research article a novel model of schizophrenia age-of

Hindawi Publishing CorporationISRN PsychiatryVolume 2013 Article ID 604587 10 pageshttpdxdoiorg1011552013604587

Research ArticleA Novel Model of Schizophrenia Age-of-Onset DataChallenges the Conventional Interpretations of the Discordancein Monozygote Twin Studies

Ivan Kramer1 and L Elliot Hong2

1 Physics Department University of Maryland Baltimore County 1000 Hilltop Circle Catonsville MD 21250 USA2Maryland Psychiatric Research Center University of Maryland School of Medicine PO Box 21247 Baltimore MD 21228 USA

Correspondence should be addressed to Ivan Kramer kramerumbcedu

Received 21 May 2013 Accepted 7 July 2013

Academic Editors S Chakrabarti and S Lawn

Copyright copy 2013 I Kramer and L E Hong This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The relative importance of genetics and the environment in causing schizophrenia is still being debated Although the highproportion of monozygote cotwins of schizophrenia patients who are discordant suggests that there may be a significantenvironmental contribution to the development of schizophrenia this discordance is predicted by an accumulative multimutationmodel of schizophrenia onset constructed here implying a genetic origin of schizophrenia In this model schizophrenics areviewed as having been born with the genetic susceptibility to develop schizophrenia As susceptible gene carriers age theyrandomly accumulate the necessary mutations to cause schizophrenia the last needed mutation coinciding with disease onsetThemutation model predicts that the concordance rate in monozygote twin studies will monotonically increase with age theoreticallyapproaching 100 given sufficient longevity In dizygote cotwins of schizophrenia patients the model predicts that at least 71 ofcotwins are incapable of developing schizophrenia even though every cotwin and their schizophrenic twin shared a similar earlyenvironment The multimutation model is shown to fit all of the monozygote and dizygote concordance rate data of the principleclassical twin studies completed before 1970 considered in this paper Thus the genetic hypothesis of schizophrenia can be testedby bringing these studies up to date

1 IntroductionSchizophrenia is a brain disease characterized by delusionshallucinations and behavioral and functional disturbancesOnce the disease develops most patientsrsquo functioning is seri-ously impaired Discovering its cause and cure remain someof the biggest challenges to modern medicine and neuro-science A fundamental debate on the etiology of schizophre-nia is the relative importance of genetics and environmentalfactors in causing the disease A consistently higher concor-dance rate of schizophrenia in monozygotic twins than indizygotic twins supports the genetic hypothesis By contrastexternal environmental factors are thought to contribute tothe development of schizophrenia because a significant pro-portion of monozygotic twins are discordant However thisinference from this fact is challenged by the mutation modelto be developed in this paper

The high rate of discordance in monozygotic twins(around 50) is typically credited to environmental factors[1ndash3] However there is also strong evidence to counter thisexplanation of this discordance In monozygote twin studiesthe risk for schizophrenia in the offspring of the schizo-phrenic twins is 168 while it is 174 in their normaldiscordant cotwinsrsquo offspringmdashvirtually identical rates [4]These results suggest that the genetic susceptibility to developthe disease is carried by both the schizophrenic and thediscordant nonill cotwin and can be transmitted to the nextgeneration with equal probability even when environmentsare no longer shared as in twin development

A common misconception in the literature is the mis-taken belief that if genes were 100 responsible for schiz-ophrenia then ldquowhen one identical twin had schizophreniathere would be 100 chance that that the other twin would

2 ISRN Psychiatry

have it as wellrdquo Perhaps the best way to demonstrate the fal-lacy of this argument is to consider the physics of radioactivenuclear decay

For example all uranium-238 nuclei are naturally radio-active and decay inmultiple steps ending in the stable nucleuslead-206 Despite the fact that uranium-238 nuclei are com-pletely indistinguishable from each other and are consideredto be identical particles about half of a sample of pure U-238nuclei remains exactly as it was 45 billion years ago while theother half has decayed into lead-206 In physics each of thesedecays or the lack thereof is considered a random eventhaving nothing to do with external triggers and is entirelydictated by the internal physics of the nucleus [5] This is theanalogue of the schizophrenic identical twin data Althoughin physics it is accepted that identical radioactive nuclei donot necessarily evolve in tandem in nature the discordancein schizophrenic identical twins is typically modeled as envi-ronmental contributions in genetic epidemiology

According to the World Health Organization (WHO)schizophrenia occurs in all the countries of the world witha prevalence rate that is very narrowly proscribed (within afactor of about two between the highest and lowest sites)While there is much local evidence of environmental triggersfor increasing prevalence in specific regions and times ameta-analysis of 188 schizophrenia studies producing a totalof 1721 prevalence estimates drawn from 46 countries led tothe conclusions that the lifetime prevalence rate was around072 worldwide and there was no significant prevalencedifference between males and females or between urbanrural and mixed sites [6] Such similar prevalences acrossdiverse climatic economic and social environments raiseserious doubts of the validity of the environmental model ofschizophrenia etiology invoked to explain traditional twinand family data The incidence as a function of age data pro-duced by diseases with known environmental contributions(eg many infectious diseases) fluctuates significantly fromcountry to country around the world and from year to yearwithin a country

The fact that somemonozygote twins susceptible to devel-oping schizophrenia remain discordant at a certain age doesnot necessarily mean that external environmental factorscontribute to the development of the disease Analogous to achain of radioactive nuclear decays a genetically driven ran-dommutationmodel is constructed here that is shown to suc-cessfully fit disparate schizophrenia age-of-onset data fromboth general population and twin studies

2 The Independent Multimutation Model ofSchizophrenia for Singleton Births

In this section the age-of-onset schizophrenia prevalencefunction that will be used to fit singleton and twin study datawill be derived from anovelmultimutationmodel (MMM)Arelatively simple model for the development of schizophreniaassumes that the brain of every person susceptible to thedeveloping this disease must chronically undergo a series ofcharacteristic changes or mutations numbering 119898 in anyorder to get it The last change occurs at the age-of-onset ofschizophrenia Assuming that every change or mutation is

independent of all the others this model will be calledthe independent mutation model What role genetic andorenvironmental factors play in causing these characteristicchanges to the brain is a question that will be discussed laterThe male and female data for a given country or region (riskpopulation) will be separately modeled

