS with a changing environmental exposure in genetically susceptible women and men (Fig 4). What these curves make clear is that both P(E) and P(MS) are changing over time, which indicates that specific environmental conditions, in addition to specific susceptible genetic combinations, are necessary for MS to develop. Thus, MS develops when the right genetic constitution is exposed to the right environmental conditions (i.e., it is fundamentally due to a gene-environment interaction).

Because, the scales for the response-curves for women and men are initially assumed to be proportional they can be plotted on the graph (see Fig 4; see also #6e, in S1 File) and, when this is done, the threshold (x-intercept) occurs at {(a,Zm) = (λm,0)} for men and {(a,Zw) = (λw,0)} for women. By the definitions of (E) and (a), one of these thresholds must occur at {(a,Z) = (0,0)} –provided this exposure level is possible (see Fig 4 & above, same section). However, because these thresholds need not be the same, we define the difference in threshold between women and men as: (λ = λw−λm) such that, if women have a higher threshold than men: (λ>0). However, as noted (above), the (a) scale (for men) may be different than the (a^app) scale for women so that in order to plot them on the same graph requires the conversion of (a^app) units into (a) units–see #6c, in S1 File.

Three final points are also worth making. First, because, as demonstrated in the (#6c) in S1 File, (λw) is independent of R, we can use the condition of (R = 1) to evaluate (λw). In this circumstance, these exponential equations can be re-arranged (#6c, in S1 File) to yield:

Consequently, basic epidemiologic data can be used to determine the difference in threshold (λ) that exists between women and men. As demonstrated in the in S1 File (#6c), this leads to the two conclusions that:

Moreover, the value of (λ) depends only upon the value of (C) and the sex-ratio change over time so that, if the hazards are proportional, men must have a lower threshold for developing MS compared to women (Fig 4; see also #6c, in S1 File). A lower threshold in men is also suggested by a report from Europe and the United States [66], which found that prior to 1922 men accounted for 58% of the MS cases (Table 5). By our definition of P(E), these thresholds indicate the exposure, at which MS becomes possible. If women required a fundamentally different kind of exposure than men, it would be very hard to rationalize a difference in threshold because, in such a circumstance, in some environments, women would be more likely and, in other environments, less likely than men to receive the correct exposure. Rather, a difference in threshold implies that men and women are responding to similar events but that men require a less extreme degree of exposure in order to develop MS. For example, perhaps, susceptible men develop MS with a lesser degree of vitamin D deficiency or with EBV infection occurring over a broader age-range compared to susceptible women. {NB: even if there were no threshold difference, proportionality, by itself, would suggest that difference in exposure was one of degree but not kind.}

Download:

• PPT
  PowerPoint slide
• PNG
  larger image
• TIFF
  original image

Table 5. Sex distribution of multiple sclerosis cases reported prior to 1922^*.

https://doi.org/10.1371/journal.pone.0246157.t005

Alternatively, there may an environment-gender interaction such that susceptible men, in any given environment (i.e., E_T), are more likely to experience a sufficient exposure than susceptible women. For example, perhaps men are more likely to engage in “risky” behaviors compared to women, or that they are more likely to be “sun-averse” than women. Having said this, however, it is not clear how (or whether) “individual” differences in behavior (even if they are biologically driven) could lead to a “population-level” difference in threshold (Fig 4). More likely, any such interactions would have to be related to physiological differences between the genders.

Another possibility is that a small percentage of MS patients (in men or women or both), have “purely genetic “MS, whereby any environment is sufficient to cause MS, given their genotypes. Such a circumstance renders the points (λw = 0), or (λm = 0) or both unobservable, as drawn in Fig 4 (see above; same section). For example, in Fig 4, if ~1.8% of both susceptible men and women had “purely genetic” MS, this would raise the zero point of the y-axis for “environmental” MS such that this threshold difference would disappear (i.e., λ = 0) and both men and women would begin their “environmental” response at the new (0, 0) point in Fig 4 –i.e., at the point of intersection of the two curves. The same would be true if only men had this percentage of “purely genetic” MS except that, in this case, men would begin their “environmental” response at the point of intersection–i.e., at (Zm = 0.018), wherease women would begin at (Zw = 0), which would define the the new onset point (0,0). If both had more than this percentage (or other combinations), the exact relationship between the curves at the start would change but, depending upon the exact situation, there still could be no difference in the threshold for “environmental” MS (Fig 4). Clearly, this example only applies to the specific conditions of Fig 4. Nevertheless, because (λ>0) and because, at every exposure at or below the exposure at the point of intersection: (0<Zm<Zm₁), in this circumstance, only a small amount of “purely genetic” MS would be necessary to eliminate the threshold difference for every condition (see also #8 below).

