Course of Laboratory Medicine
Medical School course F - Faculty of Pharmacy and Medicine
Prof. Andrea Bellelli

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     In this purely theoretical exercise, we shall investigate one among the possible causes of inter-individual variability.

The reaction scheme
      Let us suppose that the terminal part of a metabolic pathway converts irreversibly metabolyte A into metabolyte B and then into terminal product C that is excreted. The reaction scheme is as follows:
A → B → C

      Reaction 1 irreversibly converts A to B and is catalyzed by enzyme E1; reaction irreversibly 2 converts B to C and is catalyzed by enzyme E2.
      We further suppose that the concentration of metabolyte A is regulated by a negative feedback mechanism and is maintained constant. Moreover since product C is excreted and the reactions leading to its production are irreversible, we can neglect its concentration. We want to investigate the steady-state concentration of metabolyte B.
      Let us assume that both E1 and E2 operate under steady-state conditions and obey a simple Michaelis and Menten equation. The rate of change of the concentration of metabolyte B is described by the following differential kinetic equation:
          (eqn. 1)
where the first term describes the rate of formation of B from A (whose concentration is assumed to be constant) and the second term describes the rate of degradation of B. Vmax,1 and KM,1 are the Michaelis and Menten parameters for E1 and Vmax,2 and KM,2 those for E2.

      Under steady-state conditions, the concentration of B is constant, i.e. the differential equation describing its change equals zero. Thus, we can solve the above equation for [B] to obtain:
          (eqn. 2)

      The above formula allows us to calculate the concentration of metabolyte B for any set of conditions. For example, if we assume the following set of parameters:
steady-state concentration of A       KM,1       Vmax,1       KM,2       Vmax,2      
1 mM 1 mM 10 s-1 1 mM 10 s-1
we can easily calculate from eqn. 2 that the steady-state concentration of B is 1 mM.

Presence of allelic variants of the enzyme(s) and their distribution in the population
      Let us assume that in the population under study there are two genetic variants of E1 and two of E2, that we call E1, e1, E2 and e2 respectively. All variants are functional, but they differ because of slight changes in Vmax and KM, e.g.:
E1: Vmax,1=11 s-1; KM,1=0.9 mM                   e1: Vmax,1=9 s-1; KM,1=1.1 mM
E2: Vmax,2=11 s-1; KM,2=0.9 mM                   e2: Vmax,2=9 s-1; KM,2=1.1 mM

      Let us further assume that the concentration of each enzyme is constant and that function of enzyme 1 in the heterozygous individual having an equimolar mixture of E1 and e1 can be approximated by a Michaelis and Menten equation with averaged parameters (i.e. Vmax,1=10 s-1; KM,1=1 mM). The same applies to the function of enzyme 2. These approximations are very rough, but sufficient for our present purpose:

      To simulate the distribution of the concentration of B in the population we need one further information, namely the gene frequencies for E1 and e1, E2 and e2. Let us assume that the gene frequencies are 50% for each gene; the Hardy-Weinberg law allows us to calculate the phenotype frequencies as follows:
E1E1: 0.52=0.25       E1e1: 0.5 x 0.5 x 2 = 0.5       e1e1: 0.52=0.25
E2E2: 0.52=0.25       E2e2: 0.5 x 0.5 x 2 = 0.5       e2e2: 0.52=0.25

      The frequencies of the different two-gene phenotypes are then calculated by multiplying those of the pertinent single-gene phenotypes; e.g.:
E1e1 / E2e2 = 0.5 x 0.5 = 0.25

      The distribution of metabolyte B concentration in the population can be calculated as follows:

1) A very simple model of a metabolic pathway including only three chemical intermediates and two enzymes, under the assumption of two very similar allelic variants of each enzyme, can generate a surprisingly wide variability of the intermediate metabolyte.
2) The distribution of the intermediate metabolyte concentration is described by a skewed Gaussian curve.
3) As predictable, the concentration of the intermediate metabolyte is highest in subjects whose genetic constitution is such that they are homozygous for the more effective variant of the producing enzyme (E1E1), and again homozygous for the less effective variant of the degrading enzyme (e2e2).
4) If the concentration of the intermediate metabolyte is in any way related to susceptibility to a specific disease the model would partially explain the incidence of the disease.
5) The frequencies of the highest and lowest values of the intermediate metabolyte concentrations do not correlate with the frequency of any single gene variant (because they correlate with the products of the gene frequencies).