21 The Age-of-Onset Schizophrenia Distribution Curve in theMultimutation Model Consider a random sample of a riskpopulation all born in the same year with the susceptibilityto develop schizophrenia later in life The size of the samplepopulationwill be denoted by119873

119904 and the cumulative number

of people in this sample that has developed schizophrenia byage 119905 will be denoted by119873(119905) The fraction of this populationthat has not developed the 119894th mutation at age 119905 is givenby exp(minus119896

119894119905) where 119896

119894is defined as the mutation rate (a

constant) for the 119894th change or mutation The meaning ofldquomutationrdquo here generically refers to any internally drivenbiological change that contributes to the onset of schizophre-nia and does not necessarily refer to a change in genomicsequenceThus the fraction of the susceptible population thathas developed the 119894th mutation at age 119905 equivalent to theprobability of developing this mutation by age 119905 is given by

119901119894(119905) equiv 1 minus exp (minus119896

119894119905) 119894 = 1 2 3 119898 (1)

where the mutation rate 119896119894is related to the average time

119879119894necessary for this mutation to occur by 119879

119894= 1119896

119894 If

119898 independent mutations are required for schizophrenia todevelop then the probability that schizophrenia will developat age 119905 in the susceptible population a quantity to be calledthe susceptible prevalence is given by

119875119904(119905) =

119873 (119905)

119873119904

= 1199011(119905) 1199012(119905) 1199013(119905) sdot sdot sdot 119901

119898minus1(119905) 119901119898(119905) (2)

where the values of the 119898 mutation rates (constants) 1198961

1198962 119896

119898minus1 119896119898are all independent of each other in general

Thus in this model the mutations can occur in any ordersimultaneously or at completely different times Notice thatthe maximum possible value of 119875

119904(119905) is 1

Now suppose a total of 1198730infants are born in a given

year in a given region or country If a fraction 119891119904of this

population cohort is susceptible to developing schizophreniathen the number of people in the cohort that is susceptible todeveloping the disease is given by 119873

119904= 1198911199041198730 If the age of

the cohort is denoted by 119905 (birth is coincident with age 119905 = 0)then the number of people in the cohort that have developedschizophrenia by age 119905 will be denoted by119873(119905)

The population prevalence or schizophrenia risk at age 119905for the entire population or cohort is therefore given by

119875 (119905) =

119873 (119905)

1198730

equiv 119891119904119875119904(119905)

= 119891119904sdot 1199011(119905) 1199012(119905) 1199013(119905) sdot sdot sdot 119901

119898minus1(119905) 119901119898(119905)

The fraction of the risk population that develops schizo-phrenia between the ages of 119905 and 119905 + 119889119905 is given by 119889119875(119905) sothat the fractional incident rate is given by

IR (119905) = 119889119875 (119905)119889119905

= 119891119904

119889119875119904(119905)

119889119905

equiv 119891119904sdot IR119904(119905) (4)

ISRN Psychiatry 3

The best fits to schizophrenia data occurred if1198981= 119898minus1

mutation rates are all equal to the same constant rate 1198961 while

the remaining one is equal to another rate 1198962= 1198961 then the

prevalence function in (3) becomes

119875 (119905) =

119873 (119905)

1198730

= 119891119904119875119904(119905)

= 119891119904[1 minus exp(minus119896

1119905)]1198981

[1 minus exp (minus1198962119905)]

where119898 = 1198981+1 Since this model depends on 4 parameters

(119891119904 1198961 1198962 and 119898) it will be referred to as the 4-parameter

model If it turns out that 1198962= 1198961 then the number of

parameters in (5) is reduced to 3 and this simplest possiblemodel will be called the 3-parameter model

The values of the parameters in the prevalence functionin (5) or its corresponding incidence function depend onthe values of four fit parameters 119891

119904 1198961 1198962 and 119898

equivalently 119898) whose values are determined by a least-squares fit to appropriate data

If the set of 119899 consecutive data-values used in the fit aredenoted by 119889

119894 and if the correspondingmodel fit-values are

denoted by 119909119894 then the square of the error of the fit to be

called chisq (chi square) is defined as

chisq equiv119899

119894=1

[119909119894minus 119889119894]2

The better the fit to the data the smaller the value of chisqreturned by the fit

22 Results of Fitting theMMM to Schizophrenia Age-of-OnsetData for Singleton Cases Unlike point or lifetime prevalencetrue age-of-onset prevalence rate in a given population isdifficult to ascertain for schizophrenia because the definitionof ldquoonsetrdquo does not have a common consensus in many casesof insidious onset or prolonged prodromal cases A relativelyobjective estimate for age-of-onset is first hospitalization forpsychotic break especially in the earlier era where hospital-ization was still widely available and considered a standard ofcare for the first psychotic episodes in schizophrenia patientsTherefore we used the schizophrenia age-at-first admissionincidence rate data for USA hospitals by Kramer et al (seeTable 34 in [7]) These data were compiled for the historicalperiod before or at the beginning of the widespread use ofantipsychotic medications We assume that this first hospitaladmission incidence rate by age range was proportional tothe true age-of-onset by the same age range in the generalpopulationHowever the true rate should be higher because aproportion of first onset cases was assumed not hospitalized

The male and female cumulative incidence data as afunction of age curves (ie the original data in [7]) as wellas the 4-parameter model fits to them using (5) are plottedin Figure 1 The values of the male and female fit parametersappear in Table 1 and satisfy the modeling requirement thatthe number of steps or mutations 119898 necessary to cause theonset of the disease (in analogy to uranium-238 decay intolead-203 in steps or stages) is independent of sex In nuclearphysics the number of internal changes in a radioactive