Second, we note that: {P(MS|G,E,M) = c}, so that (Zm2) can be re-expressed as:

This equation can be rearranged to yield: P(E|G,M)₂ = Zm₂/c

From above: Zm₂/c = (1−e⁻¹)/{1−[P(M|MS)₁/P(M|MS)₂]*C*e⁻¹}

Therefore, in men ∀C>0.5: P(E|G,M)₂>0.83

And, similarly, in women: ∀C>0.5: P(E|G,F)₂>0.76

These results strongly suggest that the relevant environmental exposures (especially when these are multiple) are currently occurring at population-wide levels. For example, if three, equally likely and independent, environmental events (EE₁, EE₂, and EE₃)–possibly sequential [51, 52]–were sufficient to produce MS in a susceptible individual, then: so that, under the stated circumstances, more than 94% of the population would experience each environmental event. Such a conclusion is fully consistent with the same conclusion reached from studies in adopted individuals, in siblings and half-siblings raised together or apart, in conjugal couples, and in brothers and sisters of different birth order, which have generally indicated that MS-risk is unaffected by the micro-environments of families but, rather, results from population-wide exposures [67–73].

And third, it is clear that both of these response curves plateau well below 100% failure, especially in men (Fig 4). Therefore, there must be stochastic processes that partially determine whether a susceptible individual with a sufficient environmental exposure will actually develop disease (see #9, below).

8. The future fate of P(MS│IG_MS)

Conclusions: 1.

2.

Argument: As a sufficient environmental exposure {P(E)} becomes more likely, the quantity {P(MS│IGMS)} will, of necessity, change. Earlier, we described this term as having removed the impact of the shared IU and certain (especially early) post-natal environments of MZ-twins. This description, however, is not quite accurate. For example, we can break down a “sufficient” environmental exposure (see #1, in S1 File) into those factors that are shared exclusively by MZ-twins (E1), those factors that are shared by the population generally (E2), and those factors that shared exclusively within the family micro-environment (E3). As noted above, however, the family micro-environment seems not to have any impact on the likelihood of MS [67–73]. In this circumstance, assuming only factors (E1 and E2) are necessary for a sufficient exposure, then:

If an individual’s identical twin is known to have MS, it is likely that this individual, also, has experienced a “sufficient” (E1) exposure.

Conceived of in this way, the term {P(MS│IGMS)} can be rewritten as: and the adjusted penetrance {P(MS|IGMS)} hasn’t really “removed” the impact of these environmental similarities. Rather, {P(E₁|MZ_MS)} has simply been reset to its population level {P(E₁)}. Because MZ-twins share both identical genotypes and the same IU and certain post-natal environments, we expect that: P(E₁|MZ_MS) = 1. Consequently, as {P(E₁)} increases in the population to: the term {P(MS│IGMS)} will approach, and ultimately reach:

In this case, therefore, the limiting value for {P(MS,E|G)} in men (c) and women (d)–see #7 above; see also Fig 4 –must conform to the constraints of: The reason for the inequality is that, in those circumstance where: it must be that both: P(E₁) = 1 and: P(E₂|E₁) = 1. Naturally, the fact that {P(E₁)} has increased to unity does not guarantee that {P(E₂|E₁)} has done the same, so that the limiting value for P(MS,E|G) may be greater than P(MS|MZ_MS).

Nevertheless, if it is currently true (see #7 above), that: then it must also true that: c≈P(MS|M,MZ_MS) and: d≈P(MS|F,MZ_MS)}

Regardless, however, the depicted curves (Fig 4) must be inaccurate because, in the Figure:

Clearly there are several variables that can be adjusted {C, R, P(G), P(F|G) and P(MS)} to match the values for both (c) and (d) with these observed MZ-twin concordance rates. Therefore, we iteratively considered various combinations of these variables and determined which of those combinations matched these constraints. Specifically, we considered the plausible variables ranges of: (0.25≤C≤0.75), (0.20≤R≤5.0), (0.001≤P(G)≤1.0), (0.18≤P(F|G)≤0.70), and (0.002≤P(MS)≤0.006), and further required that the estimates for (c) and (d) be within (± 15%) of the observed values for their proband-wise MZ-twin concordance rates (Table 3; Fig 3). In this analysis, we found numerous combinations, which matched these constraints. The solution space covered by these matching combinations included the full range of possibilities for the parameters of (C) and (R). By contrast, the ranges for both P(F|G) and P(G) were restricted: {0.33≤P(F|G)≤0.5} and {0.02≤P(G)≤0.055}. This restricted range for P(G) fits, generally, within the framework developed previously and confirms the conclusion that developing MS is not a possibility for a large majority of the population (see #4a & #4b above). Similarly, this analysis confirms that women are less likely than men to be in the (G) subset, although the estimated range for P(F|G) is somewhat higher than the ranges developed previously (see #5 above; see also #4 in S1 File). As discussed in the in S1 File (#4), however, this could relate to an underestimate for the parameter {P(MS|M,MZ_MS)}, which is based upon only 2 observations of concordant male MZ-twins (Table 3).

