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There are two types of diabetes mellitus.

a. Type 1 diabetes mellitus b. Type 2 diabetes melliuts

Is either of them a sex linked disease? Can either one be inherited? My book says, "this disease is transmitted as a recessive genetic characteristic." What does this mean?

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You seem to be confusing inheritance with being sex linked. The two are not related. While sex linked characteristics are indeed inheritable, not all (or even most) inheritable traits are sex linked. As for what a recessive characteristic is, have a look here (the first hit when you search for "recessive genetic characteristic" in Google. –  terdon Aug 6 '13 at 2:24
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Is either of them a sex linked disease? Can either one be inherited? My book says, "this disease is transmitted as a recessive genetic characteristic." What does this mean?

Neither are sex-linked. Type 1 can be directly inherited (in a non-Mendelian fashion), but Type 2 genetic factors mostly increase risks. "Recessive genetic traits" are traits that only express themselves when ONLY the recessive alleles are present in the organism (a dominant allele, if present, will 'overpower' recessive one).

Longer info:

Diabetes mellitus Type 1 (Juvenile Diabetes) is inherited, but it is autosomal with complex dominant/recessive rules:

Type 1 diabetes is a polygenic disease, meaning many different genes contribute to its onset. Depending on locus or combination of loci, it can be dominant, recessive, or somewhere in between.

That results in some interesting, albeit complex, expressions:

The risk of a child developing type 1 diabetes is about 10% if the father has it, about 10% if a sibling has it, about 4% if the mother has type 1 diabetes and was aged 25 or younger when the child was born, and about 1% if the mother was over 25 years old when the child was born.

Diabetes mellitus Type 2 does have genetic components, but the vast majority merely increase the risk of developing Type 2 Diabetes. Environmental factors play a large role in Type 2.

Excess body fat is associated with 30% of cases in those of Chinese and Japanese descent, 60-80% of cases in those of European and African descent, and 100% of Pima Indians and Pacific Islanders.

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Fascinating and complex indeed! So even though there are different risk factors for Type 1 based on mother or father having it, you still wouldn't term it "sex-linked"? –  Oreotrephes Aug 6 '13 at 6:22
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@Oreotrephes: sex-linked just means that it depends on some gene present on the sexual chromosomes (chromosomes X and Y), so that it is not passed equally in the progeny (e.g. a gene on the Y chromosome will not influence females, who are XX). I would also like to add that epigenetics plays a very important role in the development of diabetes. –  nico Aug 6 '13 at 7:23
    
I understood from your explanation that type 1 is inherited but the defective gene is on autosome not on sex chromosome. But didn't understand what you said about type 2. Does the same explanation goes for type 2? –  Rafique Aug 6 '13 at 11:59
    
Actually the genetic component of type 2 diabetes is much higher than type 1. See this link diabetes.co.uk/diabetes-and-genetics.html –  Snipergirl Aug 7 '13 at 1:35
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@MuhammadRafique - The genetic components for Type II would be mostly autosomal, though if I recall correctly it's development is more prevalent in women so there may be some sex-linked genetic factors, but I have not read anything directly stating that. –  MCM Aug 7 '13 at 3:42
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