00 10 20 30 40 50 60 70 80

Age t (years)

enceP(t)

e as a

fs = 000157 (males)= 000164 (females)

m1 = m minus 1 = 15

Male prevalence P(t) dataFemale prevalance P(t) data

Figure 1 USAMF4P Four-parameter independent mutation modelfit to Kramer male and female USA schizophrenia first hospitaladmission cumulative incidence rate per 100000 data

nucleus leading up to its spontaneous decay is unknownand factored into the measured value of its lifetime In thesame way internal changes to the brain leading up to aschizophrenia mutation are ignored in this modeling but itpresumed that these changes or mutations could be observ-able with current or future neuroscience techniques Thevalues of the lifetime risk returned by these two fits refer tohospital admissions only Assuming that the USA hospitaladmission cohort is a perfect random sample of the USArisk population as a whole to get the actual schizophreniaprevalence of the entire USA risk population we need onlyto replace the value of the 119891

119904returned by the fit by the total

USA value for the lifetime risk obtained by accurate surveydata (119891

119904asymp 001 or about 1) This assumption will be used

in all the modeling that follows Thus in this model about99 of the USA population cannot develop schizophreniaand can be regarded as unsusceptible to it Since 119898

1= 15 of

the 16 changes or mutations necessary to cause schizophreniain this model 15 schizophrenia mutations take place at therate of 119896

1 while the remaining one occurs at the rate 119896

Thus for the USA male risk population the mean time fora schizophrenia mutation associated with 119896

1to occur is 119879

11198961= 858 years while the mean time for a schizophrenia

mutation associated with 1198962is 1198792= 1119896

2= 351 years≫ 119879

The analogous results for USA females are 1198791= 101 years

and 1198792= 300 years differing by modest 177 and minus145

from the respective male results However the difference inthe values of male and female lifetime risk 119891

119904returned by

the fits was only about 4 suggesting that the prevalenceof schizophrenia in the general USA population is largelyindependent of sexThe parameters for the fit to the aggregatemale plus female USA data also appear in Table 1

Using the values of the parameters119898 1198961 and 119896

2returned

by the fits the male and female susceptible incidence rate

4 ISRN Psychiatry

Table 1 Values of model parameters for the independent mutation model fits to USA schizophrenia first hospital admissions data (males Mfemales F and males + females M + F)

CohortNumber of

parameters inmodel

119898

mutationnumber

119896 or 1198961

mutation rate in(years)minus1 [119898

1= 119898 minus 1]

1198962

119891119904

lifetime riskchisq error in(years)minus2

USA males 3 10 008172 000137 922119890 minus 10

USA males 4 16 011653 0028465 00015737 353119890 minus 10

USA females 4 16 009859 0035728 00016428 782119890 minus 11

USA males + females 4 16 010757 0029959 00016363 226119890 minus 10

curves IR119904(119905) computed from (4) and (5) are shown in

Figure 2 As seen from this figure the peak in the incidencerate curve for USA males occurs at the age of 119905peak = 2665years while the female curve peaks at 119905peak = 3060 yearsThese results are consistent with the known delayed onset ofschizophrenia in females when compared to males [18]

The 3-parameter model fit to the USA male data alsoyields a credible fit but with a modest increase in fit erroras seen in Table 1 Since we have found that the 4-parametermodel always yields the best fit to data only the 4-parametermodel results will be presented from now on Since mostschizophrenia age-of-onset data are significantly imprecisedue to biases such as who got hospitalized there is presum-ably some noise inherent in the dataThe need to introduce asecondmutation rate to accurately fit the datamay be entirelydue to noise Thus it is possible that if the data were perfectthen only one mutation rate 119896 would be necessary to giveexcellent fits

3 Modeling Schizophrenia Twin Study Data

In this section the schizophrenia MMM constructed inSection 21 for singleton cases will be extended to describetwin births In a collection of monozygotic twins where oneof the twins is schizophrenic every member of the cohortis born with a susceptibility to develop schizophrenia andso the risk fraction or lifetime risk is 119891

119904= 1 In the anal-

ysis of all such studies each twin pair must be separatedand randomly assigned to two different subcohorts using acriterion that has nothing to dowith schizophrenia for exam-ple by the random flipping of a coin Thus two subcohortsare assembled with identical twin pairs assigned to differentsubcohorts in random fashion Thus the age-of-onset preva-lence curve 119875

119904(119905) of the two subcohorts should be identical

even though this function may have nothing to do with themodel prevalence function given in Section 2 and has a formcompletely different from that in (5)This is one of the crucialtests of the validity of the model Since the published twinstudies contain no such analysis it is essential to reanalyzethe data in these studies to test this age-of-onset prediction

In ourmodel of schizophrenia susceptibility all membersof both subcohorts are born with the susceptibility to developthe disease thus it is predicted that all monozygote cotwinswill eventually develop schizophrenia if they can live longenough

0 10 20 30 40 50 60 70 80 90 120100 110Age t (years)

m = 16

USA male IRs(t) = dPs(t)dt

USA female IRs(t) = dPs(t)dt

IRs(t) = dPs(t)dt = IR(t)fs

tpeak =

2665 y (males)3060 y (females)

s of m

IRs(t)

Figure 2 Plots of model susceptible incidence rates IR119904(119905) obtained

from fits to USA male and female schizophrenia age-of-onset data

31 Extending the Singleton Multimutation Model to DescribeTwin Age-of-Onset Data When one twin (the index twin) ineach pair has developed the disease the other twin will bereferred to as the cotwin in this paper Our model posits thatin monozygote twin pairs the cotwin has the same suscep-tibility to develop schizophrenia as the index twin Considera birth cohort of monozygotic twins all having the same age119905 Assuming that birth is coincident with age 119905 = 0 onetwin (either the first or second born) will experience theonset of schizophrenia say at age 119905 As soon as that happensone twin is randomly assigned to subcohort 1 and the otherto subcohort 2 All schizophrenia twin studies can easilyassemble subcohorts 1 and 2 in this way As these subcohortsage their cotwins start experiencing the onset of schizophre-nia The risk of developing the disease at age 119905 by membersof a subcohort is given by a susceptible prevalence function119875119904(119905) which in our modeling is defined in (5) Assuming that

genetic factors are entirely responsible for the development ofschizophrenia for any monozygote cohort both susceptiblesubcohorts will experience the same susceptible prevalence

ISRN Psychiatry 5

function 119875119904(119905) Thus when a member of subcohort 1 experi-

ences the onset of schizophrenia at age 119905 the probability thatthe cotwin in subcohort 2 will [will not] develop the diseaseby this age is given by 119875