Inter-population variability
      Human populations usually present different frequencies of the same alleles of the same genes. It is quite uncommon that an allele is present in a population and completely absent in another (if this happens, this points to a very ancient separation in the evolutionary history of Homo sapiens).
      We can easily simulate this condition using the same data we used for simulating inter-individual variability, but changing the gene frequencies. An important consequence of the above consideration is that it is very difficult to assign a single individual to any given population. What one can do is to estimate the probability that a single individual is a member of a given population. Obviously, the more allele variants and the more genes are considered, the more reliable this estimate is.

Variability within group and between groups
      An important statistical concept to be applied when a population is composed by two or more different groups is that of variability within and between. The total variability recorded in the entire population is estimated as the total deviance, DT:
DT = Σ (xi - xm)2
where x is the value of the parameter considered, and xm its average. If we divide DT by (N-1), where N represents the number of the members of the total population we obtain the total variance, σ2T of the population: σ2T = DT / (N-1).
      If the population is composed by two groups, each with its own deviance, we have:
D1 = Σ (x1,i - x1,m)2
D2 = Σ (x2,i - x2,m)2
where x1,i represents the individual values of parameter x in group 1 and x1,m the average value of parameter x in group 1; the same applies to x2,i and x2,m for group 2.
      The deviance within, DW is the sum of the deviance of each group with respect to its average value: DW = D1 + D2.
      The deviance between, DB is the deviance of the average of each group with respect to the average of parameter x in the total population times the number of elements in each group (in our example we only have two groups): DB = Σ Nj (xj,m - xm)2. Obviously the sum of the number of elements of each group equals the total population: N = Σ (Nj).
      The sum of the deviance within and the deviance between equals the total deviance of the population: DT = DB + DW.
      The above relationship is important: in some cases the DB may be large with respect to DW. For example if we compare the glycemias of a group of diabetic patients with that of a control group of healthy subjects we observe that DB >> DW, implying that there is a larger difference in the average values of glycemia in the two groups than there is in the glycemias of diabetic patients or healthy subjects. The opposite is usually found in human groups of healthy subjects, where DB << DW. For example, in cognitive tests measuring linguistic ability women usually score slightly better than men, but the deviance among women and the deviance among men are much larger than those between the averages of women and men. These two possible conditions are schematicaly represented below:

Populations are not races or breeds
      The concept of race applied to human populations has a releatively short (and infamous) history: it was initially used by De Gobineau in France and Knox in England (both circa 1850). A breed is an artificial group, obtained in zootechnology or agriculture by rigorous control over reproduction. There is some confusion on the relationship between the concepts of race and breed, and the biologist J.B.S. Haldane in 1956 pointed this out by asking whether human groups or races were similar to dog breeds. Without entering a question of terminology, it is by now absolutely clear that human populations are very different from dog breeds, and also from what is usually meant as "races", and we can state that human races do not exist. The difference between a race and a population or ethnic group is evident to genetists but quite subtle for non-specialists. We may explain this difference by comparing two examples, one for two populations, the other for two races/breeds.
      The distribution of blood groups in Italy is fairly homogeneous, except for Sardinia, which is different, as shown in the table below:
blood groups:0ABAB
Italy except Sardinia46%42%9%3%

      Clearly, the Sardinian population stands apart from the rest of Italy, but you cannot find this property in any single individual. A population differs from another because of the allelic frequencies of the same genes. Allele frequencies are properties of groups, not of individuals. Obviously, two populations differ because of the allele frequencies of a large number of genes, not only one. A consequence of the above is that we cannot assign a person to a population, if not in statistical terms.
      By contrast let's consider the case of two breeds, e.g. basset-hounds and terrier. All and every basset-hounds have a mutation of the gene FGFR3, and carry the same genetic defect of human achondroplasia, which causes a reduced growth of long bones. No terrier has this mutation. Thus breed is a genetic property of the population and of each of its members. Clearly, a breed can be maintained only if mating is strictly controlled, either by humans or by physical barriers. Again, two races of the same species differ because of allele variants of many genes, not only one.
      As a consequence of the above distinction, we can confidently state that an individual animal is a member of a specific breed (and eventually how "pure" it is), whereas in the case of individual humans we can determine which alleles he/she posesses for relevant genes and infer the probability that he/she is a member of a population. Moreover, in order to define the human populations we also need non-genetic criteria (e.g. language or geographical considerations).
      Races/breeds, but not populations, have an impoverished gene pool. Applying strict control over mating one can further reduce the gene pool of a group of animals and obtain the strain. This is only done for laboratory animals; all individuals belonging to the same strain are genetically identical or almost so and they accept transplants from individuals of the same strain without rejection (which is not the case with animal of the same race).
      The genetic of populations is important in medicine because different alleles may be differently related to susceptibility to diseases, or to the response to drugs; thus disease prevalences may vary among different populations, because of their characteristic allele frequencies. This in turn affects the reliability of our diagnostic hypotheses, because the prevalence appears as the pre-test probability in Bayes' formula. In medicine, the term race is often used as a shorthand term to indicate the probable population of which the patient is a member, but one should always remember that this term is a very rough and imprecise description of the reality it is thought to represent.