Also, the 5 potential solutions for which: {P(MS)≤0.003} accounted for 11% of the total matching combinations. By contrast, the 5 potential solutions for which: {P(MS)≥0.004} accounted for 79% of the total. This circumstance suggests that we are under-estimating P(MS) when using the observed disease prevalence in the general population (Z). Indeed, several autopsy studies have indicated that the prevalence of undiagnosed (pathological) MS is ~0.1% [46–51]. Thus, with minimally symptomatic (or asymptomatic) MS occurring in as many 0.1% of the population, this could potentially increase the estimated P(MS) by as much as 50−100%. Although, such diagnostic errors are probably less common in the modern era, many minimally symptomatic (or asymptomatic) patients are still being undiagnosed during life [59]. Moreover, any such under-ascertainment is likely to be less for MZ-twins, DZ-twins, and siblings than in the general population. For example, an initially unaffected twin or non-twin sibling of a patient with MS will, almost certainly, be more carefully monitored for possible MS symptoms (i.e., for minimally symptomatic presentations) than will an individual in the general population. In such a circumstance, these diagnostic failures will be fewer in the (MZ_MS), (DZ_MS), and (S_MS) populations than in the general population and the MZ-twin concordance rates will, thus, provide a more accurate reflection of the maximum likelihood of getting MS {i.e., P(MS|G,E₁,E₂)} than will those estimates of P(MS) derived from the MS-prevalence in the general population. Such a circumstance would help to account for this apparent discrepancy.

9. Missing heritability?

Conclusions: 1. Both “genetic” and “environmental” factors are necessary for MS

expression; Neither alone are sufficient.

2. A large portion of the “causal pathway” to MS is stochastic

3. There is no need to invoke any “missing heritability” in MS

Argument: Only a small proportion of the population seems to be genetically susceptible to developing MS, which implies that MS is a genetic disorder. In addition, a suitable environmental exposure, like a suitable genetic constitution, is also a necessary part of MS pathogenesis. Despite this, however, the combination of a susceptible genotype together with a sufficient environmental exposure, does not invariably lead to the disease of MS and, in fact, the response curves in both women and especially men plateau well below 100% (Fig 4), even when everyone receives an environmental exposure suitable for their particular genotype–i.e., when {P(E) =1}. This variance in the likelihood of getting MS for certain susceptible genotypes cannot be attributed to unidentified environmental conditions because the definition of the term {P(E)} − see #7 above − explicitly includes all such factors, both if they are known (or suspected) and also if they are completely unknown. Therefore, a large portion of the overall variance in MS disease-expression must be due to stochastic processes.

In this context, dividing the total variance in disease expression into genetic and environmental components, at least for MS, mischaracterizes the situation. This has important implications regarding current estimates for the “missing hereditability” in MS [74–76]. First, as noted above, a large portion of the variability in MS expression must be due to stochastic processes that are neither environmental nor genetic. And second, specific gene-gene combinations (likely unique to individuals or very small groups of individuals) must underlie genetic susceptibility to MS (see #6 above; see also #7, in S1 File). Thus, with over 200 MS-associated loci [14], each (potentially) having more than one “susceptible state” (e.g., the MHC), the number of possible combinations of states at these loci is so huge that, almost certainly, everyone (except MZ-twins) possesses a unique combination of these “susceptible states” (see #7, in S1 File). Indeed, considering (H+)-status together with only the first 102 of these MS-associated SNPs [13], everyone (including both cases and controls) in the WTCCC population does, in fact, possess a unique combination (#7, in S1 File). Consequently, if only a few such combinations are members of the (G) subset, even among those combinations that are quite similar to each other (see #6 above; see also #7, in S1 File), then there are more than enough genetic associations already identified to account fully for (G) subset membership. Naturally, many more loci may yet be identified, although positing their existence is unnecessary.

Alternatively, if “missing heritability” is only meant to imply that our genetic model cannot predict accurately the occurrence of MS, then, indeed, almost all of the heritability of MS remains unexplained. Thus, the environmental factors, the actual (as opposed to associated) genetic factors involved in causing disease, the necessary gene-gene combinations, the various gene-environment interactions, and the stochastic factors–all of which contribute importantly to whether MS can, or will, develop in a specific individual–are poorly understood, thereby making any accurate prediction of MS occurrence impossible at present.