119904(119905)[119876119909(119905) equiv 1 minus 119875

119904(119905)]

The probability that any member of subcohort 1 will befound [will not to be found] to have schizophrenia by age 119905will be denoted by 119875(1)

119904(119905)[119876(1)

119909(119905) equiv 1minus119875

119904(119905)] with a similar

notation for subcohort 2 Since119875(119894)119904(119905)+119876

(119894)

119909(119905) = 1 for 119894 = 1 2

we have

1 = [119875(1)

119904(119905) + 119876

119909(119905)] [119875

119904(119905) + 119876

119909(119905)]

= 119875(1)

119904(119905) 119875(2)

119904(119905) + (119875

119904(119905) 119876(2)

119909(119905) + 119875

119904(119905) 119876(1)

119909(119905))

+ 119876(1)

119909(119905) 119876(2)

119909(119905)

Thus we define subcohort concordant discordant and non-schizophrenia probabilities as

119875119904119904(119905) equiv 119875

119904(119905) 119875(2)

119904(119905)

119875119904119909(119905) equiv (119875

119904(119905) 119876(2)

119909(119905) + 119875

119904(119905) 119876(1)

119909(119905))

119875119909119909(119905) equiv 119876

119909(119905) 119876(2)

119909(119905)

respectively where

119875119904119904(119905) + 119875

119904119909(119905) + 119875

119909119909(119905) = 1 (7c)

It is important to note that subcohort concordance asdefined above for example is not the same as pairwise con-cordance as usually used in the literature Here if a memberof subcohort 1 and a member of subcohort 2 are chosen atrandom at age 119905 the probability that both will have acquiredschizophrenia is given by 119875

119904119904(119905) and the probability that they

will be found to be discordant is denoted by 119875119904119909(119905) Finally

the quantity 119875119909119909(119905) is the probability that neither one of them

will be found to be schizophrenic at age 119905 even though theyare both susceptible to developing the disease

For monozygote (MZ) twins subcohorts 1 and 2 aregenetically identical so that 119875(1)

119904(119905) = 119875

119904(119905) equiv 119875

119904(119905) and

119876(1)

119904(119905) = 119876

119904(119905) equiv 119876

119904(119905) thus the probabilities in (7b)

become

119875119904119904(119905) = 119875

119904(119905)

119875119904119909(119905) = 2119875

119904(119905) 119876119904(119905)

119875119909119909(119905) = 119876

119909(119905)

[MZ twins]

When a susceptible monozygote twin pair in the (119909 119909)state (neither twin has developed schizophrenia yet) makesa transition to the (119904 119909) state at age 119905 it means that oneof the twins has developed schizophrenia at age 119905 (the age-of-onset) The probability that such a transition would takeplace denoted by 119889119875+

119904119909(119905) is given by

119889119875+

119904119909(119905) = minus119889119875

119909119909(119905) (9a)

Integrating this result from 119905 = 0 to any age 119905 gives

119875+

119904119909(119905) = 1 minus 119875

119909119909(119905) = 1 minus [1 minus 119875

119904(119905)]2

= 119875119904(119905) [2 minus 119875

119904(119905)]

since 119875+119904119909(0) = 0 and 119875

119909119909(0) = 1 Since 119875+

119904119909(119905) is the age-

of-onset distribution curve for the first twin of a pair that issusceptible to developing schizophrenia the result in (9b) isextremely important in describing monozygote discordanceNotice that although it might have been expected that 119875+

119904119909(119905)

would turn out to be equal to 119875119904(119905) as it is in single-births

(9b) for twins shows that this is not true It is also veryimportant to note that the prevalence function 119875

119904(119905) in this

section is completely independent of the mutation modelversion of this function constructed in Section 21 above

In schizophrenia twin studies the birth cohort consistsof only the concordant and discordant twin cases since todate it remains difficult to determine susceptibility to schizo-phrenia unless the disease is emerging (as in some prodromecases) or actually developsThus referring back to the resultsin (8) the fraction 119862119872

119904119904of the monozygote birth cohort that is

concordant at age 119905 is given by

119862119872

119904119904(119905) equiv

119875119904119904(119905)

[119875119904119904(119905) + 119875

119904119909(119905)]

1198752

119904(119905)

[1198752

119904(119905) + 2119875

119904(119905) 119876119904(119905)]

119875119904(119905)

[119875119904(119905) + 2 (1 minus 119875

119904(119905))]

119862119872(119905) equiv 119862

119872

119904119904(119905) equiv

119875119904119904(119905)

[119875119904119904(119905) + 119875

119904119909(119905)]

119875119904(119905)

[2 minus 119875119904(119905)]

Notice that the monozygote concordance rate 119862119872(119905) equiv

119862119872

119904119904(119905) is a function of 119875

119904(119905) Since 119875

119904(0) = 0 and 119875

119904(infin) = 1

the monozygote concordance rate also varies between 0 and1 Inverting (11a) by solving for 119875

119904(119905) gives

119875119904(119905) =

2119862119872(119905)

[1 + 119862119872(119905)]

[Monozygotic twins] (11b)

Since the value of 119862119872(119905) is determined from twin studies the

result in (11b) is a model prediction of the value of 119875119904(119905) this

prediction can be tested by reanalyzing the data in the twinstudies to compute this quantity

For the dizygotic twin cases the formal results in (7a)(7b) and (7c) carry over here Keeping the superscript (1)to refer to the schizophrenic index twin and superscript (2)to refer to the fraternal cotwin a new expression for 119875(2)

119904(119905)

must be developed To this end we define the probabilitythat a fraternal cotwin of a schizophrenic will also inheritthe susceptibility to develop schizophrenia and denote thisprobability by 119878inher Then we can set

119875(2)

119904(119905) = 119878inher119875

119904(119905) equiv 119878inher119875119904 (119905)

where 119876(2)119904(119905) = 1 minus 119875

119904(119905) as before

6 ISRN Psychiatry

Using (12a) in (7a) (7b) and (7c) then gives analogous to (8)

119875119904119904(119905) = 119878inher119875

119904(119905) (12b)