      A practical exercise on the meaning of the concepts of race and population. Let's imagine that we have a gene presenting only two alleles. We examine two populations of the same numerical weight for allele frequencies, and we find that the frequencies of allele A is, say, 60% in population 1 and 40% in population 2; the opposite applies to the non-dominant allele a. The system having only two alleles of th esame gene, obeys a binomial distribution. If we assign value=0 to A and 1 to a, and apply the formulas of binomial distributions we obtain:
Populationfrequency of Afrequency of aaverage scorevariance
total variance0.25
variance within (mean of the variances #1 and #2)0.24
variance between (variance between means #1 and #2)0.01
      The table demonstrates two points: 1) The two populations, provided that their numerical consistency is large enough are indeed different, because they present different average values for the parameter studied, and we can apply the test of statisticall significance of difference between mean values:
Z = (mean1 - mean2) / √ (variance1/n1 + variance2/n2)
this formula tells us that we obtain P<0.001 (corresponding to Z>3.5) for a numerosity of at least 500 elements per population.
2) The variance between, that is the variance due to the difference between the two populations is only 4% of the total variance, the remaining 96% of variance being due to the variance within. This implies that the members of each population differ among themselves to a much larger extent than they differ across the population boundary. This is typical of human populations and much more complex analyses carried out on many genes with many alleles each have provided a similar estimate of human variance among populations (the pioneering study by R. Lewontin of 1972 can be read at this link). By contrast in dogs' breeds an average estimate of variance between is in the order of 30% of the total variance, for many genes. As population genetist Livingstone famously said, in humans "there are no races, only clines" (but see also this paper where L. Brace points out two major criticisms of the race concept: 1) human genes follow frequency gradients rather than groups; and 2) the frequencies of different genes do not cluster or covariate).
      In medicine the population of which a subject is member may suggest some statistically relevant consideration: indeed, as we saw in the example above, two populations may differ in the mean vaue of some measurable biological property, which may bear significance for clinical analysis or response to therapy. E.g. in populations of european descent the mean concentration of blood triglyceride and cholesterol is higher than in many populations of african descent, even though this difference is inversely related to heart disease (so called lipid paradox). Clearly human populations do exist, even though they are not demarcated enough to warrant the use of the negatively connotated term race; usually we refer to different populations by their probable descent and keep in mind that they are genetically heterogeneous.

Questions and exercises:
1) Inter-human variability is a consequence of:
the distribution of allelic variants of the same genes
the presence of non-functional genes in the population
environmental factors

2) Inter-population variability is a consequence of:
different frequencies of (the same) allelic variants of genes
environmental factors
the presence of different genes in the different populations

3) The Hardy Weinberg law
describes the genotype frequencies
correlates the phenotype frequencies to the gene frequencies
describes allelic frequencies

4) The variability of analytes' concentrations among different individuals
is determined by stochastic phenomena
is genetically determined because of allelic variants of the analytes
is genetically determined because of allelic variants of the enzymes responsible of the metabolism of analytes

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Thank you Professor (lecture on bilirubin and jaundice).

The fourth recorded part, the one on hyper and hypoglycemias is not working.
Bellelli: I checked and in my computer it seems to work. Can you better specify
the problem you observe?

This Presentation (electrolytes and blood pH) feels longer than previous lectures
Bellelli: it is indeed. Some subjects require more information than others. I was
thinking of splitting it in two nest year.

Bellelli in response to a question raised by email: when we compare the blood pH
with the standard pH we do not mean to compare the "normal" blood pH (7.4)
with the standard pH. Rather we compare the actual blood pH of the patient, with
the pH of the same blood sample equilibrated under standard conditions.
Thus, if we say that standard pH is lower than pH we mean that equilibriation with
40 mmHg CO2 has caused absorption of CO2 and has lowered the pH with respect
to its value before equilibration.

