To solve this optimization problem, we first need to translate the natural language description into a symbolic representation. Let's denote:

- $x_1$ as the number of regular games sold
- $x_2$ as the number of collector's edition games sold

The objective function aims to maximize profit. The profit per regular game is $20, and the profit per collector's edition game is $30. Thus, the objective function can be written as:

Maximize: $20x_1 + 30x_2$

The constraints are:
1. Each regular game costs $30, and each collector's edition game costs $50. The store wants to spend at most $4000.
   - $30x_1 + 50x_2 \leq 4000$
2. The store can sell at most 100 video games of either type per month.
   - $x_1 \leq 100$
   - $x_2 \leq 100$
3. Non-negativity constraints, as the number of games cannot be negative:
   - $x_1 \geq 0$
   - $x_2 \geq 0$

Thus, the symbolic representation of the problem is:

```json
{
  'sym_variables': [('x1', 'number of regular games sold'), ('x2', 'number of collector\'s edition games sold')],
  'objective_function': '20*x1 + 30*x2',
  'constraints': ['30*x1 + 50*x2 <= 4000', 'x1 <= 100', 'x2 <= 100', 'x1 >= 0', 'x2 >= 0']
}
```

Now, let's write the Gurobi code in Python to solve this optimization problem:

```python
from gurobipy import *

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

# Create variables
x1 = m.addVar(vtype=GRB.CONTINUOUS, name="regular_games", lb=0)
x2 = m.addVar(vtype=GRB.CONTINUOUS, name="collectors_edition_games", lb=0)

# Set the objective function
m.setObjective(20*x1 + 30*x2, GRB.MAXIMIZE)

# Add constraints
m.addConstr(30*x1 + 50*x2 <= 4000, "budget_constraint")
m.addConstr(x1 <= 100, "regular_games_limit")
m.addConstr(x2 <= 100, "collectors_edition_limit")

# Optimize model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Regular games: {x1.x}")
    print(f"Collector's edition games: {x2.x}")
    print(f"Max Profit: {m.objVal}")
else:
    print("No optimal solution found")
```