SME
Biochemistry · Problem type
Kₘ and Vₘₐₓ from two rates (Lineweaver–Burk)
Use this for “An enzyme runs at 2.5 μM/s with 20 μM substrate and 4 μM/s with 80 μM. Find Kₘ and Vₘₐₓ.”
Chart
Zero: 1/[S] = −0.05
y-intercept (0, 0.2)
This is an example. Type your own values to start a new problem.
[S]₁
Substrate, first reading
example · μM
v₁
Rate, first reading
example · μM/s
[S]₂
Substrate, second reading
example · μM
v₂
Rate, second reading
example · μM/s
m
Slope
calculated · s
b
Intercept
calculated · s/μM
Vₘₐₓ
Maximum rate
calculated · μM/s
Kₘ
Michaelis constant
calculated · μM
Formulas
m
=
1 ÷
v₂
− 1 ÷
v₁
1 ÷ [S]₂ − 1 ÷ [S]₁
4
=
1 ÷ 4 − 1 ÷ 2.5
1 ÷ 80 − 1 ÷ 20
b
=
1
v₁
−
m
÷
[S]₁
0.2
=
1
2.5
−
4
20
Vₘₐₓ
=
1
b
5
=
1
0.2
Kₘ
=
m
×
Vₘₐₓ
20 = 4 × 5
Assumptions
•
Both rates are initial rates of the same enzyme at the same amount, and it follows Michaelis–Menten.
•
Flipping v = Vₘₐₓ[S] ÷ (Kₘ + [S]) gives a line, 1/v = m(1/[S]) + b: slope m = Kₘ ÷ Vₘₐₓ, intercept b = 1/Vₘₐₓ.
•
The line crosses the 1/v axis at 1/Vₘₐₓ and the 1/[S] axis at −1/Kₘ.
•
More than two points: fit the best line through all of them, then read its slope and intercept.
Step-by-step
WE KNOW
[S]₁ = 20 μM, v₁ = 2.5 μM/s, [S]₂ = 80 μM, v₂ = 4 μM/s
FIND
slope (m), intercept (b), maximum rate (Vₘₐₓ), Michaelis constant (Kₘ)
Step 1 · Find slope (m)
Use m = (1 ÷ v₂ − 1 ÷ v₁) ÷ (1 ÷ [S]₂ − 1 ÷ [S]₁)
Flip
each
reading
to
(1/[S],
1
v
);
the
slope
is
the
rise
in
1
v
over
the
run
in
1/[S].
m
=
1 ÷ 4 − 1 ÷ 2.5
1 ÷ 80 − 1 ÷ 20
m
=
0.25 − 0.4
0.0125 − 0.05
m
=
−
0.15
−0.0375
m
=
4
s
Step 2 · Find intercept (b)
Use b = 1 ÷ v₁ − m ÷ [S]₁
Slide
back
along
the
line
from
the
first
point
to
1/[S]
=
0:
take
m
×
1/[S]₁
from
1
v₁
.
b
=
1
2.5
−
4
20
b
=
0.4
−
0.2
b
=
0.2
s/μM
Step 3 · Find maximum rate (Vₘₐₓ)
Use Vₘₐₓ = 1 ÷ b
The
intercept
is
1
v
at
endless
substrate,
which
is
1/Vₘₐₓ.
Vₘₐₓ
=
1
0.2
Vₘₐₓ = 5 μM/s
Step 4 · Find Michaelis constant (Kₘ)
Use Kₘ = m × Vₘₐₓ
The
slope
is
Kₘ
Vₘₐₓ
,
so
multiply
it
by
Vₘₐₓ.
Kₘ = 4 × 5
Kₘ = 20 μM
Check
Put the numbers back in. Both sides should match.
4
=
1 ÷ 4 − 1 ÷ 2.5
1 ÷ 80 − 1 ÷ 20
✓
0.2
=
1
2.5
−
4
20
✓
5
=
1
0.2
✓
20 = 4 × 5 ✓
Topic check
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Enzyme kinetics (Michaelis–Menten)
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