(Lipoproteins) Is the production of leptin an indirect cause of type 2 diabetes since
it works as a stimulus to have more adipose tissue that produces hormones?
Bellelli: in a sense yes, sustained increase of leptin causes the hypothalamus to adapt
and to stop responding. Obesity ensues and this in turn may cause an increase in the
production of resistin and other insulin-suppressing protein hormones produced by the
adipose tissue. However, this is quite an indirect link, and most probably other factors
contribute as well.

(Urea cycle) what is the meaning of "dissimilatory pathway"?
Bellelli: a dissimilatory pathway is a catabolic pathway whose function is not to produce
energy, but to produce some terminal metabolyte that must be excreted. Dissimilatory
pathways are necessary for those metabolytes that cannot be excreted as such by the
kidney or the liver because they are toxic or poorly soluble. Examples of metabolytes
that require transformation before being eliminated are heme-bilirubin, ammonia,
sulfur and nitrogen oxides, etc.

Talking about IDDM linked neuropathy can be the C peptide absence considered a cause of it??
Bellelli: The C peptide released during the maturation of insulin, besides being an indicator
of the severity of diabetes, plays some incompletely understood physiological roles. For
example it has been hypothesized that it may play a role in the reparation of the
atherosclerotic damage of the small arteries. Thus said, I am not aware that it plays a direct
role in preventing diabetic polyneuropathy. Diabetic neuropathy has at least two causes: the
microvascular damage of the arteries of the nerve (the vasa nervorum), and a direct
effect of hyperglycemia and decreased and irregular insulin supply on the nerve metabolism.
Diabetic neuropathy is observed in both IDDM and NIDDM, and requires several years to
develop. Since the levels of the C peptide differ in IDDM and NIDDM, this would suggest
that the role of the C peptide in diabetic neuropathy is not a major one. If you do have
better information please share it on this site!

In acute intermitted porphyria and congenital erythropoietic porphyria why do the end product
of the affected enzymes accumulate instead of their substrate??
Bellelli: First of all, congratulations! This is an excellent question.
Remember that a condition is which the heme is not produced is lethal in the foetus; thus
the affected enzyme(s) must maintain some functionality for the patient
to be born and to come to medical attention. All known genetic defects of heme
biosynthesis derange but do not block this metabolic pathway.
Congenital Erythropoietc Porphyria (CEP) is a genetic defect of uroporphyrinogen
III cosynthase. This protein associates to uroporphyrinogen synthase (which is present
and functional in CEP) and guarantees that the appropriate uroporphyrinogen isomer is produced
(i.e. uroporphyrinogen III). In the absence of a functional uroporphyrinogen III
cosynthase other possible isomers of uroporphyrinogen are produced together with
uroporpyrinogen III, mostly uroporphyrinogen I. The isomers of uroporphyrinogen
that are produced differ because of the positions of propionate and acetate side chains,
and this in turn is due to the pseudo symmetric structure of porphobilinogen. Only
isomer III can be further used to produce protoporphyrin IX. Thus in the
case of CEP we observe accumulation of abnormal uroporphyrinogen derivatives, which, as
you correctly observed are the products of the enzymatic synthesis operated by
uroporphyrinogen synthase.
The case of Acute Intermittent Porphyria (AIP) is similar, although there may be variants
of this disease. What happens is that either the affected enzyme is a variant that does not
properly associate with uroporphyrinogen III cosynthase or presents active site mutations
that impair the proper alignement of the phoprphobilinogen substrates. In either case
abnormal isomers of uroporphyrinogen are produced, as in CEP.
Also remark that in both AIP and CEP we observe accumulation of the porphobilinogen
precursor: this is because the overall efficiency of the biosynthesis of uroporphyrinogens is
reduced. Thus: (i) less uroporphyrinogen is produced, and (ii) only a fraction of the
uroporphyrinogen that is produced is the correct isomer (uroporphyrinogen III).

is it possible to take gulonolactone oxidase to synthesize vitamin C
instead of vitamin C supplement?
Bellelli: no, this approach does not work. The main reason is that
the biosynthesis of vitamin C, as almost all other metabolic processes, occurs intracellularly.
If you administer the enzyme it will at most reach the extracellular fluid but will not be
transported inside the cells to any significant extent. Besides, there are other problems
in this type of therapy (e.g. the enzyme if administered orally, may be degraded by digestive
proteases; if administered parenterally, may cause the immune system to react against a
non-self protein). In theory one could think of a genetic modification of the inactive human
gene of gulonolactone oxidase, but the risk and cost of this intervention would not be
justified. In addition to these considerations, except for cases of shipwreckage or
other catastrophes, a proper diet or administration of tablets of vitamin C is effective,
risk-free and unexpensive, thus no alternative therapy is reasonable. However, I express my
congratulations for your search on the biosynthesis pathway of ascorbic acid.