119875119904119909(119905) = 119875

119904(119905) [1 minus 119878inher119875119904 (119905)] + [1 minus 119875119904 (119905)] 119878inher119875119904 (119905)

= [1 + 119878inher] 119875119904 (119905) minus 2119878inher1198752

119904(119905)

for dizygotic twinsIn the same way the fraction 119862119863

119904119904(119905) of the dizygote birth

cohort that is concordant at age 119905 is given by

119862119863(119905) equiv 119862

119863

119904119904(119905) equiv

119875119904119904(119905)

[119875119904119904(119905) + 119875

119904119909(119905)]

119878inher119875119904 (119905)

[1 + 119878inher minus 119878inher119875119904 (119905)]

Notice that the dizygote concordance rate 119862119863(119905) equiv 119862

119863

119904119904(119905) is

also a function of 119875119904(119905) Since 119875

119904(0) = 0 and 119875

119904(infin) = 1 the

dizygote concordance rate varies between 0 and 119878inherSolving (13a) for the unknown probability 119878inher gives

119878inher =119862119863(119905)

[119875119904(119905) + (119875

119904(119905) minus 1) 119862

119863(119905)]

Using (11b) in (13b) gives

119878inher =[1 + 119862

119872(119905)]

[2119862119872(119905) 119862

119863(119905) minus 1 + 119862

119872(119905)]

The values of the monozygote and dizygote concordancefractions 119862

119872(119905) and 119862

119863(119905) respectively are determined by

twin studies so (13c) is a model prediction of the valueof 119878inher the probability that a fraternal cotwin of a schizo-phrenic will also inherit the susceptibility to develop schiz-ophreniaThis prediction of the model can be tested by rean-alyzing the data in classical twin studies to compute the valueof 119878inher It is again important to note that the prevalencefunction 119875

119904(119905) in this section is independent of the mutation

model version of this function thus all of the formulas from(7a) to (13c) are independent of any model

32 Results of Modeling Age-of-Onset of Schizophrenia TwinData Using the singleton USA male plus female multi-mutation model the probability curves for identical twinconcordance discordance and no-schizophrenia defined in(8) respectively are plotted in Figure 3 Notice that the con-cordance probability curve 119875

119904119904(119905) monotonically increases

with age but never reaches saturation at 100during a normallifetime In fact even at the age of 80 years old 119875

119904119904(119905) asymp 08

119875119904119909(119905) asymp 02 119875

119909119909(119905) asymp 0 so that there is only an 80 chance

that bothmembers of a susceptiblemonozygote pair will havedeveloped schizophrenia and a 20 chance that they will bediscordant

What twin studies actuallymeasure is the average value ofconcordance for a cohort made up of members with a varietyof different ages Suppose that the cohort ranges from a low

0 10 20 30 40 50 60 70 80 90Age t (years)

Pss = Ps(t) lowast Ps(t)

Psx = 2 lowast Ps(t) lowast Qs(t)

Pxx = Qs(t) lowast Qs(t)

Figure 3 Plots of concordance probability119875119904119904(119905) discordance prob-

ability 119875119904119909(119905) and nonschizophrenic probability 119875

119909119909(119905) for identical

twins susceptible to developing schizophrenia using the USA maleplus female 119875

119904(119905) curve

age 119905119871to a high age 119905

119867 where 119905

119867minus 119905119871= 119887 years Let 119905

119886denote

the age of a member of the cohort in years where the index119886 = 1 2 119887 and where 119905

1= 119905119871and 119905119887= 119905119867 If 119899119886(119905119886) denotes

the number of members of the cohort with age 119905119886and if the

total number ofmembers of the cohort is119873119879 then the average

monozygote concordancemeasured for the cohort is given by

⟨119862119872(119905)⟩ =

119887

119886=1

119862119872(119905119886)

119899119886(119905119886)

119873119879

For a uniform distribution of ages where 119899119886(119905119886) is a constant

independent of age 119905119886 the result in (14a) reduces to

⟨119862119872(119905)⟩ =

119905119867

119905119871

119862119872(119905) 119889119905

119905119867minus 119905119871

(Uniform distribution)

Similar expressions apply for the average dizygote concor-dance ⟨119862

119863(119905)⟩

The average monozygote and dizygote concordance ratesfrom representative samples of schizophrenia twin studiesfrom around the world are summarized in Table 2 [17 19]Only significant studies published before 1970 are includedhere in the hope that follow-up studies of the reporteddiscordant twins would be carried out to definitively supportor refute the predictions of the model Virtually none ofthese studies published either the age or schizophrenia age-of-onset distributions of their twin cohort so the followinganalysis will make due without these data In what followswe shall show that themonozygote and dizygote concordance

ISRN Psychiatry 7

Table 2 Concordance rate table Uncorrected concordance rates in schizophrenia twin studies from around the world and modeling resultsfrom fits to these data Only significant studies published before 1970 are included here so that the updates of these studies could definitivelytest the predictions of the model

Investigator Year Country MZ pairsconcordance

DZ pairsconcordance 119878inher

lowast

119875119904(119905 119903)

119905 (119903)

Rosanoff et al [8] 1934 USA 2841 = 0683 15101 = 0149 0190 0812Essen-Moller [9] 1941 Sweden 611 = 0545 427 = 0148 0224 0706Kallmann [10] 1946 USA 120174 = 0689 53517 = 0102 0129 0816 310 y (194)Slater [11] 1953 UK 2437 = 0648 10112 = 00892 0116 0787Inouye [12] 1961 Japan 3355 = 0600 217 = 0117 0163 0750Harvald and Hauge [13] 1965 Denmark 49 = 0444 662 = 00967 0167 0615Gottesman and Shields [14 15] 1966 UK 1024 = 0416 333 = 00909 0165 0588 410 y (1)Kringlen [16] 1966 Norway 1950 = 0380 1394 = 0138 0283 0551Hoffer and Pollin [17] 1970 USA 1180 = 0137 6145 = 00413 0197 0242 43 y (0631)lowast

