To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining the variables, the objective function, and the constraints using mathematical notation.

Let's define:
- $x_1$ as the milligrams of vitamin B4,
- $x_2$ as the milligrams of vitamin C.

The objective function to maximize is: $1 \cdot x_1 + 5 \cdot x_2$

Given the constraints:
1. The muscle growth index of milligrams of vitamin B4 is 1, and for milligrams of vitamin C is 6.
2. The total combined muscle growth index from both vitamins must be at least 52.
3. $-1 \cdot x_1 + 1 \cdot x_2 \geq 0$
4. The total combined muscle growth index should not exceed 111.
5. $x_1$ must be an integer, while $x_2$ can be any real number.

Thus, the symbolic representation of the problem is:
```json
{
    'sym_variables': [('x1', 'milligrams of vitamin B4'), ('x2', 'milligrams of vitamin C')],
    'objective_function': '1*x1 + 5*x2',
    'constraints': [
        '1*x1 + 6*x2 >= 52',
        '-1*x1 + 1*x2 >= 0',
        '1*x1 + 6*x2 <= 111'
    ]
}
```

To solve this problem using Gurobi, we'll write the Python code to model and solve the optimization problem:

```python
from gurobipy import *

# Create a new model
m = Model("Vitamin_Optimization")

# Define variables
x1 = m.addVar(vtype=GRB.INTEGER, name="milligrams_of_vitamin_B4")
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="milligrams_of_vitamin_C")

# Set the objective function
m.setObjective(1*x1 + 5*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(1*x1 + 6*x2 >= 52, "Muscle_Growth_Index_Min")
m.addConstr(-1*x1 + 1*x2 >= 0, "Vitamin_B4_Vitamin_C_Relation")
m.addConstr(1*x1 + 6*x2 <= 111, "Muscle_Growth_Index_Max")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Milligrams of Vitamin B4: {x1.x}")
    print(f"Milligrams of Vitamin C: {x2.x}")
    print(f"Objective Function Value: {m.objVal}")
else:
    print("No optimal solution found")
```