Resorption and not reabsorption would lead to hypercalcemia ie bone matrix being broken down.
Bellelli: I am not sure to interpret your question correctly. Resorption indicates destruction of the bone matrix and release of calcium and
phosphate in the blood, thus it causes an increase of calcemia. Reabsorption usually means active transport of calcium from the renal tubuli to the blood, thus
it prevents calcium loss. It prevents hypocalcemia, and thus complement bone resorption. To avoid confusion it is better use the terms "bone resorption" and "
renal reabsorption of calcium". If you have a defect in renal reabsorption, parthyroid hormone will be released to maintain a normal calcium level by means of
bone resorption; the drawback is osteoporosis.

In Reed and Frost model: I haven't understood what is the relationship
between K and R reproductive index. Thank you Professor!
Bellelli: in the Reed and Frost model K is the theoretical upper limit of
R0. R the reproductive index is the ratio (new cases)/(old cases) measured after
one serial generation time. R0 is the value of R one measures at the beginning
of the epidemics, when in principle all the population is susceptible.

What is the link between nucleotide metabolism and immunodeficiencies and mental retardation?
Bellelli: the links may be quite complex, but the principal ones are as follows:
1) the immune response requires a replication burst of granulocytes and lymphocytes, which in turn requires
a sudden increase of nucleotide production, necessary for DNA replication. Defects of nucleotide metabolism
impair this phase of the immune defense. Notice that the mechanism is similar to the one responsible of
anemia which requires a sustained biosynthesis of nucleotides at a constant rate, rather than in a burst.
2) Mental retardation is mainly due to the accumulation of nulceotide precursors in the brain of the
newborn, due to the incompletely competent blood-brain barrier.

How can ornithine transaminase defects cause hyperammonemia? Is it due to the accumulation
of ornithine that blocks the urea cycle or for other reasons?
Bellelli: ornithine transaminase is required for the reversible interconversion of ornithine
and proline, and thus participates to both the biosynthesis and degradation of ornithine. The enzyme is
synthesized in the cytoplasm and imported in the mitochondrion. Depending on the metabolic conditions
the deficiency of this enzyme may cause both excess (when degradation would be necessary) or defect
(when biosynthesis would be necessary) of ornithine; in the latter case, the urea cycle slows down. Thus
there is the paradoxical condition in which alternation may occur between episodes of hyperammonemia
and of hyperornithinemia.

When we use the Berthelot's reaction to measure BUN do we also have to
measure the concentration of free ammonia before adding urease?
Bellelli: yes, in principle you should. Berthelot's reaction detects ammonia,
thus one should take two identical volumes of serum, use one to measure free ammonia,
the other to add urease and measure free ammonia plus ammonia released by urea. BUN is
obtained by difference. However, free ammonia in our blood is so much lower than urea that
you may omit the first sample, if you only want to measure BUN.

Why do we have abnormal electrolytes in hematological neoplasia e.g.
Bellelli: I do not have a good explanation for this effect, which may have
multiple causes. However, you should consider two factors: (i) acute leukemias cause a massive
proliferation of leukocytes (or lymphocytes depending on the cell type affected) with a very
shortened lifetime; thus you observe an excess death rate of the neoplastic cells. The dying
cells release in the bloodstream their content, which has an electrolyte composition different
from that of plasma: the cell cytoplasm is rich in K and poor in Na, thus causing hyperkalemia.
(ii) the kidney may be affected by the accumulation of neoplastic white cells or their lytic products.