119878inher is defined as the probability that a fraternal cotwin of a schizophrenic will also inherit the susceptibility to develop schizophrenia119875119904(119905 119903) is the prevalence obtained from a model simulation of twin study concordance results for a susceptible age cohort at age 119905

rates computed in these studies can be reproduced by ourUSA singleton multimutation model prevalence function byconsidering the results of three studies shown in Table 2

Let us first consider the Gottesman and Shields data[14 15] The age range for the Gottesman cohort in Table 2was reported to be 19 y lt 119905 lt 64 y with the median agebeing 37 y [20] Let us assume that the schizophrenia age-of-onset distribution curve for the Gottesman cohort is identicalto that of the USA modeled in Section 22 Then the averagemonozygote concordance rate for theGottesman cohortmustfall within the range from 119862

119872(19 y) = 00278 to 119862

119872(64 y) =

0724 which it clearly does Assuming that the Gottesmancohort is close to a uniform distribution then (14b) yieldsan average monozygote concordance value of ⟨119862

119872(119905)⟩ =

04012 very close to the value of 1024 = 04166 obtained byGottesman Since a USA age cohort of age 119905 = 41 years hasa monozygote concordance of 1024 the Gottesman cohortis equivalent to a USA age cohort of 41 years old The valuesof 119875119904(41 y) = 0588 and 119878inher = 0165 shown in Table 2 are

computed using a USA age cohort that is 41 years old Noticethat two other studies inTable 2 [13 16] have data very close tothat of Gottesman and therefore these cohorts are also verylikely described by the USA age-of-onset distribution curve

We next turn our attention to the Hoffer and Pollin[17] results shown in Table 2 The Hoffer and Pollin studywas composed of 15930 US military twin pairs where bothtwins served in the armed forces Since all members acceptedinto the USA military had to pass a rigorous mental-healthexam it is very likely that many potential recruits at risk forschizophrenia were rejected and skewed results can thereforebe expected from this study The age range for this cohortwas 38 y lt 119905 lt 48 y so this cohort is very close to being anage cohort with an average age of 43 years old The averagemonozygote concordancemeasured by Hoffer and Pollin was1180 = 0137 In the USA age cohort model a monozygoteconcordance of this value occurs around 165 years of age farbelow the age range of this cohortThus the Hoffer and Pollinand USA age-of-onset distribution curves must be radicallydifferentNonetheless we shall show that both sets of data can

be described by the same risk function119875119904(119905) but with different

values for the mutation rate parametersAs we have seen in Section 22 the USA prevalence

function for an age cohort with age 119905 is given by

119875119904(119905) equiv 119875

119904(119905 1198961 1198962) = [1 minus exp (minus119896

1119905)]15

[1 minus exp (minus1198962119905)]

where 1198961= 010757 yminus1 and 119896

2= 0029959 yminus1 To represent

theHoffer andPollin prevalence functionwewill assume thatit has a modified version of the result in (15a) namely

119875119904(119905 119903) equiv 119875

119904(119905 119903 sdot 119896

1 119903 sdot 1198962)

= [1 minus exp (minus1199031198961119905)]15

[1 minus exp (minus1199031198962119905)]

where 119903 is a dimensionless scaling factor that slows down(119903 lt 1) or speeds up (119903 gt 1) the rate at which schizophrenicmutations occur (the biological clock rate) In this notation119875119904(119905 1) equiv 119875

119904(119905 1198961 1198962) given in (15a) Using (11b) we find that

for an age cohort of 119905 = 43 y (15b) must satisfy

119875119904(43 y 119903) = [1 minus exp (minus43119903119896

[1 minus exp (minus431199031198962)]

= 02417

Numerically solving (16a) yields the value

119903 = 0631 (Hoffer and Pollin) (16b)

a result that was also placed in Table 2 Thus the Hoffer andPollin age-of-onset prevalence function has exactly the sameform as that of the USA but with mutation rates 1198961015840

1= 1199031198961and

1198961015840

2= 1199031198962in place of 119896

1and 1198962 respectively Aplot of theHoffer

and Pollin and USA prevalence functions appear in Figure 4These curves differ in the value of only a single parametermdashthe biological clock rate parameter 119903

As a final example consider the largest twin study inTable 2 that of Kallmann [10] From Table 2 the measured

8 ISRN Psychiatry

monozygote concordance rate in the Kallmann study is120174 Using this value in (11b) yields 119875

119904(119905) = 0816 where

119905 is the average age of the concordant members of the cohortat this point Using the singleton USA male plus female 119875

119904(119905)

curve that results from the parameters in Table 1 we predictthat the age of the Kallmann cohort when this concordancerate was reached was 119905 = 603 years old However since theKallmann cohort ranged in age from a low of 119905

119871= 15 years

to a high of 119905119867= 45 years the prediction from the USA

data is above this range and therefore the USA prevalencefunction is inconsistent with the Kallmann data Thus theprevalence curves for the Kallmann and USA cohorts mustbe significantly different We proceed here in the same waythat we did in the Hoffer and Pollin analysis above We nowassume that the Kallmann prevalence function is given by(15b) where the parameter 119903 must be determined from thedata To determine the value of 119903 we will use the expressionfor the mean age 119905 at which schizophrenia is developed in acohort

119905 equiv

119905119867

119905119871

119905119889119875119904(119905 119903)

119875119904(119905119867 119903) minus 119875

119904(119905119871 119903)

Since 119905 = 238 y for the Kallmann cohort (17) is an equationfor 119903 Numerically solving this equation yields the solution 119903 =194 a result that was also placed in Table 2 It then remainsto solve

119875119904(119905 119903) = [1 minus exp (minus119903119896

1119905)]15

[1 minus exp (minus1199031198962119905)] = 0816

for the age 119905 since 119903 1198961and 119896

2are known The numerical

solution to (18) is 119905 = 310 y a value that is almost exactlyin the middle of the age range for the Kallmann cohortThe Kallmann prevalence function in (18) is also plotted inFigure 4 and it is now apparent that the USA prevalencecurve (119903 = 1) is approximately an average of the Hofferand Pollin Kallmann and other prevalence curves in Table 2Thus not all cohorts have prevalence curves with the samedependence on age 119905 but if we average over all of themwe expect to get the USA result Nonetheless all of theprevalence curves have the same form shown in (15b) andtherefore they are generated by the same multimutationmodel describing the development of schizophreniawith119898 =16mutationsThe exceptionally large value for the Kallmannmonozygote concordance rate (689) may be traced to thefact that his cohort largely consisted of severe or chronicschizophrenics in hospitals catering to long-stay patients [19]