Gaussian curve: If it is bimodal is it more likely to be a "certain diagnosis" than if it is
unimodal or does it only show the distinguishment from health?
Bellelli an obviously bimodal Gaussian curve indicates that the disease is clearly
separated from health: usually it is a matter of how precise and clear-cut is the definition of the disease.
For example tuberculosis is the disease caused by M. tuberculosis, thus if the culture of the sputum is
positive for this bacterium you have a "certain" diagnosis (caution: the patient may suffer of two diseases,
e.g. tuberculosis and COPD diagnosis of the first does not exclude the second). However, in order to have
a "certain" diagnosis it is not enough that distribution of the parameter is bimodal, it is also required that the
patient's parameter is out of the range of the healthy condition: this is because a distribution can be
bimodal even though it is composed by two Gaussians that present a large overlap, and the patient's
parameter may fall in the overlapping region. Thus, in order to obtain a "certain" diagnosis you need to
consider not only the distribution of the parameter(s) but also the patient's values and the extent of the
overlapping region.

Prof can you please elaborate a bit more on the interhuman variability and its difference
with the interpopulation variability please?
Bellelli: every individual is a unique combination of different alleles of the same genes;
this is the source of interindividual variability. Every population is a group of individuals who intermarry and
share the same gene pool (better: allele pool). Every allele in a population has its own frequency. Two
population may differ because of the diffferent frequencies of the same alleles; in some cases one
population may completely lack some alleles. The number and frequencies of alleles of each gene
determine the variance. If you take two populations and calculate the cumulative interindividual variance
of the population the number you obtain is the sum of two contributions: the interindividual variance within each population, plus the interpopulation variance
between the means of the allele frequencies. For example, there are human population in which the frequency of blood group B is close to 0% and other populati
ons in which it is 30% or more.

Prof can you please explain again the graph you have showed us in class about thromboplastin?
(Y axis=abs X axis= time)
Bellelli: the graph that I crudely sketched in class represented the signal
of the instrument (an absorbance spectrophotometer) used to record the turbidity of the
sample (turbidimetry). The plasma is more or less transparent, before coagulation starts.
When calcium and the tissue factor (or collagen) are added. thrombin is activated and begins
digesting fibrinogen to fibrin; then fibrin aggregates. The macroscopic fibrin aggregates cause
the sample to become turbid, which means it scatters the incident light. The instrument reads
this as a decrease of transmitted light ( an increase of the apparent absorbance) and the
time profile of the signal presents an initial lag phase, which is called the protrombin or
thromboplastin time depending on the component which was added to start coagulation
(tissue factor or collagen).

Prof can you please explain the concept you have described in class about
the simultaneous hypercoagulation and hemorrhagic syndrome? How can this occur?
Bellelli: The condition you describe is observed only in the Disseminated
Intravascular Coagulation syndrome. Suppose that the patient experiences an episode of
acute pancreatitis: tripsin and chymotripsin are reabsorbed in the blood and proteolytically
activate coagulation causing an extensive consumption of fibrinogen and other coagulation
factors. Tripsin and chymotripsin also damage the vessel walls and may cause internal
hemorrages, but at that point the consumption of fibrinogen may have been so massive that
not enough is left to form the clot where the vessel has been damaged, causing an internal
hemorrage. Pancreatitis is a very severe, potentially lethal condition, and DIC is only one of
the reasons of its severity.

You said that certain drugs (ethanol, cocaine, cannabis, opiates...) cause a
necessity of higher and higher dosage, for two reasons: the enzyme in the liver is inducible and
the receptors in the brain are expressed less and less. So, first, I am not sure I got it right, and
second I did not understand how expressing less receptors leads to a necessity of higher
Bellelli: You got it correctly, but the detailed mechanism of resistance may
vary among different substances, and not all drugs cause adaptation.
The reason why reducing the number of receptors may require an increased dosage of the drug
is as follows: suppose that a certain cell has 10,000 receptors for a drug. When bound to its
agonist/effector, each receptor produces an intracellular second messenger. Suppose that in
order for the cell to respond 1,000 receptors must be activated. The concentration of the
effector required is thus the concentration that produces 10% saturation. You can easily
calculate that this concentration is approximately 1/10 of the equilibrium dissociation constant
of the receptor-effector complex (its Kd), the law being
Fraction bound = [X] / ([X]+Kd)
where [X] is the concentration of the free drug.
After repeated administration, the subject becomes adapted to the drug, and his/her cells
express less receptors, say 5,000. The cell response will in any case require that 1,000
receptors are bound to the effector and activated, but this now represents 20% of the total
receptors, instead of 10%. The drug concentration required is now 1/4 of the Kd.
Continuing administration of the drug further reduces the cell receptors, but the absolute
number of activated receptors required to start the response is constant; thus the fewer
receptors on the cell membrane, the higher the fraction of activated receptors required.

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