In the twin data analysis we introduced the probabilitythat a fraternal cotwin of a schizophrenic will also inheritthe susceptibility to develop schizophrenia and denoted itby 119878inher (see (12a)) Then using the results in (13c) and(11b) in Section 31 the value for 119878inher predicted by themodeling can be calculated and the results also appearin Table 2 Table 2 also contains the predicted value of theschizophrenia prevalence 119875

119904(119905 119903) of a birth cohort at age 119905

when the monozygote concordance reaches the value shownin this table Since the values for these two quantities can becomputed directly from the schizophrenia twin study data

0 10 20 30 40 50 60 70 80 90 100

nsPs(tr)

Cohort age t (years)

Ps(t k1 k2) USA r = 1

Ps(t r) Kallmann r = 194

Ps(t r) Hoffer and Pollin r = 0631

Figure 4 Comparison of schizophrenia prevalence functions119875119904(119905 119903) for USA data (119903 = 1) Hoffer and Pollin twin cohort (119903 =0631) and Kallmann twin cohort (119903 = 194)

the predictions for these quantities in Table 2 constitute testsof the model

Although the monozygote concordance rates of thesestudies vary widely (from 0138 to 0689) the range in thevalue of 119878inher is found to be 0116 lt 119878inher lt 0283 withthe result of 0129 for the Kallmann data being near the lowerend of this range Since the Kallmann study had the largestcohort of monozygote twin pairs by far it is clearly the mostimportant study in this tableThe results of the Japanese studyby Inouye produced the value of 119878inher = 0163 slightly abovethe Kallmann result In fact all but the result for the Kringlenstudy appearing in Table 2 produce values for 119878inher that arewithin a factor of 2 of that obtained from the Kallmanndata Since the risk for schizophrenia in children with oneschizophrenic parent is 164 (0164) [21] the values of 119878inherin the table average out to be about this value

Using the Gottesman and Shields data as a typical exam-ple of the results we have obtained the susceptible prevalence119875119904(119905 119903 = 1) is plotted in Figure 5 (see (5) and Table 1)

Using the same model in (11a) the monozygote concordancerate curve is also plotted in Figure 5 Finally using the USAmodel coupled with the value of 119878inher for the Gottesmanand Shields study in Table 2 the dizygote concordance curvefor the this twin cohort is plotted in Figure 5 using (13a)When the monozygote concordance of this cohort reachesthe value of 1024 = 0416 the prevalence is 119875

119904(119905 1) = 0588

at the age 119905 = 410 years and the dizygote concordanceis 119862119863(119905) = 01025 all three of these points fall exactly on

their respective curves in Figure 5 Changing the value of119878inher in (13a) to match the value of the different studies wesee that each study generates a dizygote concordance rate

ISRN Psychiatry 9

0 10 20 30 40 50 60 70 80 90 100Cohort age t (years)

Cm(t) = Ps(t)[2 minus Ps(t)] r = 1

Cd(t) r = 1

Cm(41 y) dataCd(41 y) dataPs(41 y) data+ model

Figure 5 Plots of Gottesman and Shields prevalence 119875119904(119905) monozy-

gote concordance 119862119898(119905) and dizygote concordance 119862

119889(119905) curves

together with corresponding twin study data from Table 2

curve that has the same characteristic as the one plotted inFigure 5 namely it plateaus at the maximum value of 119878inheritself If thismodel prediction of the plateauing of the dizygoteconcordance curve turns out to be correct then it wouldsupport the proposition that the susceptibility to developschizophrenia is acquired by internal genetic factors notexternal environmental ones

The model predicts that both the monozygote and dizy-gote concordance rate curves are monotonically increasingfunctions of age but saturate at 1 and 119878inher ≪ 1 respectivelyvery different values as seen in Figure 5 These predictionscan easily be tested by revisiting the classical twin studiesusing the same cohorts and bringing the data up-to-date

We can find only one study that made one follow-updiagnosis of the nonill monozygote cotwins after variableyears [22] This study supports our proposition by showingincreases in both concordance rate and new psychopathologyamong previously healthy cotwins although the follow-upinterval was not long enough nor the age of the twins wereold enough to provide quantitative support to the modelFrom themonozygote concordance rate curve that appears inFigure 5 note that 100 concordance is generally not possibleto observe because again this value occurs at an age 119905 abovethe maximum human life span although recollecting twindata in their advanced age should provide sufficient test ofthe model

Now the fraction of dizygotic cotwins that has suscepti-bility to develop schizophrenia is by definition 119878inher Thusthe fraction of dizygotic twins that is unable to developschizophrenia is 1minus119878inher Using the calculated values of 119878inhershown in Table 2 we calculate that at least 71 [(1 minus 119878inher) times100] of cotwins in dizygotic twin studies is predicted to

be unable to develop schizophrenia even though the cotwinshared a similar environment as their schizophrenic twinThis prediction would not support substantial environmental(prenatal or postnatal) contribution to schizophrenia suscep-tibility

4 Conclusion

Although a wide variety of prenatal maternal infections suchas influenza herpes polio rubella and toxoplasmosis havebeen linked to schizophrenia [23 24] many investigationshave shown that prenatal exposure to infection did notsignificantly increase the risk [25] The data linking prenatalexposure to influenza and schizophrenia remain contradic-tory [26] For example in an investigation of psychiatricadmissions of people born a few months after the 1957 A2influenza epidemic in Scotland it was found that only 3children of the 945 born to mothers who actually sufferedfrom influenza during the second trimester of pregnancybecame schizophrenics this risk rate was no greater thanthat faced by children of mothers who were not infected[27] A study using Japanese government data reached theidentical conclusion that there was no relationship betweeninfluenza epidemics and schizophrenic births [28] Thusthe genetic multimutation model described here remains aviable explanation for very disparate data on schizophreniaThe multimutation model constructed here is shown to fitmonozygote and dizygote concordance rate data of importanttwin studies completed before 1970 in addition to singletonage-of-onset data Thus revisiting the historical twin studieslisted in Table 2 to reexamine the previously declared nonillcotwinrsquos diagnostic status at their advanced age would be atest of this random multimutation model

References

[1] P McGuffin A E Farmer I I Gottesman R M Murray andA M Reveley ldquoTwin concordance for operationally definedschizophrenia Confirmation of familiality and heritabilityrdquoArchives of General Psychiatry vol 41 no 6 pp 541ndash545 1984

[2] T D Cannon J Kaprio J Lonnqvist M Huttunen and MKoskenvuo ldquoThe genetic epidemiology of schizophrenia in aFinnish twin cohort a population-based modeling studyrdquo Ar-chives of General Psychiatry vol 55 no 1 pp 67ndash74 1998

[3] P F Sullivan K S Kendler and M C Neale ldquoSchizophrenia asa complex trait evidence from a meta-analysis of twin studiesrdquoArchives of General Psychiatry vol 60 no 12 pp 1187ndash11922003

[4] I I Gottesman and A Bertelsen ldquoConfirming unexpressedgenotypes for schizophrenia Risks in the offspring of FischerrsquosDanish identical and fraternal discordant twinsrdquo Archives ofGeneral Psychiatry vol 46 no 10 pp 867ndash872 1989

[5] I Kaplan Nuclear Physics Addison-Wesley 2nd edition 1962[6] S Saha D Chant J Welham and J McGrath ldquoA systematic

review of the prevalence of schizophreniardquo PLoS Medicine vol2 no 5 article e141 2005

[7] M Kramer E S Pollack R W Redick and B Z LockeMentalDisordersSuicide Harvard University Press 1972

[8] A J Rosanoff L M Handy I R Plesset and S Brush ldquoTheetiology of so-called schizophrenic psychoses with special

10 ISRN Psychiatry

reference to their occurrence in twinsrdquo American Journal ofPsychiatry vol 91 pp 247ndash286 1934

[9] E Essen-Moller ldquoPsychiatrische unter-suchungen in einer serievon zwillingenrdquo Acta Psychiatrica Scandinavica vol 16 supple-ment 23 pp 1ndash30 1941

[10] F J Kallmann ldquoThe genetic theory of schizophrenia an analysisof 691 schizophrenic twin index familiesrdquo American Journal ofPsychiatry vol 103 pp 309ndash322 1946

[11] E Slater Psychotic and Neurotic Illness in Twins MedicalResearch Council Special Report Series No 278 Her MajestyrsquosStationery Office London UK 1953

[12] I Inouye ldquoSimilarity and dissimilarity of schizophrenia intwinsrdquo in Proceedings of the 3rd World Congress of Psychiatryvol 1 pp 524ndash530 University of Toronto Press MontrealCanada June 1961

[13] B Harvald and M Hauge ldquoHereditary factors elucidated bytwin studiesrdquo in Genetics and the Epidemiology of Chronic Dis-eases J V Neel Ed Publication No 1163 Department ofHealth Education and Welfare Public Health Service 1965

[14] I I Gottesman and J Shields ldquoSchizophrenia in twins 16 yearsrsquoconsecutive admissions to a psychiatric clinicrdquo British Journalof Psychiatry vol 112 no 489 pp 809ndash818 1966

[15] I I Gottesman and J Shields ldquoContributions of twin studiesto perspectives on schizophreniardquo in Progress in ExperimentalPersonality Research B AMaher Ed pp 1ndash84Academic PressNew York NY USA 1966

[16] E Kringlen ldquoSchizophrenia in twins An epidemiological-clinical studyrdquo Psychiatry vol 29 no 2 pp 172ndash184 1966

[17] A Hoffer andW Pollin ldquoSchizophrenia in the NAS-NRC panelof 15909 veteran twin pairsrdquoArchives of General Psychiatry vol23 no 5 pp 469ndash477 1970

[18] AW Loranger ldquoSex difference in age at onset of schizophreniardquoArchives of General Psychiatry vol 41 no 2 pp 157ndash161 1984

[19] J Shields I I Gottesman and E Slater ldquoKallmannrsquos 1946schizophrenic twin study in the light of new informationrdquo ActaPsychiatrica Scandinavica vol 43 no 4 pp 385ndash396 1967

[20] I I Gottesman J Shields and D R Hanson SchizophreniaTheEpigenetic Puzzle Cambridge University Press 1982

[21] E Slater and V A Crowie The Genetics of Mental DisordersOxford University Press 1971

[22] R Belmaker W Pollin R J Wyatt and S Cohen ldquoA follow upof monozygotic twins discordant for schizophreniardquoArchives ofGeneral Psychiatry vol 30 no 2 pp 219ndash222 1974

[23] S A Mednick R A Machon M O Huttunen and DBonett ldquoAdult schizophrenia following prenatal exposure to aninfluenza epidemicrdquo Archives of General Psychiatry vol 45 no2 pp 189ndash192 1988

[24] D St Clair M Xu P Wang et al ldquoRates of adult schizophreniafollowing prenatal exposure to the Chinese famine of 1959ndash1961rdquo Journal of the American Medical Association vol 294 no5 pp 557ndash562 2005

[25] M C Clarke A Tanskanen M Huttunen J C Whittakerand M Cannon ldquoEvidence for an interaction between familialliability and prenatal exposure to infection in the causation ofschizophreniardquo American Journal of Psychiatry vol 166 no 9pp 1025ndash1030 2009

[26] T Ebert and M Kotler ldquoPrenatal exposure to influenza and therisk of subsequent development of schizophreniardquo The IsraelMedical Association Journal vol 7 no 1 pp 35ndash38 2005

[27] T J Crow and D J Done ldquoPrenatal exposure to influenza doesnot cause schizophreniardquo British Journal of Psychiatry vol 161pp 390ndash393 1992

[28] Y Mino I Oshima T Tsuda and K Okagami ldquoNo relationshipbetween schizophrenic birth and influenza epidemics in JapanrdquoJournal of Psychiatric Research vol 34 no 2 pp 133ndash138 2000

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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